Pythagorean Tuning and Medieval Polyphony

Table of Contents

  1. Introduction

    1. Remarks and acknowledgements

  2. Basic concepts

  3. Pythagorean tuning and Gothic polyphony

    1. Simple vertical intervals
      1. Ideal fifths and fourths
      2. Active thirds and sixths
      3. Milder major seconds and minor sevenths
    2. Coloristic harmony
      1. Stable trines (8/5, 8/4)
      2. The "split fifth" (5/3)
      3. Quintal/quartal sonorities (9/5, 5/4, 5/2, 7/4)
    3. Cadential action

  4. Pythagorean tuning in more detail

    1. Tuning a basic scale
      1. Additional diatonic intervals
    2. Ratios, cents, and the complete chromatic scale
      1. An aside: calculating cents
      2. Completing a 12-note scale: from apotome to "Wolf"
    3. Modes, scales, and measures
      1. Tuning data and the chromatic octave of C
      2. Medieval modes in Pythagorean tuning
    4. The two commas: bugs or features?
      1. The Pythagorean comma: mostly a bug
      2. The syntonic comma: "One era's feature ..."
    5. Pythagorean tuning modified: a transition around 1400

  5. Pythagorean tuning: a just appraisal in context

    1. Pythagorean tuning as quintal just intonation
    2. Beauty, utility, and choice
    3. From Gothic to Renaissance: a new intonational dilemma
    4. Meantone temperaments: tunings with an attitude
    5. Well-temperaments, triadic and trinic
      1. Well-temperament, Pythagorean style
      2. The just, the true, and the incisive
      3. Two sample solutions
    6. Equal temperament: a musical supplement
      1. Equal temperament as 1/11-comma meantone
      2. Exploring the meantone spectrum
      3. An aside: the meantone equations
      4. Equal temperament as a well-temperament
      5. Early history of equal temperament
      6. Equal temperament and Gothic music

  6. Bibliography

1. Introduction

One aspect of medieval music now receiving much interest is the matter of tuning. This FAQ article is intended to explain the system of tuning in perfect fifths commonly known as "Pythagorean intonation," its interaction with the stylistic traits of medieval polyphony, and its relationship to other systems of tuning.

While our focus here is on the music of medieval Europe, the concept of a tuning based on a series of twelve notes in perfect fifths also plays an important part in other world musical traditions, for example in Chinese theory and practice.

Providing a simple and elegant way of generating a musical scale, this tuning system may have a special appeal for styles of harmony where fifths and fourths are the most favored intervals, as is true in the ensemble music of Chinese and related traditions, for example, as well as in medieval European polyphony.

In the West, as the name suggests, Pythagorean tuning was credited to the ancient Greek philosopher Pythagoras, known (like many of the pre-Socratics) mainly through quotations and anecdotes in later writers. Interestingly, it is documented in guides to organ building from the post-Carolingean era (9th-10th centuries), also a period when polyphony was beginning to be recorded.

Remaining the standard theoretical approach in the High Gothic era of the 13th century, Pythagorean tuning seems very congenial to the complex polyphony and subtle harmonic continuum of composers such as Perotin, Adam de la Halle, and Petrus de Cruce. It also nicely fits the style of many 14th-century works, such as the famous Mass of Guillaume de Machaut.

By around 1420 on the Continent, however, musical style had begun to change in ways that invited new tunings. As composers such as Dufay and Binchois emulated John Dunstable, and gave their music an "English countenance" with a more and more pervasive emphasis on thirds and sixths, fashion moved in the direction of intonations that would make these intervals more smoothly blending. By the end of the century, such tunings (e.g. meantone) were becoming the norm in theory as well as practice.

The unsuitability of medieval Pythagorean intonation for Renaissance music should not be seen as a "flaw," any more than Renaissance meantone tuning is "flawed" because it is hardly suitable for the works of Wagner or Max Reger. Rather, techniques of tuning and notation interact creatively with musical style in each period, and should all be taken into consideration in understanding and recreating the music of a given age.

Section 2 presents some basic concepts of Pythagorean tuning as applied to Gothic music, while Section 3 explores how this system nicely fits in with the subtle spectrum of harmonic tension in the 13th century. Section 4 explores some aspects of the tuning in more detail, while Section 5 considers its relationship to other systems of just intonation as well as alternative approaches such as equal temperament.

1.1 Remarks and acknowledgements

Readers interested in the practical details of Pythagorean tuning are encouraged to jump directly from Section 2 to Section 4. Section 3, on stylistic considerations, is linked in many ways to a companion article on 13th-century polyphony, and owes a special debt of gratitude to studies by Vincent Corrigan on the Notre Dame conductus repertory, and by Mark Lindley on the later 13th and 14th centuries, although any flaws or infelicities are of course mine.

Also I would like very warmly to thank the many people who have exchanged ideas and perceptions with me on the topic of medieval tunings. Todd McComb, Olivier Bettens, Margaret Hasselman, Bill Alves, Jason Stoessel, Tore Lund, Ed Foote, and many others have helped spur me on to this project with their lively divergence of opinions, and I would emphasize that what follows is only one view of the matter.

Finally, it may be worth pointing out at the outset that fixed tunings, including Pythagorean intonation, are more strictly applicable to fixed-pitch instruments such as harps or keyboards than to singers or to other kinds of instruments. It seems safe to assume that medieval performers, like their modern counterparts, may have varied their tuning of intervals considerably, although we cannot be sure quite how. Especially in the case of ensemble music, any tuning on paper is the distillation of a more complex musical reality. Tuning systems, like notations, nevertheless offer us intriguing clues to the musical spirit of an age.

To Section 2 - Basic concepts.

See also, Experimental Tuning Discussions, based on the present article.

To Early Music FAQ.

Margo Schulter
Sacramento, CA
10 June 1998