Originally I was tempted to label this section "Mathematical aspects of Pythagorean tuning," but decided that such a title might discourage some readers mainly interested in practical details of implementing this tuning on various instruments ranging from medieval harps to organs and electronic synthesizers.

Please let me emphasize that it is not necessary to understand all of the fine points that follow in order to obtain the tuning in practice - and to explore it through actual music, which is the best way. A harp player may need to know only how to tune a series of perfect fifths, while a synthesizer player may pleasantly discover that Pythagorean tuning is available as a preprogrammed option, requiring only a convenient menu selection.

Other synthesizers might require the player to specify a custom tuning, and some specifications are included in Section 4.3 below. More generally, the focus here is on understanding the tuning and its qualities rather than on any specific kind of instrument.

A good way to explore Pythagorean tuning is to generate a simple scale. Here we will tune a very popular scale of the Gothic period, the Lydian mode (Mode V of Gregorian chant) consisting of the "white keys" in an octave from F to the F above. Let us choose the octave f-f', that is from F below middle C to the F above it.

Our first note is the lower f, which stands in unison with itself (1:1). Now we tune our first pure fifth (3:2) to generate the second note, c' a fifth above. With an acoustical instrument, this involves adjusting the upper note until no beats can be heard when the two notes are sounded together - a task easiest on a sustained instrument such as an organ.

f c' 1:1 3:2

Not inappropriately, our first interval is the fifth, the optimally blending interval in Gothic polyphony (see Section 3). As we continue along our chain of fifths, we will generate not only the individual notes of our scale but the various intervals making up the spectrum of concord and discord.

If we were to continue with the perfect fifth *above*
c', we would arrive at g', a major ninth above our original f.
This note would have a ratio of 3:2 squared, which we may write
(3:2)^2, or 9:4. Note that to find the size of an interval created
by adding two others - here two fifths - we *multiply* the
ratios of these intervals: 3:2 x 3:2 yields 9:4 for the resulting
major ninth.

To keep within the range of our first octave, we instead place
our third note an octave lower, at the g located a fourth (4:3)
*below* c' and a major second (9:8) above our initial f:

f g c' 1:1 9:8 3:2

Thus our second fifth (here a fourth down) gives us the new interval of the major second, regarded in the 13th century as relatively tense but somewhat compatible.

Our third fifth, taken up from g, generates d', a major sixth
above the original f. To find the ratio of this new interval,
regarded like M2 as relatively tense, we multiply 9:8 by 3:2 to
get 27:16. Note that this interval consists precisely of a fifth
plus a major second, and so is known in medieval terminology as a
*tonus cum diapente* or "whole-tone-plus-fifth":

f g c' d' 1:1 9:8 3:2 27:16

Were we to move up a fifth from d', we would arrive at a' a major 10th from f, 27:16 x 3:2 or 81:32; instead, we move down a fourth from d', arriving a major third above our original f (81:64).

f g a c' d' 1:1 9:8 81:64 3:2 27:16

As it happens, this new interval of M3 is regarded as relatively
blending and somewhat more concordant than our preceding ratios of
M2 (9:8) and M6 (27:16), although it is mathematically more complex.
This Pythagorean M3 is known as a *ditone*, since it is
equal to precisely two whole tones of 9:8; 9:8 x 9:8 = 81:64.

So far, our unstable M2, M6, and M3 as well as our stable fifths or fourths have all had some degree of "compatibility," but our next fifth takes us into the territory of strong discord, as we move from our last tone a to the e' a fifth above it, a major seventh above our original f or 243:128 (i.e. 81:64 x 3:2):

f g a c' d' e' 1:1 9:8 81:64 3:2 27:16 243:128

This interval is often known as a *ditonus cum diapente* or
major third plus fifth, since it is equal to the sum of these
intervals (or the product of their ratios).

Moving up a fifth from e' would again take us out of our octave range, so we instead move a fourth down to b, arriving at another strong discord with f, the tritone 729:512, equal as its name suggests to three whole tones or (9:8)^3:

f g a b c' d' e' 1:1 9:8 81:64 729:512 3:2 27:16 243:128

To complete our mode, we finally need the note an octave above our original f. Since no number of superimposed perfect fifths will yield precisely an even octave, we instead define this special interval independently as the pure ratio 2:1. Indeed, medieval theorists such as Johannes de Grocheio and Jacobus of Liege describe the purely blending octave as the font and source of all other intervals, including the more richly stable fifths and fourths:

f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1

We now have all eight tones of our basic mode, and our tuning process is finished for the moment. Before going on to add Bb (an integral part of the medieval gamut) and the other accidentals, we may wish optionally to consider the additional Pythagorean intervals our tuning in fifths has generated.

In addition to this octave and the seven intervals we have derived through our chain of fifths up (or fourths down), our scale includes some other important intervals arising from these.

One way to approach these intervals is to treat them as the
*difference* of two intervals already defined as part of
our chain of fifths. To find such a difference of intervals, we
*divide* their ratios. For example, we may define the fourth
c'-f' as the difference of the octave f-f' (2:1) and the fifth f-c'
(3:2). Therefore dividing 2:1 by 3:2, or equivalently multiplying
2:1 by 2:3, we get 4:3 as the ratio of the fourth. We could also
take the fourth, e.g. g-c', as the difference between the fifth
f-c' (3:2) and a major second f-g (9:8); dividing 3:2 by 9:8, i.e.
3:2 x 8:9, gives us 24:18 or 4:3.

Like the fifth, the fourth is a richly stable interval, although not so smooth and conclusive in itself.

Let us next consider a vital melodic interval in our scale: the diatonic semitone or minor second occuring at b-c' and e-f'. By taking the difference of the octave f-f' (2:1) and the major seventh f-e' (243:128), i.e. dividing 2:1 by 243:128 or multiplying 2:1 x 128:243, we find that this Pythagorean semitone has a ratio of 256:243. As a vertical interval, it is a strong discord which plays a striking role in various two-voice and multi-voice resolutions; as a melodic interval, its rather compact size gives it an expressive quality (see Sections 3.3 above, and 4.2 following). Note that an octave is equal to five whole tones of 9:8, plus two semitones of 256:243.

We could also define a Pythagorean semitone such as b-c' as the difference between the fifth f-c' (3:2) and the tritone f-b (729:512); dividing 3:2 by 729:512, i.e. 3:2 x 512:729, we would find that the result simplifies to 256:243.

Much milder as a vertical interval is the relatively blending
minor third, e.g. d'-f', which we may define as the difference
between the octave f-f' (2:1) and the major sixth f-d' (27:16).
Thus dividing 2:1 by 27:16, i.e. 2:1 x 16:27, we find that a minor
third is 32:27. This interval, like the major third, represents
the mildest level of vertical instability. Note that we could also
define a minor third as the difference between a fifth (3:2) and
a major third (81:64), i.e. 3:2 divided by 81:64 or 3:2 x 64:81,
also yielding 32:27. This interval is known as the *semiditonus*,
being smaller than the ditone or major third.

Our Lydian scale also includes a minor sixth a-f', which we
could define as the difference between the octave f-f' (2:1) and
the major third f-a (81:64); dividing 2:1 by 81:64, i.e. 2:1 x
64:81, we find the ratio of this Pythagorean m6 as 128:81. It is
often ranked as an intense discord along with m2, M7, and the
tritone, although Johannes de Garlandia more moderately places it
on par with M2. A common name for this interval is the *semitonium
cum diapente*, or semitone-plus-fifth, and in fact it is equal
to the sum of a fifth (3:2) and our diatonic semitone (256:243);
multiplying these ratios yields 128:81.

Somewhat milder is the minor seventh g-f' in our scale, conveniently
defined as difference between the octave f-f' (2:1) and the major
second f-g (9:8), i.e. 2:1 divided by 9:8 or 2:1 x 8:9, yielding
a ratio of 16:9. This interval is known as a *semiditonus cum
diapente* or "minor-third-plus-fifth," and indeed is
equal to 32:27 x 3:2, or 16:9. It is also sometimes defined as
*bis diatessaron*, two fourths, or 4:3 x 4:3, again 16:9.

We have now derived all 13 standard intervals of music as listed by Anonymous I around 1290, ranging from unison to octave. Here is a summary of these intervals and their derivation:

--------------------------------------------------------- A. Notes and intervals derived directly from the tuning --------------------------------------------------------- Tone Interval to f ratio --------------------------------------------------------- f 1 1:1 c' 5 3:2 g M2 9:8 d' M6 27:16 a M3 81:64 e' M7 243:128 b A4 729:512 f' 8 2:1 _________________________________________________________ --------------------------------------------------------- B. Other intervals derived from these scale elements --------------------------------------------------------- Tones Interval Derivations ratio --------------------------------------------------------- e'-f', b-c' m2 8 - M7; 5 - A4 256:243 d'-f', etc. m3 8 - M6; 5 - M3 32:27 g-c', etc. 4 8 - 5; 5 - M2 4:3 a-f' m6 8 - M3; 5 + m2 128:81 g-f' m7 8 - M2; 5 + m3 16:9 _________________________________________________________ ---------------------------------------------------------

In a more detailed exposition of the intervals in his *Speculum
musicae*, Jacobus additionally notes the *semitritonus*
or diminished fifth which occurs at b-f', or in other words an
octave less a tritone (i.e. augmented fourth, e.g. f-b). While
medieval writers often lump both augmented fourth and diminished
fifth under the term tritone, in fact the latter interval is not
identical to the former. By taking the difference of the octave
f-f' (2:1) and the augmented fourth f-b (729:512), i.e. 2:1 divided
by 729:512 or 2:1 x 512:729), we find that its ratio is 1024:729.
As we shall see in the next section, the Pythagorean diminished
fifth is slightly *smaller* than the augmented fourth.
While both intervals are counted as strong discords, Jacobus finds
the diminished fifth somewhat milder.

To this point, we have been describing the notes and intervals of the Pythagorean scale in a medieval manner, using tuning ratios - then conceived in terms of string ratios, and now typically conceived in terms of frequency ratios. However, especially when comparing different scales or tunings, it is useful to measure intervals having ratios such as 128:81 or 256:243 in some tidier fashion.

A standard modern measure for such ratios is the *cent*,
a unit equal to 1/100 of an equally tempered semitone, or 1/1200
of an octave. Thus if we divide an octave into twelve equal
semitones - as is done in 12-tone equal temperament, but not in
Pythagorean tuning (although these tunings are in some ways rather
kindred) - each semitone will equal 100 cents. More generally, a
pure octave (2:1) is precisely 1200 cents.

As discussed at more length for interested readers in
Section 4.2.1, cents are a *logarithmic*
measure of interval ratios based on the powers of two. The important
practical consequence is that to find the sum of two intervals, we
may either multiply their ratios or *add* their measures in
cents; to find their difference, we may either divide their ratios
or *subtract* their measures in cents.

For example, the ratios for the fifth and fourth are 3:2 and 4:3 respectively. To find the sum of these intervals, we can multiply these ratios, getting 12:6 or 2:1 - a perfect octave.

If we know that a 3:2 fifth is approximately 702 cents, and a 4:3 fourth approximately 498 cents, then we can simply add these two measures to arrive at a sum of 1200 cents, precisely a 2:1 octave.

Similarly, to find the difference between fifth and fourth - a major second - we can divide 3:2 by 4:3, or multiply 3:2 by 3:4, getting a ratio of 9:8. Alternatively, we can simply calculate (702 - 498) cents, or 204 cents, as the size of the pure 9:8 major second.

In addition to simplying calculations, especially for complex ratios, the system of cents can tell us interesting thing about a given tuning and its comparison to other tunings. For example, the Pythagorean fifth at around 702 cents is slightly larger than seven equally-tempered semitones of the kind found on many guitars (700 cents) while the fourth at around 498 cents is slightly smaller than five such semitones (500 cents).

These differences represent a slight compromise of these intervals in equal temperament; its 700-cent fifths are just a tidge smaller than a true 3:2, and its 500-cent fourths a tidge larger than a true 4:3.

Of special interest in mapping out a scale are the whole tones and semitones. The Pythagorean major second (9:8), at around 204 cents, is larger than two equally-tempered semitones (200 cents), giving it a generously wide quality as a melodic interval (Section 3.3) as well as a more mild or "concordant" quality as a vertical interval (Sections 3.1.3, 3.2.3).

The diatonic semitone (256:243), e.g. b-c' or e'-f', at 90 cents, is considerably narrower than an equally-tempered semitone (100 cents), and thus has a quality which Lindley aptly describes as "incisive" (Section 3.3).

With a generous whole-tone of 204 cents, and a rather narrow diatonic semitone of 90 cents, the Pythagorean scale offers expressive contrasts for Gothic melody and harmony alike. Familiarity with these interval sizes in cents is helpful not only in appreciating some of the artistic possibilities of the tuning, but in navigating our way around as we complete our process of generating a full 12-note chromatic scale.

Incidentally, we can now demonstrate (as promised at the end of
the previous section) that a Pythagorean diminished fifth at 1024:729
(e.g. b-f') is actually a bit *smaller* than an augmented
fourth at 729:512 (e.g. f-b). The latter interval, a tritone in
the strict sense, contains three 9:8 whole-tones (f-g, g-a, a-b)
of about 204 cents each, and thus comes to about 612 cents. The
diminished fifth, consists of a 256:243 diatonic semitone (b-c')
of about 90 cents plus a fourth (c'-f') of about 498 cents - or
about 588 cents in all. More precisely, these intervals come to
about 611.73 and 588.27 cents respectively, adding up to a perfect
2:1 octave of 1200 cents.

While 14th-century authors such as Nicholas Oresme took an interest in fractional exponents and their possible applications in areas such as mechanics, logarithms and their musical application had to wait some two centuries. The purpose of this section is briefly to explain how to determine, for example, that a 9:8 major second is indeed equal to 204 cents or thereabouts.

As noted above, the system of cents is based on powers and logarithms of two, and on the ratio of the 2:1 octave. To find the measure in cents of any interval ratio, we first express it as a power of 2 - that is, find its base-2 logarithm. Multiplying this result by 1200, we arrive at the size of the interval in cents.

The formula for the measure in cents of an interval a:b may be thus be stated:

cents = (log2 a:b) x 1200

Let us first take the trivial but illustrative case of the octave itself, 2:1. The base-2 logarithm of 2:1 is equal to 1; that is, 2 is equal to 2^1. Multiplying 1 x 1200, we confirm the definition of an octave as 1200 cents.

A calculator is helpful in our next case, that of the pure or just fifth at 3:2, as in Pythagorean tuning. Let us assume for the moment that we are fortunate enough to have a calculator at hand that directly supports base-2 logarithms. Using GNU Emacs Calc, I find that log2 3:2 = .5849625... (i.e. 3:2 = 2^.5849625...). Multiplying this result by 1200, I get 701.955... cents, or about 702 cents.

Similarly, for a 9:8 major second, I get a log2 9:8 of .169925..., and multiplying by 1200, a measure of 203.91... cents, or a rounded 204 cents.

For a 256:243 diatonic semitone, I get a log2 256:243 of .075187..., and a measure of 90.22... cents.

While some calculators can directly find base-2 logarithms, this is not a standard feature. More typically, calculators support base-10 logarithms or natural logarithms (base-e, where e = 2.71828...). Fortunately, we can convert from these bases to base-2 using a simple formula. In formal terms:

logn (a:b) log2 (a:b) = ---------- logn 2

More informally, we can find the base-10 logarithm for a ratio such as 3:2 (.176091...) and then divide this amount by log10 2 (about .30103) to find the desired log2 3:2 (.5849625...), then multiplying by 1200 as usual to find the size of a pure 3:2 fifth in cents (701.955...).

With natural logarithms (shown by the symbol ln), we find that ln 3:2 is .405465..., and divide this amount by ln 2 (.693147...), again getting .5849625, which multiplied by 1200 gives us 701.955... cents.

A more direct shortcut is simply to find the base-10 log of a ratio and multiply by 3986.31371386... to convert to cents, or to find the natural log and multiply by 1731.23404907...

For example, for a major second of 9:8, we find a base-10 log of .051152... and multiply by 3986.31371386... to get 203.91... cents; or we find a natural log of .117783..., and multiply by 1731.23404907... to get a similar result of 203.91... cents.

It may be worth noting that the system of cents also works nicely for intervals larger than an octave. Thus a Pythagorean major ninth is equal to precisely two 3:2 fifths, or 9:4. For this 9:4 ratio we get a log2 of 1.69925..., and a measure of 1403.91 cents, or about 1404 cents.

Knowing that a fifth is about 702 cents, we could also have calculated the size of the major ninth simply by adding 702 + 702 cents to arrive at 1404 cents. Alternatively, to figure the size of a major ninth built by joining two identical fifths of 702 cents, we could multiply 702 x 2 = 1404.

This latter approach of multiplication can apply to any case where we join two or more identical intervals to build a new interval. Thus, given that a major second (9:8) is about 204 cents, a Pythagorean major third or ditone, (9:8)^2, is equal to about 204 x 2 or 408 cents, and a tritone, (9:8)^3, to about 204 x 3 or 612 cents. These intervals are somewhat wider than their equally-tempered versions of four and six 100-cent semitones respectively.

This multiplication approach reflects a basic property of logarithms:

log2 (a:b)^n = n log2 (a:b)

One facet of the cents system is that it tends to take equal temperament as a frame of reference - or, at least, to represent an equally-tempered semitone as a neat 100 cents, rather than the less intuitive exponential ratio of

2^100/1200 or 2^1/12

From a historical perspective, equal temperament is a fairly neutral point of reference, standing somewhere between medieval Pythagorean intonation at one end of the spectrum and Renaissance meantone tuning, for example, at another. We consider this spectrum at more length in Section 5.

In Section 4.1, we generated a diatonic scale of f-f', known as the Lydian mode. In adding the remaining notes of a full chromatic scale, we may find it helpful to keep track not only of interval ratios, but also of measurements in cents. Here is our scale so far, with some of these measurements added (to the nearest whole cent):

f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90

To this point, our journey along the chain of fifths has been
in an "upward" direction, that is, in steps of a fifth
up or a fourth down: f-c'-g-d'-a-e'-b. Let us momentarily reverse
our direction, returning to our original note f and moving a fifth
*down* to add Bb - or, to keep within the range of our
octave, bb a fourth *up*:

4:3 498 bb90|114f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90

In relation to f, bb is a perfect fourth of 4:3, or 498 cents. At the same time, it divides the whole-tone a-b into two unequal parts.

One of these parts is our diatonic semitone of 256:243, or 90 cents, at a-bb - the same size as at b-c' or e-f'. This interval is equal to the difference between the fourth f-bb and the major third f-a, i.e. (498 - 408) or 90 cents. Using interval ratios, we could also divide 4:3 by 81:64, i.e. 4:3 x 64:81, getting 256:243.

There remains a new kind of interval at bb-b, which we may call
a "chromatic semitone," and is more formally known as a
Pythagorean *apotome*. It is equal to a whole tone (9:8 or
204 cents) minus our diatonic semitone (256:243 or 90 cents) - or
about 114 cents.

To calculate the ratio of this apotome, we note that previously our chain of six fifths extended f-c'-g-d'-a-e'-b, with the extreme notes f-b forming a tritone of (3:2)^6 or 729:512. By extending the chain to bb-f-c'-g-d'-a-e'-b, we generate a new interval bb-b of (3:2)^7, or 729:512 x 3:2, getting 2187:2048.

Using cents, we could also determine the size of the apotome by taking the 612 cents of the tritone, and adding to it the 702 cents of the new fifth, getting 1314 cents - and then subtracting an octave to arrive at 114 cents. This is an approximate rather than exact value, since a fifth is equal not precisely to 702 but to 701.955... cents, but it is generally close enough. Alternatively, rather than adding a fifth to the 612-cent tritone, we could subtract a 498-cent fourth, arriving directly at 114 cents.

As already noted, Bb is an integral part of the medieval gamut, and likely was included on organ keyboards of the 10th or 11th century before other accidentals.

This uniquely privileged accidental illustrates a general rule for distinguishing a diatonic semitone from an apotome. The rule is that the diatonic semitone falls between a flat and the note immediately below (e.g. Bb-A), or a sharp and the note immediately above (e.g. F#-G). In other words, when moving semitonally, we normally descend from a flat and ascend from a sharp, an efficient 90-cent motion.

Melodic progressions involving the 114-cent apotome, e.g. Bb-B or F-F#, are rare, although in the early 14th century Marchettus of Padua recognizes chromatic progressions such as F-F#-G in theory and such idioms do now and then occur in 14th-century practice.

Interestingly, Jacobus underscores this contrast between diatonic semitone and apotome when he remarks (c. 1325) that keyboards now typically have all the whole-tones of the octave divided into their "unequal semitones."

To complete such a chromatic octave ourselves, we now produce a fifth down from bb to the remaining flat, eb - or, to keep within our octave, up a fourth to eb':

4:3 16:9 498 996 bb eb' _90_|_114_ _90_|_114_ f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90

In the flat portion of our journey, we have moved up two pure 4:3 fourths, f-bb and now bb-eb', so that the resulting interval f-eb' is equal to precisely two fourths, i.e. 4:3 x 4:3 or (4:3)^2, in other words a just minor seventh of 16:9. Taking a fourth as a rounded 498 cents, this interval is about 996 cents. It is also equal to the major sixth f-d' (906 cents) plus the diatonic semitone d-eb' (90 cents).

We also have the expected 2187:2048 or 114-cent apotome eb'-e',
and a new and yet more complex interval between the extreme notes
of our chain of eight fifths, eb'-bb-f-c'-g-d'-a-e'-b, that is
b-eb'. Here a slight complication is that the range of our f-f'
scale has placed our Eb *above* our Bb, so that b-eb' is
actually the octave complement of the usually expected interval
such as eb-b. Let us first find the expected interval, and then
subtract it from an octave to find our actual interval in this
scale.

Taking (3:2)^8, or 2187:2048, we get a ratio of 6561:4096 for
eb-b, which might be called a Pythagorean augmented fifth. Since
two fifths (minus an octave) are equal to precisely a 9:8 whole-tone,
we can also calculate this interval as (9:8)^4, and in medieval
theory it is known as a *tetratone*, or four whole tones.
It is thus about (204 x 4) or 816 cents. Note that the tetratone
is equal to the fifth e-b (702 cents) plus the apotome eb-e (114
cents), or again 816 cents.

To find the ratio of our actual interval b-eb, a Pythagorean diminished fourth, we divide the tetratone ratio 6561:4096 by the octave ratio 2:1, or 6561:4096 x 1:2, 6561:8192 - or, conventionally placing the larger term first, 8192:6561. This interval is equal to an octave minus a tetratone, roundly (1200 - 816) or 384 cents.

We may also define this interval b-eb' as the perfect fourth b-e (498 cents) less the apotome eb'-e' (114 cents), or 384 cents.

From the viewpoint of a theorist such as Jacobus of Liege following 13th-century tradition, the tetratone and presumably the diminished fourth (Jacobus includes only the former in his surveys of the intervals) are mainly curiosities, being regarded as highly discordant; but around 1400 they take on an interesting practical application (Section 4.5).

Having added bb and eb, we now return again to b, at the other end of our chain of fifths, to complete our journey on the sharp side. Moving up a fifth, we would reach f#' - to stay within our octave range, we instead move down a fourth to f#:

2187:2048 4:3 16:9 114 498 996 f# bb eb' _114_|_90_ _90_|_114_ _90_|_114_ f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90

Our new note f# is, as expected, an apotome above f and a diatonic semitone below g, respectively 114 and 90 cents. As the new diagram shows, accidentals split a whole-tone so that the favored 90-cent semitone facilitates descent from flats (bb-a, eb'-d') and ascent from sharps (f#-g).

Our ninth fifth also gives rise to a novel interval, f#-eb'; again the range of our f-f' scale results in the octave complement of the form we would get placing the note at the "lower" or flat end of our chain below, eb-f#. The latter form has a ratio equal to (3:2)^9, or the tetratone of (3:2)^8 plus a fifth, 6561:4096 x 3:2 or 19683:8192. Since this result exceeds an octave, we divide by 2: 19683:16384. This converts to roughly 318 cents.

We might describe eb-f# as a Pythagorean augmented second: it consists of the apotome eb-e, the diatonic semitone e-f, and the apotome f-f#, and thus a rounded (114 + 90 + 114) or 318 cents.

To find the ratio of our actual interval f#-eb', the octave complement of eb-f#, we can divide 19683:16384 by 2:1, getting 19683:32768, or in conventional order 32768:19683. This Pythagorean diminished seventh, as we might call it, has a size of about (1200 - 318) or 882 cents. One way of defining it is as the sum of the fifth g-d' and the diatonic semitones f#-g below this fifth and d'-eb' above it, thus about (90 + 702 + 90) or 882 cents. Another approach is to define it as a minor sixth (128:81, 792 cents) plus a diatonic semitone, e.g. f#-d' plus d-eb', again giving (792 + 90) or 882 cents.

Like the tetratonus and diminished fourth, these intervals would seem to belong to a kind of theoretical bestiary rather than to 13th-century practice; but again, they may have more practical interest in a 15th-century context.

Our next step up the chain of fifths, our tenth, is c#':

2187:2048 4:3 6561:4096 16:9 114 498 816 996 f# bb c#' eb' _114_|_90_ _90_|_114_ _114_|_90_ _90_|_114_ f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90

Between our original f and c#' we have a tetratone, a species whose octave complement we have already encounted at b-eb'. We now add to our "musical bestiary" a new interval between the extreme notes in the chain, c#'-eb', again an octave complement of the form with the note at the flat end of the chain below, eb-c#'.

The latter form eb-c#', our Pythagorean augmented second eb-f#
plus a fifth, has an imposing ratio of (3:2)^10 or 19683:16384 x
3:2, yielding 59049:32768. This comes to about 1020 cents.
Recalling that a 9:8 major second is equal to two fifths minus an
octave, we can also define this interval as precisely equal to five
whole tones, (9:8)^5, or about (204 x 5) or 1020 cents. Thus
medieval theory describes it as a *pentatone*, and it might
also be called a Pythagorean augmented sixth. We can also define
its size as the minor sixth e-c' (792 cents) plus a 114-cent apotome
at either end - eb-e, c'-c#' - giving a total of about (114 + 792
+ 114) or 1020 cents. Alternatively, we can take the octave d-d'
less the diatonic semitones d-eb' and c#'-d', leaving eb'-c#' at
about (1200 - 180) or 1020 cents.

To find the ratio of our actual interval c#'-eb', the octave complement of the pentatone, we divide the pentatone's ratio by 2:1, getting 65536:59049. This yields a kind of small or "minor" whole-tone at (1200 - 1020) or 180 cents, or two diatonic semitones.

The pentatone and minor tone are again "strange beasts" of mainly theoretical interest in a 13th-century setting, and Jacobus describes them (along with the tetratone) as highly discordant intervals not in use. However, close analogues of these intervals occur in other kinds of just intonation systems favored in the Renaissance and later.

Finally, we are ready to add the last of our 12 tones, g#:

2187:2048 19683:16384 4:3 6561:4096 16:9 114 318 498 816 996 f# g# bb c#' eb' _114_|_90_ _114_|90__90_|_114_ _114_|_90_ _90_|_114_ f g a b c' d' e' f' 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90

Our new tone and the original f form the augmented second f-g#. Our eleventh and final step on the chain of fifths gives rise to a new species g#-eb', or placing the note at the flat end of the chain below, eb-g#. This interval has a special significance.

The eb-g# form has the not inconsiderable ratio of (3:2)^11, or of our pentatone plus a fifth, 59049:32768 x 3:2, or 177147:65536; dividing by 2:1 to bring this interval within an octave, we get 177147:131072. This comes to about 522 cents, or more precisely 521.505... cents, about 23.46 cents wider than a 4:3 perfect fourth of some 498 cents.

To understand the flaw in this not-so-perfect fourth, we may break it into whole-tones and semitones: e.g.

eb-e apotome 114 e-f diatonic 90 f-g whole-tone 204 g-g# apotome 114 --- 522

From another viewpoint, a perfect fourth is equal to a minor third (294 cents) plus a whole-tone (204 cents). However, here we have in addition to the minor third e-g a 114-cent apotome below at eb-e and another above at g-g#, adding 228 cents rather than 204. Or, if we define a fourth as a major third (408 cents) plus a diatonic semitone (90 cents), then we find here the major third eb-g plus the apotome g-g#, giving (408 + 114) or 522 cents instead of 498. Again, our rounding makes the difference appear as 24 cents, although it is actually closer to 23.46 cents.

Within the octave of our f-f' scale, we similarly find that our actual octave complement g#-eb', with a ratio of 177147:131072 divided by 2:1, or 262144:177147, has a size of about 678 cents, or 24 cents (actually 23.46...) less than a perfect 3:2 fifth (702 cents or so).

Possibly the simplest demonstration of this imperfection is to consider that a just fifth is equal to precisely a fourth plus a major second, or about 498 + 204 = 702 cents. However, g#-eb includes the fourth a-d' plus the two diatonic semitones g#-a and d'-eb', 90 cents each, or 498 + 180 = 678 cents.

This imperfectly small fifth g#-e', and likewise the imperfectly large fourth eb'-g#', is affectionately known as the "Wolf"; I am not sure just when and where this term is first documented. Legend has it that these sonorities on early organs reminded listeners of the howling of wolves.

In a Gothic context, this oddity is mostly an academic point, since G# and Eb very rarely occur together. However, during the Renaissance, while a G#-Eb Wolf fifth or fourth remained an accepted feature of some favorite tunings, theorists and keyboard designers attempted to solve this problem in various ways.

Apart from our Wolf between the extreme notes of the complete chain of fifths, Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, each fifth in the chain is pure. Thus the tuning gives us no fewer than 11 out a possible 12 perfect fifths (or fourths) in an octave.

The remark of Jacobus (c. 1325) that keyboards now customarily have all the whole-tones divided into their unequal semitones is corroborated by the Robertsbridge Codex, possibly dating from about the same epoch or slightly later. This first known source of keyboard music includes pieces using all the accidentals, and seems to fit nicely with a standard Pythagorean tuning.

Our conceptual process of building a complete chromatic scale fifth by fifth may actually be quite close to the practical process of Pythagorean tuning on many keyboard instruments. However, technologies vary and change. While manuals of a millennium ago explain how to obtain this tuning by an apt measuring of organ pipes, a keyboard player of today may need to program a synthesizer by entering measurements in cents.

Also, not all historical or modern instruments support or require a full chromatic scale. Many choice polyphonic works of the 13th century use only the seven diatonic notes, or these tones plus Bb (also a regular part of the gamut). Tuning charts for the usual medieval modes may carry at least as much significance as discussions of the more esoteric chromatic intervals in the Pythagorean tonal bestiary.

The first part of this section presents some information which may assist in the tuning of digital instruments which do not provide a predefined option for Pythagorean intonation. Also, while our step-by-step tuning used the octave f-f', many keyboardists may be more accustomed to tunings based on an octave of C. The tables and diagrams which follow may hopefully be of interest to players of acoustical and digital instruments alike.

The second portion includes Pythagorean tuning charts for the medieval modes. While the tuning system itself is identical in each case, the different modes present various aspects of its character and artistic potential.

In tuning a digital instrument, the user might be asked either to specify the frequency of each note, or to specify interval ratios in cents, possibly in terms of variations from 12-tone equal temperament (12tet).

The following table shows both kinds of information, with frequencies based on the common standard of a'=440. Curiously, this modern standard may be about as good a choice as any for music before 1600. The modest evidence available from the Renaissance suggests for some kinds of wind instruments an average tuning of a'=466, but with great variation in either direction. Pitch levels in the Gothic era would seem an even more conjectural matter. Of course, the interval ratios and measurements in cents should remain valid at any chosen pitch level, and may be used to calculate frequencies for any desired standard:

---------------------------------------------------------------------- Pythagorean tuning: 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#' 278.44 2187:2048 113.69 13.69+ | 81:128 -792.18 7.82+ d' 293.33 9:8 203.91 3.91+ | 2:3 -701.96 1.96- eb' 309.03 32:27 294.13 5.87- | 512:729 -611.73 11.73- 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#' 371.25 729:512 611.73 11.73+ | 27:32 -294.13 5.87+ g' 391.11 3:2 701.96 1.96+ | 8:9 -203.91 3.91- g#' 417.66 6561:4096 815.64 15.64+ | 243:256 - 90.22 9.78+ 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- ----------------------------------------------------------------------

Note that the middle columns of the chart show intervals in relation
to the lowest note of the octave, c', while the right-hand columns
show intervals in relation to the pitch standard, a', with negative
entries in the "cents" column showing that the note in
question is located *below* a'.

In the right-hand columns, the entry under "+/-12tet"
uses a positive value to show that the Pythagorean note is located
*higher* than its counterpoint in 12tet, and a negative
value to show that it is located *lower*. For notes below
a', this can have interesting consequences.

Thus our initial note c' is located a major sixth or 905.87
cents *below* a', larger than the 900-cent interval of 12tet.
This means that it is located 5.87 cents *lower* than its
12tet counterpart, thus a variance of -5.87. Our next note, c#',
is located below a' at a minor sixth or 792.18 cents, smaller than
the 800-cent interval of 12tet; thus it is located 7.82 cents
*higher*, a variance of +7.82.

Another way of visualizing this tuning is a diagram similar to the ones used for the step-by-step tuning in f-f':

2187:2048 32:27 729:512 6561:4096 16:9 114 294 612 816 996 c#' eb' f#' g#' bb' _114_|_90_ _90_|114__ _114_|_90__114_|_90_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

Changing the octave from f-f' to c'-c'' does not change the system of the tuning, but does throw into focus a different cross-section of that system. Thus we now have a normal Pythagorean minor third of 32:27 or 294 cents available from our fundamental c' (c'-eb'), in contrast to the curious 318-cent "augmented second" found in the F tuning (f-g#). We now have a 522-cent Wolf fourth eb'-g#' rather than a 678-cent Wolf fifth g#-eb'.

As in previous diagrams of this kind, values in cents have been rounded to the nearest whole number; the table of variances given a few paragraphs above offers more precise data.

The term "mode" can have many meanings, but here is used simply to mean a scale or octave-species with a characteristic pattern of whole-steps and semitones. For theorists such as Johannes de Grocheio (c. 1300), the term could imply further a regular formula by which one may know the beginning, middle, and end of a melody - as in Gregorian chant with its reciting tones, as opposed to the world of polyphony and secular music. Thus Grocheio prefers to say that polyphonic music is based on various octave-species, but not on "modes" proper.

Grocheio's caution is a helpful reminder that whether we use "mode" in a strict or free sense, the various octave-species here called "modes" are often mixed in Gothic songs and polyphonic compositions, and not infrequently seasoned with Bb and other accidentals of various kinds, especially in the 14th century. These musical elements combine to give a total impression of lively variety.

The six basic modes are based on the diatonic or "white-key" scales on D, E, F, G, A, and C. Each of these modes has two forms. In the authentic modes, the final or note on which a melody concludes is located at the bottom of the octave. In the plagal modes, the final is located in the middle of the octave, at the fourth. Melodically, the authentic modes often seem to divide the octave into a lower fifth and upper fourth, while in plagal modes the final often acts as a kind of demarcation point between the lower fourth and upper fifth.

The authentic and plagal modes have respectively odd and even numbers, this traditional medieval scheme for Modes I-VIII being extended by Glareanus (1547) to his newly recognized Modes IX-XII. Tuning ratios and measures in cents are shown relation to the lowest note of the octave range, the final in authentic modes but the fourth below it in plagal modes, with the final identified by the symbol "F" on the line above the tuning ratios.

One quirk of this method of listing intervals for the plagal
modes in terms of the lowest scale tone rather than the final: the
chart for Hypophrygian (Mode IV) shows the basic interval B-f', a
diminished fifth or *semitritonus* at 1024:729 (588 cents).
Both notes have more conventional intervals in reference to the
final e, the fourth B-e and the diatonic semitone e-f, while e-b
provides the expected perfect fifth above the final. Note that
the tritone f-b likewise occurs in Lydian, but that this mode also
has a perfect fifth f-c' above the final. In contrast, modes on
the final B are generally rejected because of the lack of a perfect
fifth above the final.

Mode I (Dorian) F 1:1 9:8 32:27 4:3 3:2 27:16 16:9 2:1 ----------------------------------------------- d e f g a b c' d' 0 204 294 498 702 906 996 1200 204 90 204 204 204 90 204 Mode II (Hypodorian) F 1:1 9:8 32:27 4:3 3:2 128:81 16:9 2:1 ----------------------------------------------- A B c d e f g a 0 204 294 498 702 792 996 1200 204 90 204 204 90 204 204 Mode III (Phrygian) F 1:1 256:243 32:27 4:3 3:2 128:81 16:9 2:1 ----------------------------------------------- e f g a b c' d' e' 0 90 294 498 702 792 996 1200 90 204 204 204 90 204 204 Mode IV (Hypophrygian) F 1:1 256:243 32:27 4:3 1024:729 128:81 16:9 2:1 ------------------------------------------------- B c d e f g a b 0 90 294 498 588 792 996 1200 90 204 204 90 204 204 204 Mode V (Lydian) F 1:1 9:8 81:64 729:512 3:2 27:16 243:128 2:1 ------------------------------------------------- f g a b c' d e f' 0 204 408 612 702 906 1110 1200 204 204 204 90 204 204 90 Mode VI (Hypolydian) F 1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1 ------------------------------------------------- c d e f g a b c' 0 204 408 498 702 906 1110 1200 204 204 90 204 204 204 90 Mode VII (Mixolydian) F 1:1 9:8 81:64 4:3 3:2 27:16 16:9 2:1 ------------------------------------------------- g a b c' d' e' f' g' 0 204 408 498 702 906 996 1200 204 204 90 204 204 90 204 Mode VIII (Hypomixolydian) F 1:1 9:8 32:27 4:3 3:2 27:16 16:9 2:1 ----------------------------------------------- d e f g a b c' d' 0 204 294 498 702 906 996 1200 204 90 204 204 204 90 204 Mode IX (Aeolian) F 1:1 9:8 32:27 4:3 3:2 128:81 16:9 2:1 ----------------------------------------------- a b c' d' e' f' g' a' 0 204 294 498 702 792 996 1200 204 90 204 204 90 204 204 Mode X (Hypoaeolian) F 1:1 256:243 32:27 4:3 3:2 128:81 16:9 2:1 ----------------------------------------------- e f g a b c' d' e' 0 90 294 498 702 792 996 1200 90 204 204 204 90 204 204 Mode XI (Ionian) F 1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1 ------------------------------------------------- c' d' e' f' g' a' b' c'' 0 204 408 498 702 906 1110 1200 204 204 90 204 204 204 90 Mode XII (Hypoionian) F 1:1 9:8 81:64 4:3 3:2 27:16 16:9 2:1 ------------------------------------------------- g a b c' d' e' f' g' 0 204 408 498 702 906 996 1200 204 204 90 204 204 90 204

Of course, the unique qualities of these modes are not dependent on any specific tuning; rather, they result from the characteristic patterns of whole-steps and half-steps in each mode. However, by accentuating the contrast between a generous 9:8 whole-tone and a narrow 256:243 semitone, Pythagorean tuning lends a special expressiveness to monophonic chant and secular song as well as to polyphony with its interacting harmonic and melodic dimensions (see Section 3.3).

Tuning systems, like musical styles, have their characteristic qualities and quirks, and Pythagorean intonation is no exception. Two small intervals known as "commas" define some of the distinctive features of a Pythagorean tonal universe.

One quirk of Pythgorean tuning which we encountered in
Section 4.2.2 is the "Wolf" fifth or
fourth which results between the extreme notes of our tuning chain
in fifths, g#-eb' or eb-g# in a standard scheme with Eb at one end
of the chain and G# at the other. The amount by which this Wolf
falls short of a pure fifth or exceeds a pure fourth is known as
a *Pythagorean comma*, equal to about 23.46 cents.

Another trait of the tuning is its rather wide major thirds and
sixths, and its correspondingly narrow minor thirds and sixths.
In a Gothic context, this is a feature rather than a bug, since it
gives these intervals an active quality inviting very effective
resolutions (Sections 3.1.2,
3.2.2, 3.3).
A measure of this distinctive quality is the difference between a
Pythagorean third or sixth and the same interval in its simplest
ratio - for example, the Pythagorean M3 (81:64, 408 cents) vis-a-vis
the ideal Renaissance M3 (80:64 or 5:4, 386 cents). This difference
of 81:80 (about 21.51 cents) is known as the *syntonic comma*,
or *comma of Didymus*.

Our experiment in building a chromatic scale revealed that although all notes are tuned in perfect fifths, we get only 11 out of the 12 potentially perfect fifths in a full chromatic octave. Thus in our standard tuning Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, each fifth in the chain is perfect but the two notes at extremes of the chain do not quite mesh. Rather the interval g#-eb' or eb-g# is a Wolf fifth or fourth, about 23.46 cents smaller than a pure fifth or larger than a pure fourth.

Similarly, if we extended our chain by a 12th fifth in the sharp
direction, Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#, we would find that our
final D# was *not* precisely at a unison or octave with Eb,
but rather a Pythagorean comma sharper or closer to the nearest
E.

One way to explain this apparent anomaly is to show that 12
perfect fifths do *not* quite equal any even octave: rather
they exceed it by this same Pythagorean comma. We can demonstrate
this point in two ways.

Taking the ratio of the fifth, 3:2, we can calculate the interval generated by 12 fifths as (3:2)^12, an impressive 531441:4096 - or, factoring out seven octaves, 531441:524288. To avoid the complication of multiple octaves, we can also measure the Pythagorean comma as six whole-tones or (9:8)^6, or 531441:262144, which when we subtract an octave gives the same result.

Using cents, we can easily calculate that 12 fifths of about 702 cents each will yield an interval of (702 x 12) or 8424 cents, and likewise six whole-tones of about 204 cents yield an interval of (204 x 6) or 1224 cents. Subtracting seven octaves (8400 cents) in the first case, and one octave (1200 cents) in the second, we get an approximation of 24 cents for the Pythagorean comma. Since in fact a 3:2 fifth is closer to 701.955 cents, and a 9:8 major second to 203.91 cents, a closer approximation is 23.46 cents.

Another way to demonstrate the size of this comma is to note that an octave is equal precisely to five Pythagorean whole-tones and two diatonic semitones, as can be seen in each of the modal scales in Section 4.3. Thus we have, using rounded values in cents:

5 whole-tones (9:8) = 204 x 5 = 1020 cents 2 diatonic semitones (256:243) = 90 x 2 = 180 cents ---------- 1200 cents

Each whole-tone may be divided into a diatonic semitone of 256:243 plus an apotome of 2187:2048 - about 90 and 114 cents respectively. Adding these five diatonic semitones and five apotomes to our other two diatonic semitones, we have:

7 diatonic semitones = 90 x 7 = 630 cents 5 apotomes = 114 x 5 = 570 cents ---------- 1200 cents

In contrast, if we take six whole-tones (12 fifths minus six octaves), we find that they contain in rounded cents:

6 diatonic semitones = 90 x 6 = 540 cents 6 apotomes = 114 x 6 = 684 cents ---------- 1224 cents

Again, if we used more precise values in cents, we would find that an interval built from six whole-tones exceeds an octave by 23.46 cents. This value is identical to the difference between a diatonic semitone and an apotome, i.e. (113.6850... - 90.2249...) cents, or again very close to 23.46 cents.

Jacobus of Liege notes this discrepancy, describing the
*hexatone* or interval of six whole-tones as a rough discord
*not* equivalent to a pure octave.

From a medieval perspective, the Pythagorean comma might be regarded as a minor "bug" in the tuning system. As long as we stick to a chain of fifths from Eb to G#, and the Wolf fifth or fourth between these two notes rarely occurs in actual polyphony, the bug is mostly of academic interest. There remain eleven perfect fifths or fourths per octave, happily the eleven most likely to be used in practice.

Our title for this section refers to the Pythagorean comma as "mostly" a bug, because it appears that some musicians of the epoch around 1400 were cleverly taking advantage of this quirk to adjust another aspect of the tuning system a bit less congenial to an emerging "modern" style than to traditional polyphony from Perotin to Machaut. We discuss this adjustment in Section 4.5 after first considering the aspect of the system it ingeniously modified: the syntonic comma.

While the Pythagorean comma seems to be a "bug," since it limits us to 11 perfect fifths out of 12 per octave, the syntonic comma might more justly be called an artistic feature of Gothic music and tuning: the active and unstable quality of thirds and sixths. As Carl Dahlhaus has eloquently stated, the tuning of these intervals is "to be understood as a musical phenomenon rather than a mathematically imposed acoustic blemish" (translation by Mark Lindley).

Comparing the sizes of Pythagorean thirds and sixths with their counterparts in Renaissance theory having the simplest possible ratios, we find in each case a difference of 81:80 or about 22 cents, the syntonic comma:

----------------------------------------------------------- Interval Pythagorean ratio Simplest ratio ----------------------------------------------------------- M3 81:64 (408 cents) 5:4 (386 cents) m3 32:27 (294 cents) 6:5 (316 cents) M6 27:16 (906 cents) 5:3 (884 cents) m6 128:81 (792 cents) 8:5 (814 cents) -----------------------------------------------------------

Using more precise measures for these intervals, we would find, for example, that the Pythagorean M3 is roughly 407.82 cents, and the simplest M3 of 5:4 roughly 386.31 cents, giving a syntonic comma of about 21.51 cents.

This differential can serve as a kind of index of the degree of acoustical tension in thirds and sixths. In a standard Pythagorean tuning, they are a full syntonic comma (21.5 cents) wide or narrow, producing a considerable degree of tension which fits nicely with the active role of these intervals in Gothic polyphony.

In later styles, where thirds and sixths take on a quality of stable euphony and rest, this feature of Pythagorean tuning becomes more of a "misfeature," if not an outright "bug." Thus the just intonation and meantone systems of the Renaissance aim to present thirds and sixths - or at least as many as possible - in their simplest ratios, a differential of 0 cents. The "well-tempered" tunings of the 18th century place these intervals on a kind of sliding scale of tensions, with differentials ranging in one scheme from 2/11 of a syntonic comma to a full syntonic comma, the modes or keys considered more remote having the greater acoustical tension.

In modern 12-tone equal temperament, major thirds at 400 cents and minor sixths at 800 cents have a differential of about 13.69 cents (or about .64 of a syntonic comma); minor thirds at 300 cents and major seconds at 900 cents have a differential of about 15.64 cents (or about .73 of a syntonic comma). As in Pythagorean tuning, M3 and M6 are wide while m3 and m6 are narrow.

From an acoustical or mathematical viewpoint, both the Pythagorean and syntonic commas reflect basic facts of musical geometry. It is impossible to tune 12 pure fifths so as to arrive at an even octave; and it is impossible in any fixed 12-tone tuning to achieve pure fifths and also to obtain thirds (and sixths) in their simplest ratios.

In a Gothic setting, the Pythagorean comma and the resulting Wolf fifth or fourth between Eb and G# is only a minor inconvenience or "bug," since these accidentals are rarely combined. The syntonic comma, in contrast, is a congenial feature: stable fifths and fourths in their ideal ratios, and active thirds and sixths, both mesh nicely with the harmonic style.

From another perspective, the syntonic comma also represents a fact of musical geometry noted by Mark Lindley: the expressively narrow and incisive Pythagorean diatonic semitone of 90 cents (256:243) is necessarily associated with wide M3 and M6, and narrow m3 and m6.

Happily, from a Gothic viewpoint, both incisive melodic semitones and active vertical thirds and sixths concord nicely with the artistic style. As discussed in Section 3.3, these dimensions together contribute to the expressiveness of many cadences of the period.

In other periods, the tradeoffs between acoustical necessity and musical style may perhaps be somewhat less happy. While favorite meantone tunings of the Renaissance continued to accept an Eb-G# Wolf relegated to a lair on the remote periphery of the modal system, by the 18th century it had become a stylistic imperative to domesticate this creature. Schemes of well-temperament and equal temperament, compromising many intervals slightly rather than one or a few intolerably, are one approach to this problem; keyboards with more than 12 notes per octave, proposed as early as the 15th century, are another.

Similarly, in styles where thirds and sixths serve as restful concords, there will be an inevitable compromise between vertical euphony and the desire for incisive diatonic semitones. Renaissance tunings optimize thirds and sixths, accepting the consequence of diatonic semitones considerably wider than 100 cents, typically in fact rather close to the Pythagorean apotome of 114 cents. Equal temperament yields acoustically somewhat more tense thirds differing from the Renaissance ideal by the better part of a syntonic comma, as we have seen, and semitones all measuring an even 100 cents.

The inexorable mathematics of the two commas remains constant, but stylistic parameters and artful tuning solutions change. It would seem that indeed one era's feature can be another era's bug.

By the early 14th century, keyboards with all 12 chromatic notes had become common, and the full set of accidentals had become integral to the modern practice and theory of the Ars Nova. Such accidentals served, for example, the increasingly clear preference for resolutions by contrary motion where one voice moves by a whole-step and the other by a half-step, e.g. m3-1, M3-5, M6-8 - and, for Jacobus, also m7-5. Thus:

f#'-g' f#'-g' c#'-d' d' -c' c#'-d' g# -a b -c' a -g e -d 5 5 M6 8 M6 -8 M3 1 M3 5 M3 -5 (M3-5 + m3-1) (M3-5 + M6-8) (M3-5 + M6-8)

In the first two progressions, taken from a motet by Petrus de Cruce (c. 1280?), the accidentals f#' and c#' facilitate motion from an unstable 5/M3 or M6/M3 sonority to a stable fifth or trine (see Sections 3.2.1, 3.3) by way of these resolutions. In the third example, typical of the 14th century, g# (the final accidental to be added) and c#' likewise facilitate resolutions of M3-5 and M6-8.

Another way of stating this preference is to say that a third contracting to a unison should be minor, while a third expanding to a fifth or a sixth to an octave should be major. Accidentals applied to unstable intervals and combinations assist in fulfilling this preference articulated by various theorists of the early 14th century, including even the conservative Jacobus.

In such progressions, the Pythagorean accidentals facilitate the "closest approach" of an unstable interval to its stable goal even on a microtonal level (see also Section 3.3). Let us consider again the progression:

f#'-g' c#'-d' a -g

The sharps raise each upper note of the penultimate sonority by a full apotome of 114 cents (c-c#, f-f#), placing it only a 90-cent diatonic semitone from its cadential goal. Vertically, the Pythagorean M3 at 408 cents (a-c#') and M6 at 906 cents (a-f#') need expand only 294 cents each to attain the stable fifth and octave respectfully, This expansion is brought about as the lowest voice descends by a 204-cent whole-tone and each upper voice ascends by a 90-cent semitone.

Artistically speaking, the unstable M3 and M6 are at once about
21.5 cents *wider* than their ideal Renaissance counterparts,
adding a bit of extra dynamic tension, and 21.5 cents *closer*
to their directed goal, faclitating the efficient and expressive
release of this tension.

Indeed, Marchettus of Padua (c. 1318) proposes a variation on Pythagorean tuning based on a subtle division of the whole-tone designed in part to permit the smallest possible semitones in the "perfection" of intervals such as M3-5 and M6-8. One reading of his system would actually stretch the major sixth so far as to make it a minor seventh! - although another interpretation would result in a cadential M6 not far from the traditional Pythagorean ratio and a resolving semitonal motion not far from the usual 90 cents.

For much music of the 14th century, including the famous Mass of Guillaume de Machaut, Pythagorean tuning appears to provide an excellent solution in practice as well as theory. However, musical styles change, and the period around the end of the 14th century is no exception.

Composers of this epoch such as Matteo de Perugia and the contributors to the Faenza Codex of keyboard music, Mark Lindley suggests, may have exploited a rearrangement of the traditional Pythagorean tuning in order to explore the possibilities of more blending thirds and sixths.

Rather than placing Eb at one end of the chain of fifths, and G# at the other, with the Wolf located between these notes - a tuning indicated as "Eb x G#" - many musicians now evidently placed all accidentals at the flat end of the chain. Thus tuning first the diatonic notes, and then the accidentals beginning with Bb in a flat direction, we have:

Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B (i.e. F#-C#-G#)

This tuning has two obvious consequences. The Wolf moves to a
new lair between F# and B, thus identifying this as an "F# x
B" tuning, and all the accidentals now split a whole-tone into
a diatonic semitone below and an apotome above:

256:243 32:27 1024:729 128:81 16:9 90 294 588 792 996 c#' eb' f#' g#' bb' _90_|_114_ _90_|114__ _90_|_114__90_|_114_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

Another consequence is that we get a set of curiously altered thirds and sixths for intervals such as d'-f#' and e'-g#' often occurring in the music of the epoch. Using the note locations in reference to c' as shown in the above tuning chart, and extending the range upward into the next octave where necessary by adding 1200 cents, we find examples like the following:

---------------------------------------------------------------------- Interval Altered Ratio Normal Pythagorean Simplest ---------------------------------------------------------------------- M3 8192:6561 (384) 81:64 (408) 5:4 (386) Examples: d'-f#' (588 - 204) e'-g#' (702 - 498) a'-c#'' (1290 - 906) ---------------------------------------------------------------------- M6 32768:19683 (882) 27:16 (906) 5:3 (884) Examples: e'-c#'' (1290 - 408) a'-f#'' (1788 - 906) b'-g#'' (1992 - 1110) ---------------------------------------------------------------------- m3 19683:16384 (318) 32:27 (294) 6:5 (316) Examples: c#'-e' (408 - 90) f#'-a' (906 - 588) g#'-b' (1110 - 792) ---------------------------------------------------------------------- m6 6561:4096 (816) 128:81 (792) 8:5 (814) Examples: c#'-a' (906 - 90) f#'-d'' (1404 - 588) g#'-e'' (1608 - 792) ----------------------------------------------------------------------

Strictly speaking, these intervals are not new, but occur also in our traditional Eb x G# tuning of Section 4.2.2: e.g. b-eb' (M3 or dim4, 384 cents); f#-eb' (M6 or dim7, 882 cents); eb-f# (m3 or aug2, 318 cents); and f-c#' (m6 or aug5, 816 cents). However, these quirks are from a "classic" Gothic perspective merely theoretical anomalies of little practical significance. The new tuning evidently favored around 1400 makes such intervals a prominent and very audible feature of musical practice.

The lowered position of the sharps, or the reversed order of the semitones around them (diatonic semitone below, apotome above), effectively subtracts a full Pythagorean comma (about 23.46 cents) from affected major thirds and sixths, and adds the same amount to affected minor thirds and sixths. (We will recall that this comma is equal to the difference between a diatonic semitone and an apotome.) Since M3 and M6 are normally a syntonic comma (about 21.51 cents) wider than their simplest ratios, and m3 and m6 a syntonic comma narrower, the result is something very close to these most blending ratios.

In fact, there is a slight "overcorrection," since
the Pythagorean comma is slightly larger than the syntonic comma;
altered M3 and M6 actually are about 2 cents narrower than their
ideally simple ratios, and m3 and m6 about 2 cents wider. This
small difference between the two commas, incidentally, is called
a *schisma* (more precisely 1.95 cents), and these intervals
"schisma thirds" (or sixths).

Such small discrepancies would likely be inconsequential in practice, but the strikingly different quality of these altered intervals may have interacted with musical style in various ways in the period leading from Matteo de Perugia and the Faenza Codex around 1400 to the epoch of the early Dufay and his peers around 1420-1440. Lindley's work explores some of these possible interactions in more detail.

For example, he notes the prominent use of the combinations
d-f#-a and a-c#-e' in keyboard and vocal music with a Dorian
modality, as well as e-g#-b. The almost ideally blending thirds
might acoustically reflect a new stylistic role for these 5/3
combinations somewhere between the active *quinta fissa* of
the earlier Gothic (see Section 3.2.2)
and the stable 16th-century triad.

Interestingly, for a more traditionally-inclined musician such as Prosdocimus at the beginning of the 15th-century, such an altered tuning was less suitable than the classic tuning of Eb x G#. To see the difference, let us again consider the resolution from a-c#'-f#' to g-d'-g' under both tunings:

Eb x G# tuning F# x B tuning f#'-- +90 - g' f#'-- +114 - g' (906) (1200) (882) (1200) c#'-- +90 - d' c#'-- +114 - d' (408) (702) (384) (702) a - -204 - g a - -204 - g

In the former tuning, as we have seen, the characteristically wide Pythagorean M3 and M6 of 408 cents and 906 cents can efficiently expand to the fifth and octave with 90-cent semitonal ascents in both upper voices. As Prosdocimus puts it, these intervals are fully "perfected," stretching out to approach as closely as possible their stable resolutions.

With the newer tuning, however, the inflected notes c#' and f#' must ascend a full apotome of about 114 cents to reach d' and g'. From a vertical perspective, the altered M3 and M6 of 384 cents and 882 cents are each about 23.46 cents narrower - and so must expand 23.46 cents further to attain the fifth and octave. To cover this extra distance, each upper voice must move by a not-so-incisive 114-cent apotome rather than a traditional 90-cent diatonic semitone.

Therefore Prosdocimus (1413) finds this tuning defective, and Ugolino of Orvieto in the second quarter of the century, although more favorably disposed, nevertheless observes that such an altered M3 or M6 is not "fully perfected," since its width falls a comma short of the ideal closest approach to the resolving fifth or octave.

A more glaring imperfection is the placement of the Wolf fifth or fourth at b-f#' or f#-b, intervals frequently appearing in Gothic polyphony. Lindley reasonably considers the prominent appearance of such intervals in a piece as a strong contraindication to this tuning. On this and other stylistic grounds, for example, he concludes that the keyboard music of the Robertsbridge Codex (c. 1325?) is ill-suited to f# x b, and excellently suited to the standard eb x g# tuning.

However, he is willing at least to consider the possibility of the newer tuning for certain keyboard pieces of the 15th century where the Wolf would be heard in the following very popular cadence on C:

Eb x G# tuning F# x B tuning b' - +90 - c'' b' - +90 - c'' (906) (1200) (906) (1200) f#'-- +90 - g' f#'-- +114 - g' (408) (702) (384) (702) d' - -204 - c' d' - -204 - c'

In the standard tuning, both M3 (d'-f#') at 408 cents, and M6 (d'-b') at 906 cents, have their full Pythagorean size, and both intervals resolve (M3-5, M6-8) with an efficient ascending diatonic semitone of 90 cents (f#'-g', b'-c''). The new tuning retains the full Pythagorean M6 at positions not involving sharps, here d'-b', and this interval expands to the octave with the usual 90-cent semitonal motion (b-c''). The M3 d'-f#' involves a sharp, however, and so is altered to a width of only 384 cents, requiring a 114-cent semitonal motion (f#'-g') in its expansion to the fifth.

This microtonal disparity in the semitonal motions of the upper voices correlates with the vertical disparity between the usual wide M6 of 906 cents and the narrow M3 of 384 cents - producing between these upper voices not a perfect fourth of 498 cents, but an enlarged Wolf fourth of (906 - 384) or 522 cents.

Lindley argues that the cadential context and the blending qualities of the altered M3 at d'-f#' may be deemed to mitigate the Wolf. One might also note the tendency of many systems of counterpoint and harmony to take a more lenient view of "discordant" relations between two upper voices both forming legitimate "concords" with the lowest voice, however these concepts are defined in a given period or style.

In any case, theorists of the early 15th century propose a
logical solution: an instrument with 17 notes to the octave,
including *two* accidentals a Pythagorean comma apart for
each of the five whole-tones of a modal octave, permitting one to
divide such whole-tones into either a 90-cent semitone below and
an apotome above, or the converse. To permit measures in cents
rounded to whole numbers in the following diagram, the common
liberty has been taken of representing the Pythagorean comma -
actually some 23.46 cents - as 24 cents:

b: 256:243 32:27 1024:729 128:81 16:9 #: 2187:2048 19683:16384 729:512 6561:4096 59049:32768 90 114 294 318 588 612 792 816 996 1020 db'c#' eb'd#' gb'f#' ab'g#' bb'a#' _90_|_|90_ _90_|_|90_ _90_|_|90_ _90_|_|_90_ _90_|_|90_ c' 24 d' 24 e' f' 24 g' 24 a' 24 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

Such a keyboard makes available all the accidentals of the traditional Eb x G# and newer F# x B tunings. Following the first tuning, we could play the cadential sonority in the previous example as d'-f#'-b', obtaining the usual Pythagorean M3 (408 cents) and M6 (906 cents) for efficient expansion to the fifth and octave - and the expected perfect fourth (498 cents) between the upper voices.

In contrast, we could obtain the altered thirds of the new tuning
for a "coloristic" sonority such as d'-f#'-a' by playing
it as d'-gb'-a', getting a lower M3 of 384 cents and an upper m3
of 318 cents. Playing d'-f#'-a' would yield the traditional Gothic
*quinta fissa* (Section 3.2.2)
with its more active M3 of 408 cents and m3 of 294 cents.

When designing keyboards of only 12 notes per octave, it would appear that early 15th-century musicians often leaned toward the new tuning: Lindley quotes sources showing Gb as a standard keyboard accidental and F# as a note a comma above which would be useful but is not found on such keyboards.

Both technically and artistically, the F# x B tuning seems to mediate between the traditional Gothic world of sound with its active thirds and sixths and concise diatonic semitones, and the Renaissance world with its restful thirds and sixths and large diatonic semitones. For vertical sonorities and progressions involving only natural notes or flats, a Gothic ambience still prevails. The lowered position of the sharps, however, at once transforms an interval such as e-g# from an assertive and expansive 408 cents to a more blending 384 cents, and works a role reversal so that the chromatic semitone g-g# becomes a narrow 90 cents, and the diatonic semitone g#-a a 114-cent apotome.

Such a tuning in some ways resembles the well-temperaments of the 18th century, which also feature thirds and sixths of diverse ratios and expressive qualities. Just as these later tunings give acoustically distinct colors to the various major and minor keys, so Lindley suggests that the F# x B tuning may have added a dimension to the distinction between modes for Dufay and his colleagues. For example, he observes that Dorian on D or G (with Bb) would tend to use altered thirds and sixths involving sharped notes more often than F Lydian, possibly influencing stylistic evolution in various ways.

Whatever the actual pathways, by around 1500 the transformation of style and tuning systems was becoming more and more complete. Ideally blending thirds and sixths, now sometimes used by composers such as Josquin and Isaac even in closing sonorities, had become prime measures for the new just intonation and meantone systems. Inevitably this involved a generalization of the semitonal "role-reversal" brought about locally around sharps by the now dated F# x B tuning.

Thus Renaissance just intonation theory typically defined a large diatonic semitone of 16:15 or about 112 cents, e.g. e-f, as the difference between the 4:3 fourth c-f (498 cents) and the pure 5:4 major third c-e (386 cents); the small chromatic semitone of 25:24 or about 70 cents, e.g. eb-e, measured the difference of the 5:4 major third c-e (386 cents) and the 6:5 minor third c-eb (316 cents).

Traditionally-minded theorists such as Gaffurius might long for the concise diatonic semitones of an older era, but the new stylistic imperatives of the Renaissance required a different result.

Just what role the F# x B tuning may have played in the transition from Gothic to Renaissance remains a moot question. Lindley rightly raises as matters of open conjecture the possibility that this keyboard tuning may have had some influence on the vocal music of the young Dufay and his peers, or that some actual keyboards of the period 1370-1450 may have had more than 12 notes per octave.

At the least, this tuning offers one attractive hint on the intriguing question of how early 15th-century musicians may have combined a Pythagorean outlook with an increasingly pervasive "coloristic" use of thirds and sixths. It is curious to reflect that Pythagorean tuning might have played an instrumental role in what the scholar Olivier Bettens charmingly terms "the seduction of the third."

To Section 5 - Pythagorean tuning: a just appraisal in context.

Margo Schulter