Pythagorean Tuning

5. Pythagorean tuning: a just appraisal in context

To this point, we have focused on Pythagorean tuning as a system of just intonation well suited to the complex Gothic polyphony of the 13th and 14th centuries, as well as to medieval monophony. We now consider the relationship of this system to other forms of just intonation, and to alternatives such as meantone and equal temperament.

5.1. Pythagorean tuning as quintal just intonation

Like many musical terms, "just intonation" can have various meanings and implications depending on the context. Generally we might take it to mean a system of tuning where all intervals are derived from integer or whole number ratios, and where, in a given stylistic context, the stable concords are presented in their ideally pure or simple form.

In a system such as 13th-century polyphony where fifths and fourths are the favored stable concords, Pythagorean tuning meets both criteria admirably. All intervals, from the most simple to the most complex, are derived either from the ratio of the pure fifth (3:2), or from that of the octave (2:1). Furthermore, not only stable fifths and fourths (4:3), but also major seconds (9:8) and minor sevenths (16:9), are presented in their ideally just ratios.

As we move from period to period or from style to style, the implications attached to "just intonation" change. Thus to a 16th-century theorist such as Gioseffo Zarlino, such a tuning must provide ideal ratios for the major third (5:4) and minor third (6:5), now the choice concords. For some 20th-century theorists, a "just" tuning system must additionally provide what they consider an ideally concordant ratio for the minor seventh (7:1), or for even more complex intervals (e.g. 11:1, 13:1).

One way to identify a given system of just intonation is to consider both the largest prime number used in building its interval ratios, and the most complex intervals it includes based on multiplex (n:1) or superparticular (n+1:n) ratios.

Applying the first part of this concept, some scholars refer to Pythagorean tuning as "3-limit just intonation," since all intervals are derived either from fifths (3:2) or octaves (2:1), ratios involving 3 as the largest prime.

However, we might preferably speak of "3(9)-limit just intonation" to indicate that the tuning includes ideal integer ratios based on 9 as well as on 3, although only the latter is a prime number. The major second (9:8) is thus based on an ideal superparticular (n+1:n) ratio, and as observed by Jacobus of Liege, the major 23rd or "whole-tone-plus-triple-octave" is based on the yet more ideal multiplex (n:1) ratio of 9:1, which he presents as the limit of his theoretical system.

While Jacobus himself duly notes that intervals larger than a twelfth have little use in actual polyphony, he very interestingly classifies M23 (9:1) in theory as a "perfect concord," a musical judgment which to my ear is quite congenial. More practically, he discusses the role of smaller intervals with ratios based in 9 as unstable but useful intervals: M2 (9:8) and m7 (16:9) are "imperfect concords" in his system, while M9 (9:4) is a somewhat more blending "intermediate concord."

These 9-based intervals draw a certain dignity from their close association with the stable concords. Thus M2 is precisely equal to the difference between the fifth and fourth; m7 to two perfect fourths; and M9 to two perfect fifths. Even as simple intervals, they have some degree of "compatibility" (Section 3.1.3), and in settings for three or four voices they can combine with fifths and fourths to form sonorities with a relatively blending quality such as 5/4 or 5/2, 9/5, and 7/4 (Section 3.2.3). Both Anonymous I (c. 1290) and Jacobus note the pleasing quality of M9 if "a fifth is placed in the middle," thus "splitting" the outer interval (9:4) into two euphonious fifths (9:6:4), and making the extreme voices "seem to concord better."

This affinity between the fifth and the major second or ninth is not necessarily specific to the medieval Western Europe of Perotin or Machaut. In an analysis of Japanese koto music, for example, David Loeb observes: "At these points of apparent repose, the vertical intervals include, not surprisingly, perfect fifths and octaves. Quite unexpected, however, is the similar use of major ninths which possess an equally strong sense of resolution. The ninth is constructed with a fifth included in nearly all instances." (Loeb, at 347).

While the technical description "3(9)-limit just intonation" may convey this affinity as manifested in Gothic polyphony and other world musics, a more intuitive term for Pythagorean tuning might be simply "quintal just intonation." That is, the system is founded primarily on the fifth (or fourth), with the corollary that the 9-based intervals (M2, m7, M9) derive their just proportions from the prime 3-based fifths and fourths.

Here we must note a vital caution: intervals outside the "limit" of a system of just intonation can and do have great musical value, and Pythagorean tuning in a Gothic context is a fine illustration. Theorists of the 13th century in fact regard M3 (81:64) and m3 (32:27) as relatively concordant, more blending in fact than M2 (9:8) and m7 (16:9). Jacobus likewise notes the popularity and pleasing qualities of the quinta fissa or fifth "split" into two thirds - 5/M3 or 5/m3 - citing a motet opening with such a sonority. (See Sections 3.1.2, 3.2.2). In practice, even a strong discord such as M7 (243:128) can play a vital role in the cadential action and sheer vertical color of this music.

What changes in just intonation systems is not so much whether a given interval is used, but how it is used.

Thus in the Renaissance, Zarlino's system of 1558 might be described as "5-limit just intonation," with 5 as the highest prime number used in defining concords; M3 (5:4), m3 (6:5), M6 (5:3), and m6 (8:5) are the most complex intervals participating in stable sonorities. Since these intervals are now used as points of resolution and repose, their more complex and "active" Pythagorean ratios would no longer fit the musical context. At the same time, "dissonances" outside the new limits of restful concord such as the minor seventh (now 9:5) are not without musical utility, but indeed, as Zarlino notes, add grace to the concords when aptly used in suspensions, for example. As is well known, Monteverdi and Gesualdo along with other composers around 1600 make bold use indeed of such unstable intervals.

We may refer to such "5-limit" systems more informally as "tertian just intonation": thirds and sixths with their 5-based ratios are primary intervals of the system. Other intervals tend to take their measure from these: thus m7 (9:5) is equal to a pure minor third plus a fifth (6:5 x 3:2), while M2 alternates between 9:8 and 10:9, these two ratios together yielding a pure major third of 5:4.

From a 20th-century perspective, both the quintal and tertian systems of just intonation may seem artificially "limited," and contemporary performers and theorists are exploring the possibilities of 7-limit and even more complex systems. Just as Renaissance musicians found a 3(9)-limit Pythagorean tuning "incomplete" because it did not provide ratios such as 5:4 or 6:5, these contemporary musicians find 5-limit tunings incomplete because they do not include such ratios as 7:1, 11:1, or 13:1.

In its own terms, however, each system is a complete whole supporting music of great expressive power and beauty. Moving from Gothic to Renaissance, we not only encounter new possibilities opened by stable thirds and sixths, but lose the expressive power of the active Pythagorean thirds and sixths (with the modified Pythagorean tuning of around 1400 possibly representing a bridge between these worlds, see Section 4.5). Moving from the Renaissance-Romantic eras to the new world of 7-limit and higher systems, we again expand the universe of restful concords but lose the expressive power of an interval such as the minor seventh when treated as definitely active and unstable.

Any system of harmony or tuning represents a process of selection based on sheer artistic taste and more or less elegant consistency. Around 1300, the English theorist Walter Odington proposes that major and minor thirds (81:64, 32:27) have ratios close to 5:4 and 6:5 respectively, and that singers lean toward these simpler ratios. His observation may reflect the style of much English polyphony of the period, where thirds often have a pervasive role in the texture and may even serve as closing sonorities. However, he is much less friendly to the 9-based M2 and M9, finding these intervals quite "incompatible" - much in contrast to Anonymous I and Jacobus in roughly the same era, not to mention modern Japanese koto players.

Jacobus observes that it would be possible to build instruments using ratios such as 5:1 and 7:1, which he demonstrates would differ from the Pythagorean M3 (81:64) and m7 (16:9) by 81:80 and 64:63. However, he concludes, such intervals would not be well-formed in terms of proper whole-tones and semitones. Evidently he did not perceive any great musical utility in these intervals, as opposed to the familiar M3 and m7 of the standard musical language - or insufficient utility to outweigh the inconsistencies with the rest of the system.

Likewise, it is easy to see how the proposal of some 18th-century musicians for a concordant "harmonic minor seventh" with a pure ratio of 7:1 might seem an incongruous complication to composers and theorists oriented to a musical language and tertian tuning system where all sevenths are discords.

Each system has its own sense of logic as well as beauty. In quintal just intonation, the minor seventh (16:9) divides neatly into the two concordant fourths of 7/4 (16:12:9), a favored sonority of Perotin's era also recommended by Jacobus. Neither the 9:5 ratio of tertian just intonation, derived from the pure minor third and fifth, nor the 7:1 ratio of today's 7-limit and higher systems, offers this advantage.

In tertian styles of the Renaissance-Romantic eras, where 7/4 makes sense only as a restricted suspension dissonance or the like, the issue of its ideal tuning seems of little moment; a modern system of just intonation might offer multiple flavors of m7 (16:9, 9:5, 7:1) to accommodate quintal/quartal, tertian, and more novel harmonies. Such a solution brings more flexibility, and also more complexity both in conception and in implementation on fixed-pitch instruments.

Just intonation is therefore not a single system of tuning, but an open set of systems: quintal, tertian, and beyond. Such systems share in common a desire for intervals based on integer ratios, and for stable concords in their ideally blending form; their manifold differences echo the varying musical styles and tastes that not only reflect but shape tuning systems of all kinds. Pythagorean tuning in the context of Gothic polyphony offers one eloquent example.

5.2. Beauty, utility, and choice

In contrast to the various systems of just intonation, temperament involves small and deliberate deviations from ideal interval ratios (as defined in a given period or style). Since at least the 15th century, the two approaches have coexisted in European music, and indeed systems such as meantone and well-temperament may combine both approaches, tuning some intervals in a just manner and tempering others.

This final section compares Pythagorean just intonation as discussed so far with three overlapping families of temperaments: meantone, well-temperament, and equal temperament.

The history of tuning systems, it is often said, has revolved around the two goals of "beauty" and "utility." While beauty aims at the presentation of intervals in their most just or pleasing proportions, utility aims at the maximum number of "tolerable" or "playable" intervals in an octave.

A standard Pythagorean tuning, for example, seems to offer a great degree of "beauty" in the context of Gothic polyphony from Perotin to Machaut (as well as for earlier music). Its pure fifths and fourths, active thirds and sixths, just major seconds and minor sevenths, and small diatonic semitones all concord well with the stylistic qualities of harmonic color and efficient cadential action (Section 3).

At the same time, this tuning offers a rather high although not quite optimal level of "utility": in return for 11 just fifths or fourths in an octave, we accept a Wolf at G#-Eb. Since these two notes rarely occur together, the practical loss of utility is slight.

To perfect the utility of this tuning without any compromise in its beauty, we might add a 13th key to the keyboard at Ab, a Pythagorean comma below G#. Then we would have a pure fifth Ab-Eb or fourth Eb-Ab, completing our set of 12 perfect fifths or fourths in an octave.

To domesticate the Wolf within the limits of a 12-note octave, we could implement a Pythagorean well-temperament of the kind to be proposed in Section 5.5. Such a tuning is in many ways a fascinating mirror-image of the triadic well-temperaments of the late 17th and 18th centuries. The symmetries of design reflect a common goal of utility, while the contrasts reflect the rather different ideals of harmonic beauty prevailing in medieval and Baroque music.

5.3. From Gothic to Renaissance: a new intonational dilemma

Around 1400, musicians experimented with a modified form of Pythagorean tuning evidently signalling a shift in the ideal of beauty toward more blending thirds. Tuning the sharps at the flat end of the chain of fifths, a Pythagorean comma lower than usual, altered major and minor thirds involving sharps to 384 and 318 cents respectively - just 2 cents from their ideally blending ratios of 5:4 and 6:5. Other thirds retained their usual active quality.

As discussed at more length in Section 4.5, this tuning with its two contrasting flavors of thirds and sixths involved some delicate compromises in the areas of beauty and utility. Mark Lindley suggests that it may have been quite influential in the epoch of Dufay's youth around 1420, an epoch often seen as marking the musical transition from Gothic to Renaissance.

As thirds and sixths assumed a more and more pervasive role in the texture over the next few decades, early Renaissance musicians sought new tunings that would make ideally smooth and resonant forms of these intervals available at as many scale locations as possible. By 1482, Bartolome Ramos had documented a solution that remained standard for keyboard instruments of the 16th and 17th centuries: meantone temperament. Other theorists of the Renaissance described alternative solutions, now known as well-temperament and equal temperament. Before defining and exploring these overlapping categories, we should briefly consider the problem they address.

The new Renaissance sensibility favoring restful and even conclusive 5/3 and 6/3 combinations - by the end of the century, composers such as Josquin des Prez and Heinrich Isaac had begun to use the third in closing sonorities - raised a basic intonational dilemma. It is impossible, on a fixed pitch instrument having only 12 notes per octave, to achieve pure fifths and pure thirds at the same time at all positions of the chromatic scale.

In the Gothic era, when pure fifths and active thirds happily coexisted both stylistically and mathematically, this facet of tuning may have been more of a congenial feature than a bug (Section 4.4.2). The emerging tertian harmony of the 15th century, ideally calling for pure fifths and thirds, raised the urgent question which has remained perennial for triadic music: which fifths and/or thirds should be compromised, and by how much?

Meantone temperaments of the Renaissance and Baroque characteristically answer: "By all means let us have as many pure thirds as possible, even if this means compromising the fifths and breeding some Wolves in the bargain." Well-temperaments, coming into ascendency in the 18th century, more moderately reply: "Let us have a continuum of triadic colors ranging from pure thirds to pure fifths, but all intervals playable and no Wolves."

Equal temperament, curiously a special case of both other categories, offers a third answer: "Let us gently temper each fifth by an equal amount just sufficient to disperse the Pythagorean comma (Section 4.4.1), leaving rather vibrant thirds just subdued enough to support stable triads."

One might say that meantone favors beauty, equal temperament favors utility, and well-temperament leans toward utility with variety.

Here we consider each kind of temperament in its historical context, and then consider its qualities as a possible alternative to pure Pythagorean just intonation in the performance of Gothic music and related modern styles.

5.4. Meantone temperaments: tunings with an attitude

Tuning a chain of four pure fifths (3:2), such as F-C-G-A, results in a Pythagorean major third of 81:64, as opposed to a pure ratio of 5:4 or 80:64. Meantone tunings address this discrepancy of 81:80 or 21.51 cents - the syntonic comma (Section 4.4.2) - by narrowing the fifths slightly to optimize the thirds.

Technically speaking, a meantone tuning is any tuning which narrows all fifths equally, or all but one. Each major third will then consist of two equal whole-tones, or "meantones" - as opposed, for example, to the unequal 9:8 and 10:9 whole-tones favored in Renaissance and later systems of tertian just intonation (Section 5.1). If all fifths are equally narrowed, we have the special and uncharacteristic case of equal temperament (Section 5.6). Typically, however, the process of narrowing 11 fifths to achieve pure or nearly pure thirds results in leaving what has been called a "catastrophically" wide Wolf fifth, as we shall see.

In its most typical Renaissance form, meantone temperament seeks uncompromisingly pure major thirds. To achieve this goal, we must narrow each third by a syntonic comma - and thus narrow each of the four fifths generating the third by a quarter of this amount, or about 5.38 cents.

This most characteristic tuning of the family is therefore known as "1/4-comma meantone," and other meantone tunings are likewise identified by the amount of tempering or narrowing applied to each fifth - "1/6-comma meantone," "2/7-comma meantone," etc. Normally the syntonic comma is understood, unless the larger Pythagorean comma is expressly specified. The use of the term "meantone tuning" without qualification is often taken to imply 1/4-comma meantone, which produces pure major thirds.

In effect, 1/4-comma meantone is a variety of Renaissance just intonation mapped to a conventional keyboard. All major thirds are a just 5:4 (about 386.31 cents), and minor thirds are a slightly compressed 310.26 cents, quite close to a pure 6:5 (315.64 cents). This compression of 1/4 comma is equal to that of the fifth, since a major third and minor third together make a fifth.

Meantone temperament sets thirds, the choicest concords of the Renaissance, in their most pure or nearly pure aspect, celebrating as it were the new standard of harmonic beauty. By 1558, Zarlino refers to the 5/3 combination as harmonia perfetta, the perfect sonority which every competent composer strives to realize as often as possible; in 1612, Johannes Lippius describes this trias harmonica or triad as at once the manifestation of the Trinity in music and the key to the practical craft of composition. Meantone tuning embraces this sonority with due reveence.

As with the cartographer's maps of the Renaissance and later periods, the meantone mapping of tertian just intonation onto a 12-note keyboard inevitably involves some distortions. The uncompromising pursuit of just major thirds mandates a more compromising stance toward fifths, compressed from a just 701.96 cents to about 696.58 cents. Since variations in the tuning of fifths tend be perceived more acutely than is the case with thirds, 1/4 comma of tempering is a nontrivial although acceptable compromise; the identical compression of minor thirds is less significant.

From a melodic and cadential point of view, meantone results in a somewhat compressed major second of precisely half a 5:4 major third - the mean-tone - or about 193.16 cents, as compared to the expansive 203.9 cents of Pythagorean tuning. As in tertian just intonation, the diatonic semitone is larger than the chromatic semitone, 117.11 and 76.05 cents. These intervals are equal respectively to the difference between the pure major third and the fourth (enlarged by 1/4 comma) of 503.42 cents, and between the major and minor third.

Mark Lindley remarks that theorists of the earlier Renaissance such as Gaffurius found it difficult to give up the Gothic taste for small diatonic semitones, but by 1637, Mersenne extolled the large semitones of meantone as one of the "beauties" of music. While Renaissance vocal and string ensembles, like their modern counterparts, may have struck varying balances between pure thirds and high leading tones, meantone suggests at the least a sonorous ideal leaning toward the former.

All in all, meantone tuning very nicely reflects and realizes the Renaissance ideal of harmonic beauty based on just triadic ratios. It is in the area of utility that the more serious compromises occur.

In our Pythagorean tuning (Sections 4.2.2, 4.4.1), we found that a series of 12 pure fifths would exceed an even octave by the Pythagorean comma of 23.46 cents. Thus tuning a chain of 11 such fifths leaves a remaining Wolf fifth of only 678 cents, occuring in the standard tuning between the notes at the ends of the chain, G#-Eb.

If this fifth (more naturally spelled Ab-Eb) had come to play a more significant role in Gothic harmony, then utility might have suggested the compromise of slightly narrowing a number of fifths to disperse the "error term" of the comma and keep all 12 fifths within the range of "playability." Equal temperament (Section 5.6) offers one strategy for doing this, and a kind of "Pythagorean well-temperament" proposed below offers another (Section 5.5).

The disutility of 1/4-comma meantone for Wolf-domestication is that in order to achieve its primary purpose, just thirds, it narrows far too much. All 12 fifths together exceed the octave by a Pythagorean comma, so the average correction factor for each fifth would be 1/12 of this amount, or about 1.96 cents. However, the meantone imperative of pure 5:4 major thirds requires a narrowing of 5.38 cents per fifth, a discrepancy of 3.42 cents. Repeated over a chain of 11 fifths, this disparity produces a drastic overcorrection of about 35.69 cents, the amount by which the remaining fifth of around 737.65 cents must exceed a pure ratio of 3:2 (701.96 cents). The result is indeed a Wolf of "catastrophically" enlarged proportions.

At this point we should note that while ideals of "beauty" obviously change along with style, standards of "utility" are also tied to stylistic contexts. Whether a "near fifth" of 738 cents is rejected as an "unplayable Wolf" or embraced as an equal member of the microtonal democracy of intervals is a matter of context, in the 16th century or the 20th century, not of universal acoustical laws.

Nicola Vicentino (1511-1576), composer and microtonal theorist whose tuning system has been neatly summarized by Bill Alves, included in his set of intervals a "greater-than-fifth" of 20:13 (745.65 cents), an interval serviceably approximated by the enlarged fifth we have just been considering (737.65 cents). This sonority is therefore a "Wolf" in the setting of conventional Renaissance music, but a possible variant on a recognized microtonal interval in a music such as Vicentino's.

Like the narrow Wolf fifth in the standard Pythagorean tuning, this wide Wolf was placed at G#-Eb, an interval still rarely in use except in adventurous chromatic works, and deliberately avoided in keyboard compositions designed around the meantone system.

Meantone also generates Wolf diminished fourths such as Ab-C. Since two pure major thirds such as C-E and E-G# together take up about 772.63 cents, this leaves an interval G#-C (or Ab-C) of 427.37 cents (32:25), or a full 41.06 cents larger than a just M3. From Vicentino's microtonalist perspective, this interval might be described as a very close approximation of his "greater-than-major-third" of 50:39 or about 430.14 cents. In other Renaissance contexts, it is a most unsuitable substitute for an expected major third - in other words, a Wolf.

Despite its compromises in utility, meantone seems an ideal reflection of the sonorous world of the Renaissance, rather as Pythagorean tuning reflects the world of Gothic harmony. Each system is based on a commitment to just ratios for the choice concords - pure quintal just intonation in the case of Pythagorean tuning, and the closest approach to pure tertian just intonation possible on a conventional keyboard in the case of meantone. The pure fifths and assertively tense and vibrant thirds of Pythagorean, and the ideally blending thirds and compressed fifths of meantone, seem to represent two poles on the continuum of mathematically possible tunings. Our remaining systems, well-temperament and equal temperament, represent a kind of middle ground.

While admiring the curious antipodal kinship of Pythagorean and meantone as "tunings with an attitude," we can recognize that their ideals of beauty are quite divergent. The compressed fifths of meantone and its large diatonic semitones can hardly do justice to the stable trinic sonorities and incisive cadences of Gothic music (Section 3), nor can the unequivocally active thirds of Pythagorean do justice to the restful triadic sonorities and subtle shadings of Renaissance music.

Thus meantone would appear to be one of the more unlikely choices for Gothic music or modern compositions or improvisations in a similar vein.

5.5. Well-temperaments, triadic and trinic

A well-temperament, in the broadest sense, is any temperament that permits all relevant intervals in a given style to be playable in any transposition - in other words, a temperament avoiding all "Wolves." By this test, equal temperament is a special and uncharacteristic case of well-temperament (Section 5.6). More characteristically, a well-temperament involves tempering different fifths by differing amounts, thus producing a variety of sizes and "colors" for thirds and other derivative intervals. This systematic microtonal shading is the distinguishing feature of well-temperaments in their classic form.

From a historical perspective, "well-temperaments" are a family of tunings spanning the period of about 1680-1885, with qualities closely tied to stylistic features of the triadic major-minor key system which also, interestingly, prevails during roughly this same period. More broadly, we might view well-temperament as a philosophy or strategy of "Wolf-domestication" which can be applied to other harmonic styles and tuning schemes, including those of the Gothic era.

While forerunners of well-temperament might be traced back as far as an English organ tuning scheme of 1373 and the proposal of Henricus Grammateus (Heinrich Schreiber) in 1518, these systems seem more akin to equal temperament, and so will be discussed under that heading; also, such systems partially tame rather than fully domesticate the Wolf, leaving at least one fifth more than 10 cents from just.

The emergence of "well-temperament" in its classic or canonical form around 1680 reflected the demands of the new major-minor key system with its transpositions of triads to remote locations which usual meantone systems (Section 5.4) could not support. Confronted with the demand that all transpositions of the modes or keys be playable, musicians such as Andreas Werckmeister and his colleagues came up with an admirable solution.

As we have seen, the favorite meantone tunings of the Renaissance and early Baroque had sought as many just or near-just thirds in an octave as possible. Unfortunately, this required narrowing 11 fifths to a far greater extent than would be required to correct for the Pythagorean comma, thus leaving a drastically enlarged 12th fifth to become a "Wolf in reverse." Likewise, two ideally just 5:4 major thirds in an octave such as C-E and E-G# left a diminished fourth G#-C far too large to be playable as a major third. For extraordinarily demanding compositions, Renaissance composers evidently turned to extraordinary keyboards with more than 12 notes per octave.

To make remote transpositions possible on a conventional keyboard, Werckmeister and his peers hit on the solution of narrowing some of the fifths by varying amounts by while leaving others pure, so that the average adjustment would balance out to disperse the Pythagorean comma, domesticating all Wolves. Fifths involving the diatonic notes (e.g. C-G, D-A) may be tempered by a full 1/4 Pythagorean comma or 5.87 cents, while fifths among the black keys (e.g. F#-C#, Eb-Bb) are left untempered or just.

The result, in effect, is an approximation of meantone with its pure or nearly pure thirds for triads such as C-E-G or G-B-D, and an approximation of Pythagorean tuning with its tense and active thirds for remote triads such as F#-A#-C#. Triads at other points on the continuum of transpositions or keys will have their thirds built from various mixtures of tempered and untempered fifths, and thus will have intermediate degrees of tension. The total effect is a graduated spectrum of "key color."

From a purely utilitarian point of view, well-temperaments "split the difference" of meantone. Rather than a large number of pure or almost pure triads plus a sizable contingent of unuseable Wolves, we have a compromise in which the most commonly used modes or keys sound almost as harmonious as in meantone, and the more remote keys are rather tense but tolerable.

From an artistic standpoint, however, well-temperament transforms utility into beauty by giving the composer or performer a tonal palette of contrasts which itself can serve as a new expressive dimension. As scholar and tuning technician Ed Foote puts it, the "dissonance" of Pythagorean or near-Pythagorean thirds in the more remote keys becomes a part of the musical language of the 18th century at least partially reflected in theories of "key character."

This history has led to many modern interpretations and controversies. For example, scholars still debate whether Bach set out in his Well-Tempered Clavier specifically to take advantage of contrasting key colors, or mainly to demonstrate the freedom of transposition afforded by the new tunings. While modern equal temperament would fulfill the second purpose, obviously only a "well-temperament" in the classic sense could also fulfill the first.

From a medievalist point of view, well-temperaments in this historical sense present a paradox. They offer the pure Pythagorean fifths and active thirds that we want for Gothic music - but only in remote transpositions, since in 1700 as opposed to 1300 these intervals were regarded not as the sonorous ideal but as an artful as well as necessary deviation from that ideal.

Indeed such a keyboard might be an ideal incentive for a performer who loves medieval music and would like practice in the art of free transposition! Two favorite Gothic cadences may illustrate this point as presented first in a standard Pythagorean tuning, and then in Werckmeister I (1691) with transposition by a semitone:

Pythagorean (F# x B)             Werckmeister I

  e'--  +90 -  f'             d#'--  +96 -  e'
(906)        (1200)           (906)        (1200)
  b -  +90 -  c'             a# -  +96 -  b
(408)         (702)           (408)         (702)
  g - -204 -  f              f# - -198 -  e

  d'-- +204 -  e'             eb'-- +204 -  f'
(906)        (1200)           (906)        (1200)
  a - +204 -  b'             bb - +204 -  c'
(408)         (702)           (408)         (702)
  f -  -90 -  e              gb -  -90 -  f

Like the Pythagorean versions, the transposed Werckmeister I versions give expansive major thirds of 408 cents and major sixths of 906 cents, efficiently resolving to 8/5 trines with pure fifths and fourths (Sections 3.3, 4.5). As it happens, the tempering process has slightly altered the melodic progressions in our first example, contracting the descending whole-step in the lowest voice from 204 to 198 cents, and enlarging the ascending half-steps in the upper voices from 90 cents to a still-quite-incisive 96 cents. In the second cadence, where the lowest voice descends by a semitone and the upper voices ascend by whole-steps, the translation works perfectly.

In the absence of such transpositions, well-temperaments of this traditional kind present some of the same disadvantages for Gothic polyphony as meantone: unduly compromised fifths and fourths, subdued thirds and sixths, and large rather than keen diatonic semitones.

5.5.1. Well-temperament, Pythagorean style

However, our last examples suggest a possible solution: rather than transposing notes and cadences, why not transpose a well-temperament itself so that we have Pythagorean intervals in the most frequently used positions, and less Pythagorean sounds in positions involving less-used accidentals?

Such a "Pythagorean well-temperament" might invite the metaphor of a chess opening with colors reversed: our general plan is to tune the white keys in just fifths, but to temper some of the fifths involving black keys enough to avoid any Wolf. In fact, Owen Jorgensen has documented just a temperament scheme originally published by Anton Bemetzrieder in 1808!

Before considering the how of such a well-temperament, let us first ask why one would ever wish to use such a compromise rather than pure Pythagorean tuning either for Gothic polyphony, or for new music in a similar style.

The main reason is the obvious one: Wolf-domestication on a keyboard with 12 notes per octave. For certain adventurous medieval pieces, one might desire a proper fifth at ab-eb'; and modern compositions or improvisations in a "neo-Gothic" style might freely use all 12 fifths or fourths in an octave. Under such circumstances, Pythagorean well-temperament is to a pure Pythagorean tuning rather as 18th-century well-temperament is to meantone: a compromise of ideal sonority for the sake of utility.

Such a compromise is hardly necessary in the case of 13th-century polyphony, which scarcely requires a G# key, let alone the interval g#-eb' (i.e. ab-eb'). Nor is this interval very significant in 14th-century music, apart from some avant garde works. However, modern composers or improvisers might wish to experiment with a clavier, electronic or otherwise, "well-tempered" to a Gothic flavor but with the Wolf domesticated. Synthesizers with options making it easy to define and switch between tunings make experiments of this kind more attractive; we can use a custom temperament when desired, and return to pure Pythagorean for the usual medieval repertory.

5.5.2. The just, the true, and the incisive

To domesticate the Wolf, we will need to narrow some of the fifths of our tuning chain (Section 4.2.2) enough to disperse the Pythagorean comma (Section 4.4.1) while keeping all fifths within reasonable limits. Here we shall follow the custom for well-temperaments of measuring adjustments in terms of this comma, as opposed to the smaller syntonic comma (Section 4.4.2) of prime importance to meantone temperaments.

At one extreme, we could try the rough and ready remedy which Jorgensen calls "bisecting the Wolf," simply dividing the comma among two fifths - leaving them both 1/2 Pythagorean comma or 11.73 cents from just, very partially tamed "near-Wolves" at best.

At the other extreme, we could spread the comma equally among all of our fifths, tempering each by 1/12 comma or about 1.95 cents. This is, in fact, the solution of equal temperament (Section 5.6).

Between these extremes, we can seek to spread the comma among a number of fifths while striking various balances between three goals for Pythagorean well-temperaments: the just, the true, and the incisive.

Justness involves preserving as much as possible the pure ratios of the stable fifths and fourths which make up euphonious trines (Sections 3.1.1, 3.2.1), and also of the unstable intervals with simple ratios, major seconds and minor sevenths (Sections 3.1.3, 3.2.3). We would like as many of these intervals as possible to be just, and the remainder to be as close to just as possible.

Trueness is a quality applying especially to our thirds and sixths: we want them to be active and vibrant, as close as possible to their usual Pythagorean proportions. However, since these intervals are not so critical to our basic sound as fifths and fourths, and also take more kindly to inexact tuning, we can afford considerable compromise here. Curiously, our compromised thirds and sixths will become not yet more tense, but rather a bit "subdued." This is a side-effect of our Wolf-domestication for fifths, not an intentional goal as in meantone and triadic well-temperament. In fact, our goal of "trueness" is the opposite of what triadic tunings seek.

Incisiveness has two aspects, vertical and melodic (Section 3.3): efficient expansion or contraction of unstable intervals in various resolutions by contrary motion (e.g. m3-1, M3-5, M6-8, m7-5, M2-4); and concise diatonic semitones for expressive melody at other points as well.

Trueness and incisiveness are directly related: a Pythagorean major third at 408 cents has its full quantum of tension (trueness), and can expand to a fifth more efficiently (incisiveness), with the usual 90-cent semitone motion in one of the voices. Similarly, justness and incisiveness are related in the case of major seconds and minor sevenths: a just Pythagorean m7 at 996 cents can contract with equal efficiency to a fifth.

Since all three goals are ideally met by a pure Pythagorean tuning - apart from the matter of the Wolf - our first rule is to temper our fifths as little as possible.

Given the importance of our fifths and fourths as the choicest concords of our harmonic texture, let us resolve that all fifths stay within 1/4 comma of just, or 5.87 cents. This means that a major second or minor seventh (derived from two fifths or fourths) might be as much as 1/2 comma or 11.73 cents from just, another good reason to prefer smaller adjustments.

In the area of trueness, we would like our most frequently used thirds and sixths to remain at their ideal Pythagorean size, and others occurring with fair frequency to be at least as active as in equal temperament. Thus major thirds should ideally be 408 cents, and preferably at least 400 cents; major sixths, ideally 906 cents or at least 900 cents, etc.

As for incisiveness, we would like diatonic semitones to be as close to the usual 90 cents as possible, and preferably no worse than the 100 cents of equal temperament. From a vertical point of view, resolutions such as M3-5 or m7-5 by contrary motion should ideally involve only 294 cents of motion (a 204-cent whole-step plus a 90-cent half-step), and preferably no more than the 300 cents of equal temperament.

Our general strategy, as already mentioned, will be a kind of mirror image of an 18th-century triadic well-temperament. The idea is to leave the most frequently used fifths untempered, and to spread the adjustment of the comma through a number of the less-used fifths. Here I present first a solution which may be original - always a hazardous first impression in the area of tunings, where reinventing the wheel (or circle of fifths) seems a perennial occupation. Then follows Bemetzrieder's temperament of 1808, a somewhat more balanced solution.

5.5.3. Two sample solutions

One approach in deciding which fifths to temper is to follow the medieval distinction between the regular tones of the gamut including the diatonic notes plus Bb (musica recta), and the remaining accidentals (musica ficta).

Our first tuning accordingly leaves the traditional gamut in its pure Pythagorean form, tempering only those fifths involving the remaining accidentals and altering only the positions of these four tones (F#, C#, G#, Eb). Here is the general scheme in terms of the chain of fifths, with numbers representing adjustments in Pythagorean commas:

  (-1/4)                (-1/6)  (-1/6)  (-1/6)
\                                             /
  \                                         /
    --------------- (-1/4) ----------------

The three fifths at the sharp end of the chain are narrowed by 1/6 comma each, accounting for 1/2 comma in all; the fifth at the flat end, Eb-Bb, is narrowed by a full 1/4 comma. Thus our two ends of the chain have dispersed 3/4 comma in all, leaving 1/4 comma for the remaining fifth at G#-Eb, a "Wolf" now reasonably well domesticated.

Having two fifths (Eb-Bb and G#-Eb) a full 1/4 comma from pure is not the happiest condition, but the compromise does have some advantages. First, it would be my guess that fifths or fourths involving Eb-Bb may be less common in 13th-century music than B-F#, and less common in 14th-century music than cadences regularly involving fourths or fifths on B-F#, F#-C#, or C#-G#, here tempered 1/6 comma.

Another argument for keeping the most dramatic adjustments at the very end of the chain is that such adjustments have cumulative effects on the placement of accidentals, and thus the incisiveness of cadences (especially in 14th-century styles, where cadences with sharps are very common). Tempering a fifth means lowering the new note added in a sharp direction (e.g. F# in B-F#), or raising the new noted added in a flat direction (e.g. Eb in Eb-Bb), in turn affecting the placement of any additional notes added in that direction. The following diagram uses positive numbers below notes to show displacement in a sharp direction, and negative numbers for displacement in a flat direction:

        (-1/4)                 (-1/6)    (-1/6)    (-1/6)
    +1/4      0  0 0 0 0 0 0 0       -1/6      -1/3      -1/2

Here the greatest displacement is for G#, 1/2 comma flat due to the three 1/6-comma adjustements at the sharp end of the chain. At the flat end, although the fifth Eb-Bb has been tempered a full 1/4 comma, there is no cumulative displacement, so that Eb is less displaced than G# (1/2 comma) or even C# (1/3 comma). The bright side of the picture is that Eb has moved up 1/4 comma, and G# down 1/2 comma - reducing the full comma of the "Wolf gap" between them to a tenable 1/4 comma.

This is a well-temperament somewhat on the asymmetrical side: two fifths (Eb-Bb, G#-Eb) are asked to bear a full half of the comma adjustment, while seven of the 12 fifths remain just. A tuning chart of the octave c-c' may give a fuller view of the situation:

      106      300              608       804      996
      c#'      eb'              f#'       g#'      bb'
 _106_|_98_ _96_|108__      _110_|_94__102_|_102_90_|_114_
 c'        d'        e'    f'        g'         a'       b'    c''
1:1       9:8      81:64  4:3       3:2       27:16   243:128  2:1
 0        204       408   498       702        906     1110    1200
     204       204      90     204       204       204      90

All of the regular notes and intervals of the gamut remain in just intonation, as the familiar diatonic steps and ratios show. However, we note that the fourth eb'-g#' is now only about 504 cents wide - roughly 6 cents (1/4 comma) from a just 498 cents. At what cost have we domesticated this Wolf (formerly 522 cents, a full comma wide)?

To get an overview of incisiveness, we can inspect the sizes of the diatonic semitones around the four repositioned accidentals: c#'-d' (98 cents), d'-eb' (96 cents), f#'-g' (94 cents), and g#'-a' (102 cents). Such a graduated scale of variations is typical of well-temperaments, and we are pleased to find that all these semitones except the last remain smaller than the 100 cents of equal temperament. Of course, a'-bb' remains an ideally incisive 90 cents, since Bb was left unaltered.

In the special case of g#', its diatonic semitone g#'-a' and chromatic semitone g'-g#' are an equal 102 cents. This is not the unhappiest compromise, since one motivation for making G#-Eb (i.e. Ab-Eb) a proper fifth or fourth is use in cadential idioms with a descending semitonal motion Ab-G (a more natural spelling than G#-G):

f''-eb'' g''
c''      d''
ab       g

M6- 5    8
M3       5

With the whole-tone G-A divided into two "neutral" or Janus-faced semitones of 102 cents, the motions g#'-a' and ab'-g' are equally efficient. These "meansemitones" (Jorgensen) in a Pythagorean setting are slightly larger than their 100-cent counterparts in equal temperament, where a whole-tone is precisely 200 cents rather than a just 9:8 (204 cents). However, a 102-cent semitone is very close to our preference of "100 cents or less," and is considerably smaller than the diatonic semitones of tertian just intonation (112 cents) or Renaissance meantone (117 cents with pure major thirds).

All in all, our diatonic semitones have remained incisive, with some shading as we move toward either extreme in the chain of fifths. Apart from the special case of G# (or Ab), the asymmetry between small diatonic and large chromatic semitones is preserved.

Intimately related to incisiveness is the trueness of thirds and sixths. One way to get an overview is to look at a series of cadences with the very common formula where a major third expands to a fifth and a major sixth to the octave, moving in the direction of more remote accidentals on either side of the chain. For the sake of symmetry, we begin on the flat side with Bb, left unaltered:

 Cadences with sharps          Cadences with flats

  b' -  +90 - c''             g''-- +204 - a''
(906)        (1200)           (906)        (1200)
  f#'--  +94 - g'              d''-- +204 - e''
(404)         (702)           (408)         (702)
  d' - -204 - c'              bb'--  -90 - a'

  f#''--  +94 - g''             c''-- +204 - d''
(902)         (1200)           (900)        (1200)
  c#''--  +98 - d''             g' - +204 - a''
(400)          (702)           (402)         (702)
  a'  - -204 - g'              eb'--  -96 - d'

  c#''--  +98 - d''             f''-- +204 - g''
(898)         (1200)           (894)        (1200)
  g#' - +102 - a'              c''-- +204 - d''
(396)          (702)           (396)         (702)
  e'  - -204 - d'              ab'-- -102 - g'

Our major thirds and sixths are remaining fairly close to their ideal sizes of 408 and 906 cents, only falling below our preferred standard of 400 and 900 cents respectively as we reach the far end of the chain of usual cadences. From another perspective, M3-5 and M6-8 involve substantially more than 300 cents of expansion only in our cadences at this far end involving the "Janus-faced" G#/Ab (g#'-a' or ab'-a'). Note how as vertical M3 and M6 become smaller or less true, melodic semitones become larger or less incisive.

Looking at the size of the upper fourths in the cadential sonorities with sharps (the difference between M6 and M3) reveals our 1/6-comma compromise with justness: these fourths measure about 502 cents rather than a pure 498 cents. (Narrowing fifths involving sharps is the same as enlarging the corresponding fourths by the same amount.) Our keyboard octave chart above shows the fifth c#'-g#' at 698 cents rather than a just 702 cents - 1/6 comma of tempering - and eb'-bb' at only 696 cents because of our more drastic tempering of 1/4 comma.

Given that pure fifths and fourths are the quintessence of Pythagorean tuning, we might well have some mixed feelings about the justice of this tempering. As long as the accidentals are used mainly for unstable cadential sonorities, where tension is expected, we might argue that slightly out-of-tune fifths or fourths are not so crucial. However, if these intervals are used in stable sonorities such as b-f#' or f#-c#'-f#', we find ourselves in a situation where our prime concords are being tangibly compromised.

Compromising fifths and fourths also compromises the justness of major seconds or minor sevenths. In one less happy scenario, we might find ourselves trying an adventurous transposition of a common Perotinian cadence with oblique motion above a stationary lowest part:

  ab'-- 192 - bb'
(1008)       (1200)
  eb'-- 198 - f'
 (504)        (702)

  m7     -     8
  4      -     5

The upper fourths of the colorful 7/4 sonority are, unfortunately, the two we both happened to leave tempered by a full 1/4 comma: each is enlarged to 504 cents, an "injustice" three times that of equal temperament. These errors cumulatively stretch the outer minor seventh to 1008 cents, a full 1/2 comma or almost 12 cents beyond its just 16:9 ratio (996 cents). The melodic whole-tones in the upper voices are also compressed from a just 9:8 (204 cents) to 198 cents and 192 cents, the latter being more typical of meantone tuning than of Pythagorean intonation with its spacious major seconds. We are hardly doing justice to this majestic cadence.

With a standard Pythagorean tuning, of course, we could not even attempt such a transposition: the Wolf fourth eb'-ab' (or eb'-g#') would preclude it. With well-temperament, our more mildly enlarged fourths are "playable," but how pleasingly is an open question.

Using meantone-quality fifths and fourths in a Gothic or neo-medieval setting is perhaps akin to using Pythagorean thirds in a late Baroque setting: it is an inherent consequence of well-temperament which may be considered a bug, a feature, or an artistic resource. In the case of triadic well-temperaments, the 18th and 19th centuries provide a body of historical experience. In one viewpoint, these temperaments replaced the Wolves of meantone with many smaller Wolves scattered over the keyboard; but composers learned how to howl with them most artfully.

Applying the well-temperament principle to a medieval Pythagorean tuning, in contrast, moves beyond historical experience and into the realm of modern experimentation, whether we use such a temperament for Gothic polyphony or for modern compositions or improvisations in similar styles.

For experimentally minded readers, here is a table of our above temperament similar to that for a standard Pythagorean tuning given in Section 4.3.1:

Pythagorean temperament 1: B-F# F#-C# C#-G# (-1/6 PC); Eb-Bb (-1/4 PC)
Temperament table: frequencies with a'=440, and variances from 12tet
                |     with respect to c'    |    with respect to a'
Note  Hz a'=440 |  ratio    cents  +/-12tet |  ratio   cents  +/-12tet
c'     260.74       1:1       0.00    0.00  |  16:27   -905.87   5.87-
c#'    277.18    tempered   105.87    5.87+ | tempered -800.00   0.00
d'     293.33       9:8     203.91    3.91+ |   2:3    -701.96   1.96-
eb'    310.07    tempered   300.00    0.00  | tempered -605.87   5.87-
e      330.00      81:64    407.82    7.82+ |   3:4    -498.04   1.96+
f'     347.65       4:3     498.04    1.96- |  64:81   -407.82   7.82-
f#'    370.41    tempered   607.83    7.83+ | tempered -298.04   1.96+
g'     391.11       3:2     701.96    1.96+ |   8:9    -203.91   3.91-
g#'    414.83    tempered   803.91    3.91+ | tempered -101.96   1.96-
a'     440.00      27:16    905.87    5.87+ |   1:1       0.00   0.00
bb'    463.54      16:9     996.09    3.91- | 256:243    90.22   9.78-
b'     495.00     243:128  1109.78    9.78+ |   9:8     203.91   3.91+
c''    521.48       2:1    1200.00    0.00  |  32:27    294.13   5.87-

One way of making our first temperament more gentle and symmetrical would be to distribute the Pythagorean comma equally among all six fifths involving accidentals, including Bb-F. Each of these fifths will therefore be tempered by 1/6 comma.

After this scheme occurred to me, I discovered as mentioned above that an essentially identical scheme had been published in 1808 by Anton Bemetzrieder, whom Jorgensen describes as "a French music teacher and theorist who settled in London in 1781." Bemetzrieder favored tunings with a Pythagorean flavor, and in the case of present interest he "instructed that the six fifths involving the five accidentals must each be tempered to a similar degree." Other fifths are left just.

Jorgensen identifies this scheme as a temperament of Francescantonio Vallotti and Thomas Young (1799) transposed so as totally to reverse the key colors, making it the antithesis of a "well-temperament" in the traditional sense. However, it seems curiously apt as an adaption of the well-temperament concept to Gothic music, whatever its original intent may have been.

Our basic temperament plan looks like this:

  (-1/6)  (-1/6)             (-1/6)  (-1/6)  (-1/6)
\                                                  /
  \                                              /
    ----------------- (-1/6) -------------------

Note that the sharp end of the chain is unchanged, carrying 1/2 comma of the correction, while the flat end now carries 1/3 comma, leaving 1/6 comma for the twelfth fifth, our domesticated Wolf G#-Eb. This fifth and the fifth Eb-Bb are now narrowed by only 1/6 comma instead of 1/4 comma. In return for the milder treatment of these intervals, we have compromised Bb-F (previously unaltered) by the same 1/6 comma. Also, while the adjustment at the flat end is now in two gentler steps of 1/6 comma, the cumulative displacement of Eb in the sharp direction has increased from 1/4 comma to 2/6 or 1/3 comma:

       (-1/6)   (-1/6)               (-1/6)    (-1/6)    (-1/6)
  +1/3      +1/6       0 0 0 0 0 0 0       -1/6      -1/3      -1/2

The overall result is one of balance and symmetry, as we may see by looking at a keyboard octave chart for this temperament:

      106      302              608       804     1000
      c#'      eb'              f#'       g#'      bb'
 _106_|_98_ _98_|106__      _110_|_94__102_|_102_94_|_110_
 c'        d'        e'    f'        g'         a'       b'    c''
1:1       9:8      81:64  4:3       3:2       27:16   243:128  2:1
 0        204       408   498       702        906     1110    1200
     204       204      90     204       204       204      90

From the viewpoint of incisiveness, we find that the diatonic semitones f#'-g' and a'-bb' are an identical 94 cents; c#'-d' and d'-eb' are both 98 cents; and the "Janus-faced" g#'-a' and g'-ab' are an equal 102 cents. As with our previous tuning, these results seem reasonably close to our ideal of 90 cents, and our general preference for "100 cents or less."

The symmetry of this well-temperament becomes even clearer when we look at a series of cadences on the sharp and flat sides of the chain of fifths. Both the trueness of vertical thirds and sixths, and the incisiveness of melodic semitones, show a mirrorlike gradation:

Cadences with sharps          Cadences with flats

  b' -  +90 - c''             g''-- +204 - a''
(906)        (1200)           (902)        (1200)
  f#'--  +94 - g'              d''-- +204 - e''
(404)         (702)           (404)         (702)
  d' - -204 - c'              bb'--  -94 - a'

  f#''--  +94 - g''             c''-- +204 - d''
(902)         (1200)           (898)        (1200)
  c#''--  +98 - d''             g' - +204 - a''
(400)          (702)           (400)         (702)
  a'  - -204 - g'              eb'--  -98 - d'

  c#''--  +98 - d''             f''-- +204 - g''
(898)         (1200)           (894)        (1200)
  g#' - +102 - a'              c''-- +204 - d''
(396)          (702)           (396)         (702)
  e'  - -204 - d'              ab'-- -102 - g'

Both the major thirds and melodic semitones are shaded in an exactly symmetrical fashion; although there are slight differences in the size of the major sixths, they also change by identical 4-cent (1/6-comma) increments. Again, it is only at or near the end of the chain of usual cadences that M3 and M6 become smaller than our preferred sizes of 400 cents and 900 cents respectively - or that M3-5 and M6-8 require substantially more than 300 cents of motion.

What happens if we try more remote transpositions, exploring as it were an intonational "far side of the moon"? Here are a few sample cadences using our two well-temperaments:

    Temperament I                Temperament II

 eb''-- +198 - f''            eb''-- +196 - f''
(892)        (1200)           (894)        (1200)
 bb' - +204 - c''            bb''-- +200 - c''
(388)         (702)           (392)         (702)
 gb' - -110 - f''            gb' - -110 - f'

 d#''-- +108 - e''            d#''-- +106 - e''
(892)         (1200)          (894)        (1200)
 a#' - +114 - b'             a#' - +110 - b'
(388)          (702)          (392)         (702)
 f#' - -200 - e'             f#' - -200 - e'

 db''-- +194 - eb''           db''-- +196 - eb''
(898)         (1200)          (898)        (1200)
 ab' - +192 - bb'            ab' - +196 - bb'
(396)          (696)          (396)         (698)
 fb' - -108 - eb'            fb' - -106 - eb'

 g'' - +102 - ab''           g'' - +102 - ab''
(906)         (1200)          (902)        (1200)
 d'' -  +96 - eb''           d'' -  +98 - eb''
(408)          (696)          (404)         (698)
 bb' - -192 - ab'            bb' - -196 - ab'

In the first two cases, we land on usual trinic centers retaining pure fifths and fourths; but thirds and sixths are uncharacteristically small and "subdued," with large semitonal motions.

In the third case, our unstable intervals and semitones are a bit closer to the norm - but we arrive at a trine with tempered fifths and fourths rather than usual pure concords. In the fourth case, the thirds and sixths are at or quite near to their ideal sizes, and melodic semitones are around 100 cents or smaller - but again we arrive at a trine with its choice concords tempered by 1/4 or 1/6 comma. The last two examples also involve compression of melodic whole-tone motions from their ideal 204 cents to 196 cents or less.

These compromises are analogous to those of triadic well-temperaments: all transpositions are "playable," and along with utility comes what may be called either compromised beauty or stimulating variety. Within the usual Gothic range of accidentals, cadences retain their "bright" color and clarity. As our transpositions become more and more remote, cadences become more "blurred" as thirds and sixths shade toward a "subdued" quality, voices progress by larger semitones, or we arrive at not-so-restful sonorities with compromised fifths and fourths.

For the experimentally inclined, here is a table of our symmetrical 1/6-comma temperament, a curious neo-Gothic counterpart to various well-temperaments of the 18th century described by J. Murray Barbour which feature six fifths tempered by 1/6 comma and the others pure:

Pythagorean temperament II: B-F# F#-C# C#-G# Eb-Bb Bb-F (-1/6 PC)
Identical to Bemetzrieder's #2 of 1808 (Jorgensen 1991: 328-340)
Temperament table: frequencies with a'=440, and variances from 12tet
                |     with respect to c'    |    with respect to a'
Note  Hz a'=440 |  ratio    cents  +/-12tet |  ratio   cents  +/-12tet
c'     260.74       1:1       0.00    0.00  |  16:27   -905.87   5.87-
c#'    277.18    tempered   105.87    5.87+ | tempered -800.00   0.00
d'     293.33       9:8     203.91    3.91+ |   2:3    -701.96   1.96-
eb'    310.43    tempered   301.96    1.96+ | tempered -603.91   3.91-
e      330.00      81:64    407.82    7.82+ |   3:4    -498.04   1.96+
f'     347.65       4:3     498.04    1.96- |  64:81   -407.82   7.82-
f#'    370.41    tempered   607.83    7.83+ | tempered -298.04   1.96+
g'     391.11       3:2     701.96    1.96+ |   8:9    -203.91   3.91-
g#'    414.83    tempered   803.91    3.91+ | tempered -101.96   1.96-
a'     440.00      27:16    905.87    5.87+ |   1:1       0.00   0.00
bb'    464.59    tempered  1000.00    0.00  | 256:243    94.13   5.87-
b'     495.00     243:128  1109.78    9.78+ |   9:8     203.91   3.91+
c''    521.48       2:1    1200.00    0.00  |  32:27    294.13   5.87-

In conclusion, "well-temperaments" in their historical form present many of the same problems for Gothic polyphony as meantone. However, it is possible to use the broader strategy of well-temperament in a manner which seems rather happily to avoid the Wolf fifth of pure Pythagorean tuning while largely retaining the qualities of this tuning for the most frequently used sonorities.

For rare Gothic pieces which would raise a Wolf problem on a conventional keyboard, or for experimental purposes, such a "trinic well-temperament" might be an attractive option - although I am aware of no exact medieval precedent for this graduated shading of intervals (for a tuning of around 1400 which does exploit color contrasts, see Section 4.5). Otherwise, the usual Pythagorean just intonation offers the advantages of simplicity and uncompromised beauty.

5.6. Equal temperament: a musical supplement

In equal temperament, each of the 12 fifths is narrowed by just the small amount needed - 1/12 Pythagorean comma, or about 1.95 cents - to disperse the comma equally among them all. This solution not only domesticates the Wolf, but creates a "level playing field" in which modes, keys, or cadences may be freely transposed among the 12 tones without changing interval sizes. In 1588, Zarlino reports that Abbot Girolamo Roselli praised this symmetrical temperament as a "spherical music" permitting free navigation; we might also describe it as "isochromatic," since interval sizes and colors remain consistent regardless of our location in the system.

Although no replacement for true Pythagorean tuning, this offshoot shares some kindred qualities congenial to Gothic polyphony: almost pure fifths and fourths, rather active thirds and sixths, and fairly small diatonic semitones. Curiously, as we shall see, the favorite keyboard tuning of the 20th century may be more happily suited to medieval music than to much of the music of the intervening Renaissance and Baroque eras.

To see why this should be so, let us compare equal temperament with the two main families of tunings prevailing in these eras: meantone (Section 5.4) and well-temperament (Section 5.5).

Technically speaking, equal temperament belongs to both categories: at least 11 of the 12 fifths are tempered by an identical amount, and all intervals remain playable (no Wolves). However, it is characteristic of neither category: its active thirds and sixths are closer to Pythagorean than to Renaissance meantone, while its uniform color stands in contrast to the shadings of canonical well-temperaments.

5.6.1. Equal temperament as 1/11-comma meantone

Placing equal temperament on the larger spectrum of meantone tunings may help to illustrate the first point. As we may recall, meantone temperaments are customarily identified by specifying the amount by which each fifth is narrowed, taking as our unit the syntonic comma (Section 4.4.2) of 81:80, or about 21.51 cents. This comma measures the difference between active Pythagorean thirds and sixths and their ideally restful counterparts with the simplest ratios: e.g. M3, at 81:64 (407.82 cents) vs. 5:4 (386.31 cents).

In our meantone discussion, we saw that each major third derives from a chain of four fifths, so that narrowing each fifth by 1/4 of this comma - 1/4 comma meantone - results in pure 5:4 major thirds. Other meantone tunings of the Renaissance and early Baroque tend to remain fairly close to this value. Zarlino and Francisco de Salinas recommend 2/7-comma and 1/3-comma tunings respectively, which actually make major thirds slightly small while granting to minor thirds a special grace. Some musicians may have preferred milder 1/5-comma or 1/6-comma temperaments, which might more easily permit ad hoc adjustments to convert Wolf intervals into marginally playable ones (a kind of rough and ready "quasi-well-temperament").

Where is equal temperament located on this meantone spectrum? As it happens, the syntonic comma (21.51 cents) is equal to almost exactly 11/12 of the Pythagorean comma (23.46 cents). To make our 12 fifths equal, we must narrow each of them by 1/12 Pythagorean comma, or 1/11 syntonic comma: in other words, 1/11-comma meantone. This is obviously a much gentler tempering of the fifths than the 1/6-comma to 1/3-comma meantone systems prevailing during the Renaissance.

In 1/11-comma meantone, each major third becomes 4/11 of a syntonic comma or about 7.82 cents smaller than Pythagorean, arriving at its equally tempered value of 400 cents, exactly one third of an octave. From another viewpoint, it remains 7/11 of a syntonic comma or about 13.69 cents larger than its pure 5:4 counterpart at 386.31 cents.

Equal temperament therefore offers a kind of compromise between the active thirds of the Gothic and the restful thirds of the Renaissance, but a compromise leaning decidedly in the active direction. In contrast, favorite historical meantone temperaments are on the restful side. Even in 1/6-comma tuning, major thirds move 2/3 of the way toward justness; in 1/4-comma tuning, they attain it; and in the 2/7-comma and 1/3-comma tunings, they become slightly smaller than just.

Following the terminology of our discussion on well-temperaments, we might say that equally tempered thirds and sixths incline more to Pythagorean "trueness" than to Renaissance "justness."

This facet of equal temperament, pleasant to 20th-century medievalists and not necessarily so pleasant to 16th-century lutenists and their latter-day followers, reflects a simple fact of mathematics. The task of dispersing the Pythagorean comma equitably among our fifths - and creating in the process Roselli's "spherical music" - requires much less of a tempering adjustment than moving thirds all or even most of the way from "true" to "just."

To explain this disparity, let us use round numbers. If we temper each of 11 fifths by 2 cents, reducing them deftly from a pure 702 cents to 700 cents, we have opened 22 cents of space, permitting the remaining Wolf fifth to grow from a narrow 678 cents to an identical 700 cents: an elevenfold efficiency. However, each major third claims only four of these fifths in its chain of descent, so it is reduced by 8 cents only, from a true 408 cents to an even 400 cents: a mere fourfold efficiency.

To narrow our major thirds by a full 22 cents, bringing them all the way to a just 386 cents, we must increase our tempering of the fifths by this factor of 11:4 (or 22:8), moving from 1/11-comma meantone (equal temperament) to 1/4-comma meantone. Now these thirds will be pure, but our 12th fifth will receive 11/4 of the needed adjustment, an "embarrassment of riches" causing it to expand not only to a just size but far beyond it, becoming a wide Wolf.

5.6.2. Exploring the meantone spectrum

A more graphical overview of the meantone spectrum may illustrate this unequal leverage of our tempering engine. Because of the exigencies of space, some decimal numbers are broken into two lines, with the integer portion on the first line and the decimal portion on the second. Our continuum ranges from pure Pythagorean tuning, with no tempering of fifths, to the 1/3-comma meantone of Salinas:

       Pyth       "Well-tempered"        Characteristic meantone
                   ..----------..     -----------------------------
                     |  ET    |       |    |        |     |       |
SC:      0        1/14 1/11  1/8     1/6  1/5      1/4   2/7     1/3
      (cents)     1.54 1.96  2.69    3.58 4.30     5.38  6.14   7.17
tempered |-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:
cents   -0  .5   1  1.5  2  2.5  3  3.5  4  4.5  5  5.5  6  6.5  7

         -20 -15  -9  -4  +1   6  12  17  23  28  34  39  45  50  56
fifth12   .7  .2  .7  .2  .3  .8  .3  .8  .3  .8  .3  .8  .3  .8  .3
cents    |-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:
from   -23 -18 -12  -7  -1  +4   9  15  20  26  31  37  42  48  53
3:2     .5  .0  .5  .0  .5  .0  .5  .0  .5  .0  .5  .0  .5  .0  .5

M3     +21  19  17  15  13  11   9   7   5   3   1  -0  -2  -4  -6
cents   .5  .5  .5  .5  .5  .5  .5  .5  .5  .5  .5  .5  .5  .5  .5
from     |-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:-|-:
5:4      |              |                           |
       True            ET                          Just

As shown at the top of the chart, this spectrum includes two discrete regions: a "Well-tempered" zone surrounding 1/11-comma meantone or equal temperament, and ranging from about 1/14-comma to 1/8-comma, where all 12 fifths are playable; and a "Characteristic meantone" zone from about 1/6-comma to 1/3-comma, where thirds are at or fairly close to their just sizes. Tempering values for some meantone tunings of interest are shown both in their usual form as fractions of the syntonic comma (SC), and in cents.

The rest of the chart shows how, as we move along this continuum, tempering each of 11 fifths by a given number of cents affects two important aspects of a tuning, also shown in cents: the amount by which our 12th fifth varies from a pure 3:2, and the amount by which major thirds vary from a pure 5:4. Decimal values for the last two items, as mentioned above, are split into two lines.

Starting at the left end of the chart, Pythagorean tuning or 0 cents of tempering for our fifths, we see that our 12th fifth varies from its pure size by "-23.5" cents; that is, it is a narrow Wolf. Major thirds vary from 5:4 by "+21.5" cents; that is, they have their wide and active Pythagorean size, labelled at the bottom of the chart as "True."

As we move to the right, gradually increasing the tempering of our fifths, the size of the 12th fifth increases very rapidly, while the size of the major thirds decreases much more gradually. For example, moving from 1 cent to 2 cents of tempering causes this fifth to grow from "-12.5" to "-1.5" - a difference of 11 cents, bringing it from a "semi-Wolf" status to near-purity. Major thirds much less dramatically narrow from "+17.5" to "+13.5" - a difference of 4 cents, leaving them still on the active side of the spectrum.

Glancing at some other portions of the spectrum will show how, more generally, these disparate leverages of 11:1 and 4:1 operate. The size of the 12th fifth changes so rapidly that it is shown for every 1/4-cent increment of tempering, and even so it moves in rather large steps of 2.75 cents. The size of major thirds, with its more moderate rate of change, is shown at 1/2-cent increments of tempering, thus moving in 2-cent steps. We can find this value at 1/4-cent tempering points such as 4.25 cents by subtracting an even cent from the next value to the left (+5.5 at 4 cents) or adding it to the next value on the right (+3.5 at 4.5 cents), here getting +4.5, i.e. a major third about 4.5 cents wider than just. For interested readers, a more precise method of calculating these values for any arbitrary tempering amount is explained below.

An especially instructive exercise is to follow the chart from left to right, seeing how quickly the 12th fifth expands from a narrow Wolf to a just size, and from there to a wide Wolf, long before we have reached the region of pure thirds. A fifth more than 10 cents from just is often considered a Wolf, although Jorgensen describes certain fifths in the 15-cent range of variance as somewhat serviceable "semi-Wolves." Whatever our standard, it is rapidly exceeded as we move into the region of characteristic meantone.

Looking at the whole chart, we see that our "Well-tempered" and "Characteristic" zones indicated at the top each have a central point of interest indicated on the bottom line. For the well-tempered region, this point is "ET" or equal temperament at 1/11-comma, where all 12 fifths have an equal and near-just size; for the characteristic region, it is the point of "Just" or pure major thirds at 1/4-comma.

While equal temperament makes an appealing conceptual center for our well-tempered zone, shown as extending from about 1/14-comma to 1/8-comma, both center and boundaries are open to interpretation, as suggested by the dotted lines shading off at either end. Another possible center occurs at 1/11 Pythagorean comma (about 2.13 cents) - 12/121-comma meantone - correcting over 11 fifths for precisely the entire Pythagorean comma of 23.46 cents, so that the 12th fifth is pure! Starting at this alternative center of interest, we find the quality of our 12th fifth varies symmetrically from just as we move in either direction, eventually reaching the limits of comfort and "playability." From this perspective, equal temperament's location within the region is slightly to the "left of center" on the Pythagorean side (less tempering), as the chart shows.

Our boundaries of 1/14-comma and 1/8-comma are meant to define a "comfort zone" rather than outer limits of playability: the 12th fifth will be about 6.56 cents narrow at the former value, and 6.11 cents wide at the latter. In a Renaissance context, these variances seem quite acceptable when compared with the 6.14 cents of tempering routinely applied to fifths in Zarlino's 2/7-comma meantone, or the 7.17 cents of Salinas's 1/3-comma meantone. The dotted lines indicate outskirts to this zone, shading off at about 9 or 10 cents from just.

While 1/4-comma meantone, where the prime Renaissance concord of the major third becomes just, is a natural conceptual center of the "characteristic meantone" region, the minor third presents an alternative focus of interest. At 1/4-comma, it is a bit smaller than just, because the major third has reached its pure size but the fifth has contracted, and a minor third is equal to the difference of these intervals. Between 1/4-comma and 1/3-comma, the major third shrinks in size, allowing the minor third to grow from about 310.27 cents to a just 315.64 cents. Around 2/7-comma there is a balance between the two thirds.

As we move to the other side of 1/4-comma, we approach a region where the wide Wolf becomes more "finessable" by ad hoc adjustments, and also where the fifths are less compromised while thirds become somewhat less restful, although still decidedly closer to just than to Pythagorean true. While Renaissance tuning instructions often leave great room for discretion in tempering to taste, 1/6-comma seems a fair demarcation line for classic meantone: the point at which major thirds are 1/3 comma (7.17 cents) wider than just.

In sum, 1/11-comma meantone or equal temperament is situated at a point on the meantone spectrum where nearly pure fifths and active thirds prevail, and thus has qualities more akin to Pythagorean intonation than to the historical tunings usually associated with the term "meantone." It is more specifically located in a relatively narrow "well-tempered" band of the spectrum where all 12 fifths are playable, its distinctive advantage vis-a-vis either Pythagorean with its narrow Wolf or characteristic meantone with its wide Wolf.

Our chart, incidentally, suggests a possible compromise tuning for certain Renaissance pieces with chromatic passages of the kind which Mark Lindley suggests may have been intended for a keyboard with more than 12 notes per octave. Assuming that such a special keyboard is not available, we have the obvious choices of playing in Renaissance meantone and colliding with Wolves, or playing in equal temperament and producing a decidedly uncharacteristic sound.

Looking at our chart, we see that the "well-tempered" region extends somewhat to the right of 1/11-comma meantone or equal temperament, permitting us to move in the direction of more restful thirds while keeping all 12 fifths playable - and likewise intervals such as the diminished fourths c#-f and f#-a#. A tuning of 1/8-comma meantone (tempering each fifth about 2.69 cents) produces a 12th fifth only about 6.11 cents wide of just, while offering major thirds of about 397.07 cents or 10.75 cents from just, poised exactly at the halfway point between Pythagorean and pure. This tuning, mentioned by Jean Baptiste Romieu (1758), might be interesting to try for especially demanding Renaissance pieces.

5.6.3. An aside: the meantone equations

For readers interested in determining the variances of the 12th fifth and of major thirds from just in any meantone tuning, including the special case of equal temperament, two formulas may help. Both formulas use cents as the unit of measure, with "t" equal to the tempering or narrowing of each fifth in cents:

       Variance of 12th fifth from just 3:2 = (11t - 23.46)
               Variance of M3 from just 5:4 = (21.51 - 4t)

In other words, for the 12th fifth, we multiply the tempering amount in cents by 11, and then subtract the Pythagorean comma of about 23.46 cents. For the major thirds, we begin with the syntonic comma of about 21.51 cents (or 21.506 cents to three decimal places), and subtract four times the tempering amount. A positive result shows an interval wide of just, while a negative result is narrow of just.

Revising these formulas slightly, we can calculate the absolute sizes of these intervals rather than their variance from just:

       Size of 12th fifth = (678.49 + 11t)
       Size of M3 = (407.82 - 4t)

The simplest case is Pythagorean tuning, where t = 0. Our formulas tell us that we have a 12th fifth of -23.46 or 23.46 cents narrow, the Wolf of 678.49 cents, and major thirds of +21.51 or 21.51 cents wide, our true Pythagorean thirds of 407.82 cents.

Suppose we are interested in finding these values for 1/7-comma meantone, a tuning in the intriguing transitional territory between "well-tempered" and "characteristic" meantone. Fifths are tempered by 1/7 syntonic comma or about 3.07 cents, meaning that t = 3.07; thus 11t = 33.77, and 4t = 12.28.

Our formulas tell us that the 12th fifth is about (33.77 - 23.46) cents, or +10.31 cents from just (wide), having a size of (678.49 + 33.77) cents, or 712.26 cents. Major thirds are about (21.51 - 12.28) cents or +9.23 cents from just (wide), with a size of (407.82 - 12.28) cents or 395.54 cents. These results are subject to slight rounding errors (e.g. in approximating 1/7 syntonic comma as 3.07 cents); GNU Emacs Calc shows a 12th fifth of 10.33514... cents wide rather than 10.31 cents, for example.

However, our results are accurate enough for most purposes: the 12th fifth will be uncomfortable but marginally playable, and the major thirds somewhat on the restful side (4/7 of the way from Pythagorean to just). While I might incline more to the 1/8-comma tuning mentioned above for Renaissance pieces raising Wolf problems, since it treats the 12th fifth much more kindly, 16th-century keyboard enthusiasts might well argue that even 1/7-comma stretches the thirds to the very limits of stylistically-informed taste.

Such poignant dilemmas, although fortunately an exception to the rule that even Renaissance pieces with chromatic passages often work very nicely with the characteristic meantone tunings of the period, do lend a certain drama to the simultaneous equations of meantone. For this music, moving substantially to the left of the favored 1/4-comma region is a real compromise, and going as far as the region of 1/7-comma or 1/8-comma to avoid Wolves, let alone all the way to equal temperament, is a severe one.

If we start from the Pythagorean end of the spectrum, these same equations provide a near and reasonably hospitable refuge from Wolves: 1/11th-comma meantone, i.e. equal temperament. However, for typical Gothic pieces presenting no Wolf conflicts, Pythagorean just intonation may be best of all.

5.6.4. Equal temperament as a well-temperament

In addition to being an uncharacteristic meantone tuning, equal temperament is also a well-temperament in the broadest sense: all intervals and transpositions are playable. By the later 16th century, lutenists and composers were taking advantage of the availability on equally-tempered lutes of intervals which would be "false" in more conventional tunings; and by 1588, Roselli had advocated this free transposition of the gamut to any desired step as a major advantage of the temperament for voices and keyboards also (on this history, see Section 5.6.5).

As in the case of Renaissance music intended for typical meantone tunings, however, the uncharacteristc features of equal temperament make it a debatable choice for triadic music of the historical "well-temperament" era (late Baroque, Classic, and early Romantic periods). These same features seem rather benign or even advantageous for medieval music, assuming that Pythagorean tuning is unavailable or impractical.

Lovers of "well-temperament" in the canonical 17th-18th century sense have two obvious complaints about equal temperament: it has a single unvaried color for all keys and transpositions, and this happens to be the wrong color.

As we have seen (Section 5.5), historical well-temperaments feature a subtle gradation of interval colors ranging from pure or near-pure thirds like those of Renaissance meantone for the most common modes or keys to Pythagorean or near-Pythagorean thirds for the most remote ones. Equal temperament, with its "isochromatic" properties, not only levels out these fine shadings, but unobligingly fills the keyboard with near-Pythagorean thirds for the most common and remote sonorities alike.

The advocate of 18th-century well-temperaments, no less than the lover of classic 16th-century meantone, desires pure and restful thirds in the most prominent positions. These intervals not only present the triad in its most acoustically blending form, but provide a kind of contrast or resolution to the subtle "microtonal dissonance" of tense thirds and triads in remote transpositions.

For Gothic polyphony, in contrast, a keyboard filled with active thirds is exactly what we want, and a uniform color close to that of pure Pythagorean intonation is hardly a disadvantage. Since the standard Pythagorean tuning with the Wolf at Eb-G# offers the usual interval ratios except at remote transpositions rarely occurring in medieval practice, it may be more akin to the isochromatic qualities of equal temperament than to schemes based on graduated shading.

Indeed, although it is quite possible to devise schemes of shaded well-temperament for Gothic or neo-Gothic music (Section 5.5.3), equal temperament actually seems to me a more conservative compromise for the few 14th-century works raising Wolf problems. It has the advantage of leaving all fifths and fourths, and also major seconds and minor sevenths, in nearly just form; and provides a consistent interval color which may better fit medieval aesthetics than a system of graduated variation.

Music of the transitional period around 1400 presents a special case, since Lindley persuasively argues that much of it may have been based on the modified Pythagorean tuning with a Wolf at F#-B (Section 4.5), a tuning which does produce dramatic color contrasts. For pieces indeed premised on this scheme, just about any other tuning is likely to be a serious compromise, including classic Pythagorean and equal temperament.

A due caution is that some musicians of this epoch had more conservative tastes: Prosdocimus, for example, seemed mainly concerned that the usual Pythagorean intervals be available at the expected places. The traditional Eb-G# tuning would be ideal from this viewpoint, with equal temperament a tenable approximation (as it is for earlier works of the 13th and 14th centuries).

Here it seems fair to add that some theorists of the 18th century were quite prepared to accept equal temperament as one alternative, including Werckmeister, while others actively advocated it. Our purpose has been simply to suggest that the arguable disadvantages of equal temperament vis-a-vis canonical well-temperaments for the triadic music of this era may hardly be disadvantages in a Gothic context.

5.6.5. Early history of equal temperament

Given the affinity between Pythagorean just intonation and equal temperament, we should not be surprised that the first recorded approximations to equal temperament in medieval and early Renaissance theory are variations on the Pythagorean scale. Interestingly, the pragmatics of organ-pipe measurement in the 14th century may have prompted such developments.

Around 1373, an English source on organ design suggests taking the average length of two pipes a whole-tone (9:8) apart to add a pipe at the semitone between them. This simple rule produces a string ratio (or pipe ratio) of 18:17:16, dividing a Pythagorean whole-tone of about 203.9 cents into a lower semitone of 98.95 cents and an upper semitone of 104.95 cents.

Since a ratio such as 18:17:16 with equal differences between adjacent terms is known as an arithmetical division, we might refer to this tuning as an "arithmetical temperament." While still unequal, the semitones are much less disparate than in classic Pythagorean tuning with its concise diatonic semitone of 90 cents and large chromatic semitone of 114 cents. A chromatic keyboard octave in this arithmetic Pythagorean temperament might look like this:

     99        303              597       801     1005
     c#'       eb'              f#'       g#'      bb'
 _99_|_105_ _99_|105_       _99_|_105__99_|_105__99_|_105_
 c'        d'        e'    f'        g'         a'       b'    c''
1:1       9:8      81:64  4:3       3:2       27:16   243:128  2:1
 0        204       408   498       702        906     1110    1200
     204       204      90     204       204       204      90

Unfortunately, although this temperament leaves the diatonic Pythagorean scale unaltered and indeed converts the traditional Wolf G#-Eb into a perfect fifth or fourth, it places two "near-Wolves" at much more prominent locations: Bb-F and B-F#. These unhappy intervals, shown as fourths in our c'-c'' octave segment, occur at the weak links in the chain between diatonic notes and newly retuned accidentals. Thus f'-bb' is 507 cents, 9 cents wider than a just 498-cent fourth; and f#'-b' is no less than 513 cents, or 15 cents wider than just.

Curiously, the 18:17:16 division of the whole-tone is mentioned by early 14th-century writers including Jacobus of Liege, who offers it as a proof that this interval cannot be divided into two equal semitones. Jacobus is, of course, quite correct as long as we restrict ourselves to divisions based on integer ratios.

By the later 14th century, around the same time as the above organ building plan, Nicholas Oresme was developing a system of rational exponents which would permit one to divide a proportion such as 9:8 into two equal ratios. However, it was not until the early 16th century that such temperaments were proposed in theory and documented in practice.

In 1518, Henricus Grammateus (Heinrich Schreiber) published an "amusing reckoning" using Euclidean geometry to calculate the length of organ pipes for a Pythagorean temperament with diatonic whole-tones divided into two equal semitones; Jacques Lefevre d'Etaples had described this Euclidean method in a music treatise of 1496. We might describe the result as a "meansemitone" temperament. Each accidental has equal semitones of 102 cents above and below, while diatonic semitones retain their usual size of 90 cents:

     102         306             600       804      1008
     c#'         eb'             f#'       g#'       bb'
 _102_|_102_ _102_|102_      _102_|_102_102_|_102_102_|_102_
 c'         d'        e'    f'         g'        a'        b'    c''
1:1        9:8      81:64  4:3        3:2      27:16    243:128  2:1
 0         204       408   498        702       906       1110  1200
      204       204     90      204        204       204      90

As with the 14th-century arithmetical temperament, the intervals Bb-Fb and B-F# constitute the weak links in the chain of fifths or fourths. These intervals again appear on our octave segment as fourths, now measuring an equal 510 cents, or about 12 cents wide of just.

The next critical step, actually bringing equal temperament or a very close approximation into practical use, was apparently taken not so much by theorists as by pragmatic builders of fretted instruments such as the lute. Renaissance lutenists playing a burgeoning polyphonic repertory often encountered vertical progressions moving a diatonic semitone on one fret and a chromatic semitone on another. Equalizing all semitones would simplify life for builders and players alike.

Happily, the lower semitone of the earlier 18:17:16 division (18:17 or about 99 cents), repeated 12 times, yields something quite close to a perfect equally-tempered octave; the ratio for a 100-cent semitone is actually around 1800:1699. Lute designers and builders evidently found it easy enough in practice to adjust the frets slightly in order to make octaves pure.

By the middle of the 16th century, various theorists were reporting that fretted instruments were tempered in equal semitones, quite unlike the voices of singers (often assumed to follow tertian just intonation) or keyboards (tuned in characteristic meantone).

Anticipating Bach by 165 years, composer and virtuoso lutenist Giacomo Gorzanis had by 1567 written a collection of 24 dance suites, two on each of the 12 steps of the chromatic scale: one in a 16th-century modality with a major third above the final, and the other with the minor third (a distinction recognized in the harmonic theory of Zarlino). Some "major" suites have an Ionian/Mixolydian feeling, and some "minor" suites an Aeolian/Dorian feeling; each suite consists of a passamezzo-saltarello pair.

Paradoxically, while Bach's Well-Tempered Clavier may well have been intended for some unequal "well-temperament" (Sections 5.5, 5.6.4), this "well-tempered lute" collection of Gorzanis seems indeed a demonstration of equal temperament in action.

By 1588, Roselli had shared with Zarlino his treatise advocating this tuning as a solution for composers, singers, and instrumentalists alike. Not only could musicians circumnavigate this "spherical music" at will, "as if in a perpetual motion"; but ensembles with diverse instruments could maintain a satisfying accord, with the organs neither too high nor too low.

Roselli's proposal faced a technical problem: it is much easier to achieve an essentially equal temperament on a fretted instrument using known measurements than to tune such a subtle form of meantone on a keyboard, unless one can use a fretted instrument as a standard. In the early 17th century, such an accomplishment was considered a rare feat, although the great composer Girolamo Frescobaldi reportedly advocated equal temperament as an option for the organ, and Lindley finds signs of its acceptance in the harpsichord music of his pupil Froberger.

Technical complications aside, Roselli's arguments from utility were outweighed for many musicians by considerations of beauty: the near-Pythagorean thirds of equal temperament, an advantage for Gothic music, seemed quite unsuited to Renaissance music with its ideal of restful triadic concords.

In the special case of the lute, happily, the design of the instrument not only made the utility of equal temperament compelling but also served to mitigate its impact on beauty. While the exigencies of the fretboard seemed almost to cry out for equal semitones, the lute strings had a quality perceived somewhat to "soften" the color of wide major thirds, in contrast to the harpsichord's "brighter" color which would give these intervals an accentuated edge.

Even on the lute, there were some misgivings. Vincenzo Galilei, in an amusing as well as edifying portion of his dialogue Fronimo (1584), available in a translation by Carol McClintock, gives his own views on the tuning of this instrument while reporting the modifications introduced by some lutenists to obtain purer thirds, an effort analogous to current designs for just intonation guitars.

Unlike Zarlino's friend Roselli, Galilei takes it for granted that the lute with its equal semitones differs in its tuning from keyboard instruments, and that both differ from the natural intonation of singers. His purpose is to defend equal temperament as necessary and appropriate for the lute while directing some satirical barbs at those who feel a need to alter this arrangement with special frets to take away some of the "sharpness" from major thirds and tenths.

Having shown that a fretting in unequal semitones would embarrass a lutenist playing on several strings at once, Galilei turns to the matter of the tastini ("little frets") added by some people to permit smaller major thirds. While acknowledging that these intervals are "truly stretched a little," he pokes fun of claims that they hurt the ears, and argues against the tastini on two grounds.

First, he objects generally to people who "reform" the instrument with added strings or frets when it is evident that a true lutenist has no need for such gadgets (an argument providing Galilei an opportunity for a bit of self-promotion regarding his own published and forthcoming works).

Secondly, he argues that the tastini not only introduce intervals unknown to ancient Greek theorists such as Aristoxenos (who proposed a division of the octave into six equal tones and 12 semitones), but fill the fretboard with imperfect fifths and fourths (i.e. Wolves). By objecting to the slightly stretched thirds of the usual fretting but making it difficult for the player to avoid such "intolerable" fifths and fourths, the advocates of the tastini demonstrate in Galilei's ironic view that "the exquisiteness of their hearing is not so delicate that every minimal prick offends it."

Although often an iconoclastic theorist, Galilei defends equal temperament on the lute as a time-honored arrangement: if it were indeed possible to obtain purer thirds without compromising many other intervals more seriously, it would have been done long ago. In any case, Galilei along with Zarlino and other more conventional theorists bears witness that by 1584 this temperament was taken for granted in theory and practice, as was its variance from vocal or keyboard tunings. Nevertheless, some dissenting lutenists sought to modify the instrument in the direction of tertian just intonation, while a few advocates of equal temperament such as Roselli favored its extension to music for voices and keyboards also.

This Renaissance history illustrates both the advantages of equal temperament and its kinship with Pythagorean as opposed to tertian just intonation, a quality then as now evoking mixed reactions in the context of triadic music. Today, four centuries later, the debate continues.

In a Gothic context, however, the near-Pythagorean qualities of equal temperament seem far more propitious from the standpoint of beauty as well as utility.

5.6.6. Equal temperament and Gothic music

If we were to seek a "reformed" Pythagorean tuning solving the Wolf problem while treating all 12 fifths as gently as possible, then equal temperament would seem an ideal solution. Fifths and fourths remain almost pure, thirds and sixths quite vibrant and active, and diatonic semitones reasonably small.

At the outset, it would seem wise to emphasize that subtle but significant compromises of vertical and melodic intervals do occur, so that equal temperament should be seen as a "variant" on Pythagorean tuning rather than an "improvement" or replacement. Two considerations of utility might motivate such a compromise solution.

First, many modern fixed-pitch instruments such as keyboards and sets of bells in equal temperament may be difficult or impossible to retune to Pythagorean just intonation. With the modern guitar as with the Renaissance lute, it would be difficult to adapt the fretting scheme to an intonation (Pythagorean or otherwise) with unequal semitones. With such instruments, our tuning is a matter more of necessity than of choice.

Secondly, and much more rarely, a 14th-century composition with very adventurous accidentalism may require Ab-Eb as a concordant sonority, running into the problem of the Wolf at G#-Eb. In some cases it might be possible to choose an alternative Pythagorean tuning placing the Wolf at another fifth left unused, or to try a "Pythagorean well-temperament" (Sections 5.5.1-5.5.3). However, if all 12 fifths are indeed required, equal temperament has the appeal of solving the problem in a conceptually simple and musically rather unobtrusive way on a standard keyboard.

In both cases, an ideal intonation is compromised owing to the limitations of a fixed-pitch instrument, as is the usual case in the long history of temperaments. Happily, the use of equal temperament as an approximation of Pythagorean tuning is one of the more pleasant compromises a musician may face in this arena of choice and necessity.

Our task of evaluating equal temperament as a tuning for Gothic polyphony is simplified by the fact that its intervals have consistent sizes regardless of transposition. Let us consider our criteria of the just, the true, and the incisive (Section 5.5.2).

In equal temperament, or 1/11-comma meantone (Section 5.6.1), all fifths and fourths are an even 700 cents and 500 cents respectively, varying only 2 cents (actually about 1.96 cents) from their just Pythagorean ratios of 3:2 (701.96 cents) and 4:3 (498.04 cents). The choice concords of Gothic harmony remain nearly pure.

Major seconds and minor sevenths, at an even 200 cents and 1000 cents, also stay very close to their just ratios of 9:8 (203.91 cents) and 16:9 (996.09 cents), a variance of 4 cents (about 3.91 cents). Since major seconds and minor sevenths derive from two fifths or fourths, they vary twice as much from just as their parent intervals, but still retain a quality quite close to pure.

These nearly just intervals preserve many of the advantages of Pythagorean tuning both for stable Gothic sonorities involving fifths and fourths (Sections 3.1.1, 3.2.1), and for unstable sonorities involving major seconds and minor sevenths (Sections 3.1.3, 3.2.3). Incidentally, these aspects of equal temperament also fit nicely with 20th-century styles of harmony involving superimposed fifths and fourths.

Moving from justness to trueness, we find that thirds and sixths retain much of their active Pythagorean quality despite some compromise. Major thirds at an even 400 cents are about 8 cents smaller than their true Pythagorean ratio of 81:64 (407.82 cents), and minor thirds, at 300 cents, about 6 cents larger than their true ratio of 32:27 (294.13 cents). Major sixths at 900 cents are likewise about 6 cents smaller than a true 27:16 (905.87 cents), and minor sixths at 800 cents about 8 cents larger than a true 128:81 (792.18 cents).

To place these variances in perspective, we should measure them in terms of the syntonic comma (Section 4.4.2) of about 21.51 cents by which the true Pythagorean forms of these intervals differ from their simplest and most blending ratios (M3 5:4, m3 6:5, M6 5:3, m6 8:5). We find that equally-tempered major thirds and minor sixths remain 7/11 of a syntonic comma from pure, and minor thirds and major sixths approach Pythagorean tuning even more closely at 8/11 comma from pure. (See also Sections 5.6.1-5.6.3 on equal temperament and the meantone spectrum.)

In musical terms, this means that these thirds and sixths, although not having quite the dynamic edge and tension of their true Pythagorean counterparts, nevertheless retain an active flavor comporting well with harmonic color and cadential action in a Gothic context (Sections 3.1.2, 3.2.2).

Looking at incisiveness first from a melodic point of view, we find that the equal semitone of 100 cents is somewhat larger than the ideal Pythagorean diatonic semitone at a keen 90 cents, but still quite concise by comparison with systems based on pure or near-pure thirds featuring diatonic semitones of 112 cents or larger. Indeed, some Renaissance theorists described the semitones of equal temperament as "minor" or "small," as opposed to the large diatonic semitones of the favorite keyboard tunings.

From a harmonic viewpoint, Gothic progressions from an unstable to a stable interval by conjunct contrary motion featuring whole-tone motion in one voice and semitonal motion in another now require an even 300 cents of motion rather the ideally efficient 294 cents of pure Pythagorean tuning (e.g. m3-1, M3-5, M6-8, m7-5, M2-4), see Section 3.3. While our diatonic semitone has grown 10 cents larger, the tempering process has slightly compressed our whole-tone from 204 cents to an even 200 cents, making up 4 cents of this difference so that these resolutions require only an additional 6 cents of motion.

All in all, we can say that equal temperament is reasonably although not quite ideally incisive as viewed in terms of either the horizontal or vertical dimension.

Equal temperament remains a reasonable facsimile of Pythagorean tuning, as opposed to a replacement for the original. Just intonation advocates emphasize the importance of striving for absolutely pure and beatless fifths and fourths, and some feel it a substantial compromise even to use synthesizers supporting a Pythagorean scale but able to tune it only to a resolution of the nearest cent or so. Pythagorean thirds and sixths deserve to be heard in their true and distinctive character and color, lending their full impetus to Gothic textures. The expressive contrast between the expansive Pythagorean whole-tone and the incisive diatonic semitone lends a special beauty to medieval chant and secular monophonic song as well as polyphony.

More generally, equal temperament might best be viewed as one historical tuning among many, a tuning now approaching 500 years of practical use. Rather than a solution making all other tunings obsolete, it is better seen as a musical supplement - a description inspired by Zarlino's Sopplimenti Musicali of 1588, the treatise reporting Roselli's praise of this "spherical" tuning. Pythagorean intonation, with a history in Western European music a bit more than twice as long, retains its unique and irreplaceable character.

These cautions duly noted, it seems somehow fitting that the newest standard keyboard tuning should have such a close affinity to the oldest.

To Bibliography.

To Table of Contents.

Margo Schulter