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Pythagorean Tuning

Pythagorean Tuning and Medieval Polyphony Margo Schulter

[email protected] • Sacramento, CA 10 June 1998

Copied from the Early Music America online FAQ

29/09/15

1

Pythagorean Tuning Table of Contents

Introduction —————————————————————————— 1.1 Remarks and acknowledgements ——————————————— 2. Basic concepts —————————————————————————— 3. Pythagorean tuning and Gothic polyphony ——————————————— 3.1 Simple vertical intervals 1.

3.1.1 3.1.2 3.1.3

3.2 3.2.1 3.2.2 3.2.3

3.3

Ideal fifths and fourths Active thirds and sixths Milder major seconds and minor sevenths

3 4 4 5 6 6 6

Coloristic harmony

Stable trines (8/5, 8/4) The "split fifth" (5/3) Quintal/quartal sonorities (9/5, 5/4, 5/2, 7/4)

Cadential action

4. Pythagorean tuning in more detail Tuning a basic scale 4.1 4.1.1

4.2 4.2.1 4.2.2

4.3 4.3.1 4.3.2

4.4 4.4.1 4.4.2

Additional diatonic intervals

Ratios, cents, and the complete chromatic scale

An aside: calculating cents Completing a 12-note scale: from apotome to "Wolf"

Modes, scales, and measures

Tuning data and the chromatic octave of C Medieval modes in Pythagorean tuning

The two commas: bugs or features?

The Pythagorean comma: mostly a bug The syntonic comma: "One era's feature ..."

4.5

Pythagorean tuning modified: a transition around 1400

5.1

Pythagorean tuning as quintal just intonation

5. Pythagorean tuning: a just appraisal in context 5.2 5.3 5.4 5.5 5.5.1 5.5.2 5.5.3 5.6 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5

Beauty, utility, and choice

From Gothic to Renaissance: a new intonational dilemma Meantone temperaments: tunings with an attitude Well-temperaments, triadic and trinic

Well-temperament, Pythagorean style The just, the true, and the incisive Two sample solutions Equal temperament: a musical supplement Equal temperament as 1/11-comma meantone Exploring the meantone spectrum An aside: the meantone equations Equal temperament as a well-temperament Early history of equal temperament

5.6.6 6.0 Bibliography

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Equal temperament and Gothic music

2

Pythagorean Tuning 1. Introduction One aspect of medieval music now receiving much interest is the matter of tuning; this FAQ1 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.

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Pythagorean Tuning 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.

2. Basic concepts As mentioned above, Pythagorean tuning defines all notes and intervals of a scale from a series of pure fifths with a ratio of 3:2. Thus it is not only a mathematically elegant system, but also one of the easiest to tune by ear. To derive a complete chromatic scale of the kind common on keyboards by around 1300, we take a series of 11 perfect fifths: E♭

B♭

F

C

G

D

A

E

B

F♯

C♯

G♯

The one potential flaw of this system is that the fourth or fifth between the extreme notes of the series, E♭-G♯, will be out of tune: in the colorful language of intonation, a "wolf" interval. This complication arises because 12 perfect fifths do not round off to precisely an even octave, but exceed it by a small ratio known as a Pythagorean comma (see Section 4). Happily, since E♭ and G♯ rarely get used together in medieval harmony, this is hardly a practical problem. Since all intervals have integer (whole number) ratios based on the powers of two and three, Pythagorean tuning is a form of just intonation (see Section 5). More specifically, it is a form of just intonation based on the numbers 3 and 9. Thus we get just or ideally blending fifths (3:2), fourths (4:3), major seconds (9:8), and minor sevenths (16:9).

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4

Pythagorean Tuning In fact, Pythagorean tuning is described in the medieval sources as being based on four numbers: 12:9:8:6. Jacobus of Liege (c. 1325) describes a "quadrichord" with four strings having these lengths: we get an octave (12:6) between the outer notes, two fifths (12:8, 9:6), two fourths (12:9, 8:6), and a tonus or major second between the two middle notes (9:8). Other intervals can be derived from these, and the result in a medieval context is, by the 13th century, a subtle spectrum of interval tensions in practice and theory. The following table shows how the standard intervals of Pythagorean tuning except the pure unison (1:1) and octave (2:1) are derived primarily from superimposed fifths (3:2), thus having ratios which are powers of 3:2, or secondarily from the differences between these primary intervals and the octave. We show the 13 usual intervals of medieval music from unison to octave as listed by Anonymous I around 1290, and by Jacobus of Liege around 1325. (On some other intervals generated in tuning a complete chromatic scale, see Section 4.2.2.) Interval Unison minor Major minor Major Augmented minor Major minor Major Octave

Ratio nd

2 2nd 3rd 3rd 4th 4th 5th 6th 6th 7th 7th

Derivation

1:1

Unison 1:1

256:243

Octave M7

9:8 32:27 81:64 4:3 729:512 3:2 128:81 27:16 16:9 243:128 2:1

Cents* 0

90.22

(3:2) Octave M6

2

(3:2) Octave 5

4

(3:2)

6

611.73

(3:2) Octave M3

1

701.96

(3:2) Octave M2

3

(3:2) Octave 2:1

5

203.91 294.13 407.82 498.04

792.18 905.87 996.09 1109.78 1200

* For an explanation of cents, see Section 4.2

Figure 1:

Pythagorean intervals and their derivations

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.

3. Pythagorean tuning and Gothic polyphony In describing the musical practice of their time, theorists such as Johannes de Garlandia (c. 1240?), Franco of Cologne (c. 1260?), and Jacobus of Liege developed sophisticated scales of vertical tension. (For a fuller account, see also Thirteenth-century polyphony.) Fifths and fourths are the most complex stable intervals; major and minor thirds are relatively blending but unstable; major seconds and minor sevenths, along with major sixths, are rather more tense but somewhat compatible; and minor seconds, major sevenths, and tritones, often along with minor sixths, are regarded as strong discords. Here's a table showing the Pythagorean ratios of intervals on this spectrum: 29/09/15

5

Pythagorean Tuning Stability stable Marginal

Category 1 purely blending 5 Optimally blending M3 Relatively blending Relatively tense m7, M2 Most tense M7 (2

Unstable

Intervals (1:1)

8 (2.1)

(3.2)

4 (4.3)

(81:64)

m3 (32:27)

(16:9), ( 9:8)

M6 (27.16)

43:128)

m2 (256:243)

A4 (7 29:512)

d5 ( 1024:729)

m6 (128:81)

Figure 2:

Table of Dissonances

It bears emphasis that this theoretical scheme is based not only on mathematical logic but on pragmatic musical experience. Thus while the stable concords have the simplest ratios, it should be noted that relatively blending major and minor thirds (81:64, 32:27) actually have more complex ratios than the rather more tense major second and minor seventh (9:8, 16:9). In exploring how Pythagorean tuning interacts with and enhances harmonic style in this period, we may consider first the simple intervals, then "coloristic" harmony involving mildly unstable combinations, and finally the melodic and vertical aspects of cadential action.

3.1 Simple vertical intervals Pythagorean tuning involves three main nuances nicely tying in with the 13th century spectrum of intervallic tensions: it makes fifths and fourths ideally euphonious, unstable but relatively blending major and minor thirds somewhat more tense, and relatively tense major seconds and minor sevenths somewhat more blending.

3.1.1

Ideal fifths and fourths Being based on just or pure fifths, Pythagorean tuning optimizes fifths (3:2) and fourths (4:3), making it possible to obtain these intervals in their most harmonious ratios. Given that these intervals, with their opulence and clarity, define the stable basis of Gothic polyphony and lend their prevalent color to the texture, Pythagorean intonation well concords (pun intended) with this artistic style.

3.1.2

Active thirds and sixths In addition to presenting fifths and fourths in their ideal just ratios, Pythagorean tuning makes mildly unstable major thirds (81:64) and minor thirds (32:27) somewhat more active or tense. In a style where these intervals represents points of instability and motion standing in contrast to stable fifths and fourths, this extra bit of tension may be seen not only as a tolerable compromise, but indeed as an expressive nuance. The major sixth (27:16) and minor sixth (128:81) also take on a bit of an extra "edge," adding emphasis to cadential resolutions involving these intervals.

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6

Pythagorean Tuning 3.1.3

Milder major seconds and minor sevenths Pythagorean tuning makes the relatively tense major seconds (9:8) and minor sevenths (16:9) as blending as possible by presenting them in their ideal just ratios. Thus a major second is equal to precisely two pure (3:2) fifths less an octave, while a minor seventh is equal to precisely two pure (4:3) fourths. As early as the 11th century, Guido d'Arezzo recognizes M2 as a useful interval for polyphony, and both M2 and m7 play a prominent role in 13thcentury practice. While theorists of the period typically describe M2 and m7 as "imperfect discords" having some degree of "compatibility," Jacobus of Liege actually classifies them as "imperfect concords." By presenting these intervals in an ideal just ratio, Pythagorean tuning tends to bring out their more "compatible" or "concordant" side.

3.2 Coloristic harmony By making major and minor thirds a bit more tense, and major seconds and minor sevenths a bit more blending, Pythagorean tuning also affects the quality of some typical 13th-century multi-voice sonorities built by combining these intervals. The following points might be read in connection with a more general discussion of these combinations. Please note that symbols such as "8/5" or "5/4" are used here in their continuo meaning to identify a sonority as a set of intervals in relation to the lowest part, rather than to specify tuning ratios (here written as 8:5, 5:4, etc.).

3.2.1

Stable trines (8/5, 8/4) In 13th-century polyphony, the unit of complete three-voice harmony is the 8/5 trine consisting of an outer octave, lower fifth, and upper fourth. In the variant 8/4 form, the two adjacent intervals are arranged "conversely," with the fourth below and the fifth above. Pythagorean tuning yields ideal ratios for all intervals in these stable combinations. Using the medieval approach of string ratios, we have 6:4:3 for 8/5 and 4:3:2 for 8/4; using modern frequency ratios, we have 2:3:4 and 3:4:6 respectively. Thus Pythagorean tuning in a 13th-century context, like the rather different systems of just intonation favored in the Renaissance, presents the stable harmonies of the period in their most pure and restful aspect.

3.2.2

The "split fifth" (5/3) By adding a bit of extra tension to simple thirds (Section 3.1.2), Pythagorean intonation also influences the color of one of the most popular unstable combinations, the quinta fissa or "split fifth" of Jacobus with its outer fifth "split" by a third voice into a major third below and minor third above, or vice versa (5/M3 or 5/m3).

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7

Pythagorean Tuning While the outer fifths of these relatively blending combinations will be a just 3:2, the Pythagorean thirds somewhat accentuate the sense of instability: we get string ratios of 81:64:54 (5/M3) and 96:81:64 (5/M3), or frequency ratios of 64:81:96 and 54:64:81 respectively. Jacobus finds these combinations pleasing when aptly used, and Pythagorean tuning tends to bring out their relatively concordant but active nature.

3.2.3

Quintal/quartal sonorities (9/5, 5/4, 5/2, 7/4) While making the "split fifth" more tense, Pythagorean intonation makes another group of mildly unstable sonorities somewhat more blending: quintal/quartal sonorities combining fifths and/or fourths with a relatively tense M2, m7, or M9. All intervals in these relatively blending combinations will be presented in their just ratios: not only the ideally euphonious fifths (3:2) and fourths (4:3), but the unstable M2 (9:8), m7 (16:9), and M9 (9:4). Thus for 9/5 and 7/4, we get respective ratios of 9:6:4 and 16:12:9 (here the string and frequency ratios are the same); for 5/4, string and frequency ratios of 12:9:8 and 6:8:9; and for 5/2, ratios of 9:8:6 and 8:9:12. These sonorities, common in practice and endorsed by Jacobus in theory (both he and Anonymous I, possibly Jacobus at an earlier age, much recommend 9/5), present their most concordant face under a Pythagorean system of intonation.

3.3 Cadential action As in many periods and styles of European music, cadences in Gothic polyphony have both a melodic and a vertical dimension. Especially in progressions by contrary motion such as m3-1, M3-5, and M6-8, Pythagorean tuning affects both dimensions to make the resolution more dramatic and "incisive," as Mark Lindley has pointed out. For example, let us consider a very popular 13th-century cadence which becomes the most common close of the 14th century: e'- f' b - c' g -f M6- 8 M3- 5 (M6- 8+M3-5)

Figure 3a:

13th -14th century cadential motion

From a melodic point of view, Pythagorean tuning makes the descending major second (9:8) in the lowest voice handsomely wide, while also providing decisively narrow semitones (256:243) in both upper voices. 29/09/15

8

Pythagorean Tuning At the same time, in the vertical dimension, the wide unstable intervals of M3 (81:64) and M6 (27:16) not only gain a bit of extra tension but "stretch" more closely toward the fifth and octave respectively. Both the heightened sense of instability and the incisive resolution to a complete trine contribute to the total effect. While in this case the unstable intervals expand, Pythagorean tuning also enhances cadences where such intervals contract, as in the following resolution of a seventh combination common in the 13th century and also used by Machaut in the following century: d'- c' b - c' e'- f' M7- 5 M3- 5 (M7- 5+m3-1)

Figure 3b:

Cadential motion used by Guillame Machaut

In this case the lower two voices both ascend by decisive Pythagorean semitones, while the upper voice descends by a generous whole tone. The outer minor seventh (16:9) efficiently contracts to a stable fifth, while the upper minor third (32:27) likewise contracts to a unison. As it happens, the tuning characteristically makes the minor third more tense and the minor seventh more mild, while letting both intervals resolve incisively to stability. Thus while Pythagorean tuning in the West far predates the advent of the sophisticated multi-voice cadential formulae of the 13th and 14th centuries, it admirably fits the expressive nature of these progressions.

4. Pythagorean tuning in more detail 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.

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9

Pythagorean Tuning 4.1 Tuning a basic scale 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

Figure 4a:

3:2

1st Fifth in Lydian Mode: c'

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 × 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➺ 1:1

Figure 4b:

➺g➺ 9:8

➺c' 3:2

2nd Fifth in Lydian Mode: g'

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 "wholetone-plus-fifth": f➺ 1:1

Figure 4c:

29/09/15

➺g➺ 9:8

➺c'➺ ➺d' 3:2

27:16

3rd Fifth in Lydian Mode: d'

10

Pythagorean Tuning Were we to move up a fifth from d', we would arrive at a' a major 10th from f, 27:16 × 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➺ 1:1

Figure 4d:

9:8

➺a➺ 81:64

➺c'➺ ➺d' 3:2

27:16

4th Fifth in Lydian Mode: a'

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 × 3:2): f➺ ➺g➺ 1:1

Figure 4e:

9:8

➺a➺

➺c'➺

81:64

3:2

➺d'➺ ➺e' 27:16

243:128

5th Fifth in Lydian Mode: e'

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➺ 1:1

Figure 4f:

9:8

➺a➺

➺b➺

➺c'➺

81:64

729:512

3:2

➺d'➺ ➺e' 27:16 243:128

6th Fifth in Lydian Mode: b

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➺ 1:1

Figure 4g:

9:8

➺a➺

➺b➺

➺c'➺

81:64

729:512

3:2

➺d'➺

➺e'➺ ➺f'

27:16 243:128

2:1

7th Fifth in Lydian Mode: f'

We now have all eight tones of our basic mode, and our tuning process is finished for the moment. Before going on to add B♭ (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.

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11

Pythagorean Tuning 4.1.1

Additional Diatonic Intervals 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 × 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 × 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 multivoice 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 × 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 × 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 × 64:81, also yielding 32:27. This interval is known as the semiditonus, being smaller than the ditone or major third.

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12

Pythagorean Tuning 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 × 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 fg (9:8), i.e. 2:1 divided by 9:8 or 2:1 × 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 × 3:2, or 16:9. It is also sometimes defined as bis diatessaron, two fourths, or 4:3 × 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:

Tone

Interval to f

Ratio

f c' g d' a e' b

1 5 M2 M6 M3 M7 A4 8

1:1 3:2 9:8 27:16 81:64 243:128 729:512 2:1

f'

Figure 5a:

Notes f - f' and intervals derived directly from the tuning Tone

e'-f'

b-c

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

etc.

Figure 5b:

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etc.

Interval m2 m3 4 m6 m7

Derivations 8 8 8 8 8

-

M7・5 M6・5 5・5 M3・5 M2・5

+ +

Ratio A4 M3 M2 m2 m3

256:243 32:27 4:3 128:81 16:9

intervals derived from these scale element (f-f' )

13

Pythagorean Tuning 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 2:1×512:729), 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.

4.2 Ratios, cents, and the complete chromatic scale 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.

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14

Pythagorean Tuning In addition to simplifying 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 500cent 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 equallytempered 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.

4.2.1

An aside: calculating 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.

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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. 15

Pythagorean Tuning 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 21. Multiplying 1×1200, 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 = 25849625...). 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: log2 (a:b) = [logn (a:b)/log2 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 (or Naperian 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 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.

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16

Pythagorean Tuning 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 702×2 = 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 204×2 or 408 cents, and a tritone, (9:8)3, to about 204×3 or 612 cents. These intervals are somewhat wider than their equallytempered versions of four and six 100-cent semitones respectively. This multiplication approach reflects a basic property of logarithms: [log2 (a:b)n] = n [logn (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 2100/1200 or 21/12 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.

4.2.2 Completing a 12-tone scale: from apotome to "Wolf" 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

1:1

9:8

81:64

1.00

1.13

1.33

0

204 204

Figure 6a:

29/09/15

204

408

c'

d'

e'

729:512

3:2

27:16

243:128

2:1

1.42

1.50

1.69

1.90

2.00

612 204

90

702

906 204

204

f '

1110

1200 90

Whole Tones f-g-a-b-c'-d'-e'-f'

17

Pythagorean Tuning 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 B♭ - or, to keep within the range of our octave, b♭ a fourth up: f

g

a

b♭

b

c'

d'

e'

f '

729:512

3:2

27:16

243:128

2:1

1.42

1.50

1.69

1.90

1:1

9:8

81:64

1.00

1.13 1.27 204

1.33

0

204 204

Figure 6b:

408

498 90

114

612

702 90

204

906

1110

90

2.00

1200

204

f-g-a-b♭-b-c'-d'-e'-f'

In relation to f, b♭ 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-b♭ the same size as at b-c' or e-f'. This interval is equal to the difference between the fourth f-b♭ 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 × 64:81, getting 256:243. There remains a new kind of interval at b♭-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 b♭-f-c'-gd'-a-e'-b, we generate a new interval b♭-b of (3:2)7, or 729:512×3:2, 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, B♭ is an integral part of the medieval gamut, and likely was included on organ keyboards of the 10th or 11th century before other accidentals.

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18

Pythagorean Tuning 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. B♭-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. B♭-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 b♭ to the remaining flat, e♭ - or, to keep within our octave, up a fourth to e♭': f

g

1:1

1.00

0

a

9:8 81:64

b♭

b

4:3

729:512

1.13 1.27 204

1.33

204

498

114

Figure 6c:

408 90

114

c'

d'

e♭'

e'

3:2

27:16

16:9↓

243:128

1.42

1.50

612

702

204

1.69

906

90↑ 114↓

1.78

996

90↑ 114↓

114

f ' 2:1

1.90

2.00

1110

1200 90

f-g-a-b♭-b-c'-d'-e♭'-e-f'

In the flat portion of our journey, we have moved up two pure 4:3 fourths, fb♭ and now b♭-e♭', so that the resulting interval f-e♭' 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-e♭' (90 cents). We also have the expected 2187:2048 or 114-cent apotome e♭'-e', and a new and yet more complex interval between the extreme notes of our chain of eight fifths, e♭'-b♭-f-c'-g-d'-a-e'-b, that is b-e♭'. Here a slight complication is that the range of our f-f' scale has placed our E♭ above our B♭, so that b-e♭' is actually the octave complement of the usually expected interval such as e♭-b. Let us first find the expected interval, and then subtract it from an octave to find our actual interval in this scale.

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19

Pythagorean Tuning Taking (3:2)8, or 2187:2048, we get a ratio of 6561:4096 for e♭-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 e♭-e (114 cents), or again 816 cents. To find the ratio of our actual interval b-e♭, 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-e♭' as the perfect fourth b-e (498 cents) less the apotome e♭'-e' (114 cents), or 384 cents. From the viewpoint of a theorist such as Jacobus of Liege following 13thcentury 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 b♭ and e♭, 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♯: f

f♯

1:1

2187:2048 1.07 90

1.13

1.27

114

204

408

1.00

0 114

g

a

9:8 81:64

204

Figure 6d:

90

b♭

b

c'

d'

e♭'

e'

4:3

729:512

3:2

27:16

16:9↓

243:128

1.33

498

1.4238 1.50 90↑ 114↓

612

114↑ 90↑

1.6875 1.7778 90↑ 114↓

702

906 204

996

1.8984

1110

f ' 2:1 2.00

90

1200

114↓ 90↑

f-f♯-g-a-b♭-b-c'-d'-e♭'-e'-f'

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 (b♭-a, e♭'-d') and ascent from sharps (f♯-g). Our ninth fifth also gives rise to a novel interval, f♯-e♭'; 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, e♭-f♯. The latter form has a ratio equal to (3:2)9, or the tetratone of (3:2)8 plus a fifth, 6561:4096 × 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 e♭-f♯ as a Pythagorean augmented second: it consists of the apotome e♭-e, the diatonic semitone e-f, and the apotome f-f♯, and thus a rounded (114 + 90 + 114) or 318 cents. 29/09/15

20

Pythagorean Tuning To find the ratio of our actual interval f♯-e♭', the octave complement of e♭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'-e♭' 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-e♭', 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♯': f 1:1

1.00

0

f♯/g♭

g

a

2187:2048 9:8 1.0678711 1.125 90↑ 114↓

114↑•90↓

114↑ 90↓

204

b♭

81:64 4:3

1.2656 1.333 90

408

204

Figure 6e:

498

b

c'

c♯'

d'

e♭'

e'

729:512

3:2

4:3

27:16

16:9↓

243:128

996↑ 1020↓

1110

1.423828

612 114

90

1.50

1.3333

702

816 114

90

1.6875

906

f ' 2:1

1.777778 1.898438 114↑ 90↓

90↑ 114↓

2.00

1200 90

f-f♯-g-a-b♭-c'-c♯'-d'-e♭'-e'-f'

Between our original f and c♯' we have a tetratone, a species whose octave complement we have already encounted at b-e♭'. We now add to our "musical bestiary" a new interval between the extreme notes in the chain, c♯'-e♭', again an octave complement of the form with the note at the flat end of the chain below, e♭-c♯'. The latter form e♭-c♯', our Pythagorean augmented second e♭-f♯ plus a fifth, has an imposing ratio of (3:2)10 or 19683:16384 × 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 - e♭-e, c'c♯' - giving a total of about (114 + 792 + 114) or 1020 cents. Alternatively, we To find the ratio of our actual interval c♯'-e♭', 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. 29/09/15

21

Pythagorean Tuning Finally, we are ready to add the last of our 12 tones, g♯: f

f♯/g♭

g

1:1

2187:2048 9:8 1.00 1.07 1.13 90↑ 114↓ 0

114↑•90↓

114↑ 90↓

g♯

a

b♭

19683:1638481:64 4:3 1.20 1.27 1.33 90

204

318 114

408

498 90

Figure 6f:

b

c'

c♯'

d'

e♭'

e'

729:512

3:2

4:3

27:16

16:9↓

243:128

906

996↑ 1020↓

1110

1.42 114

1.50

612

702

114

1.33

1.69 90•114

816

90

90

f ' 2:1

1.90 1.90 114↑ 90↓

114↓

2.00 1200

90

step-by-step tuning f-f'

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♯e♭', or placing the note at the flat end of the chain below, e♭-g♯. This interval has a special significance The e♭-g♯ form has the not inconsiderable ratio of (3:2)11, or of our pentatone plus a fifth, 59049:32768×3:2, 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. e♭-e e-f

apotome diatonic

114

f-g

whole-tone

204

apotome

114

g-g♯

Figure 7:

••••••••••••••••••••

90

522

The Not-So-Perfect Fourth

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 e♭-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 e♭-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♯-e♭', 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)

29/09/15

22

Pythagorean Tuning 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♯-e♭ includes the fourth a-d' plus the two diatonic semitones g♯-a and d'-e♭', 90 cents each, or 498 + 180 = 678 cents. This imperfectly small fifth g♯-e', and likewise the imperfectly large fourth e♭'-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 E♭ very rarely occur together. However, during the Renaissance, while a G♯-E♭ 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, E♭-B♭-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.

4.3 Modes, scales, and measures 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 B♭ (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. 29/09/15

23

Pythagorean Tuning 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.

4.3.1 Tuning data and the chromatic octave of C 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 with respect to c'

with respect to a'

Note

Herz

c' c♯' d' e♭' e' f' f♯' g' g♯' a' b♭' b' c'

260.74

1:1

278.44

2187:2048

113.69

293.33

9:8

203.91

3.91+

309.03

32:27

294.13

330.00

81:64

347.65

ratio

cents

ratio

cents

16:27

-905.87

5.87-

13.69+ 81:128

-792.18

7.82+

-701.96

1.96-

5.87- 512:729

-611.73

11.73-

407.82

7.82+

3:4

-498.04

1.96+

4:3

498.04

1.96-

64:81

-407.82

7.82-

371.25

759:512

611.73

11.73+

27:32

-294.13

5.87+

391.11

3:2

701.96

1.96+

8:9

-203.91

3.91-

417.66

6561:4096

815.64

-90.22

9.78+

440.00

27:16

905.87

5.87+

0.00

0.00±

463.54

16:9

996.09

3.91- 256:243

90.22

9.78-

495.00

243:128

1109.78

9.78+

9:8

203.91

3.91+

521.48

2:1

1200.00

0.00±

32:27

294.13

5.87-

0.00

±12tet 0.00±

2:3

15.64+ 243:256 1:1

±12tet

frequencies with a'=440, and variances from 12tet

Figure 8:

Pythagorean Tuning with variances with 12 Tone Equal Temperament

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'.

29/09/15

24

Pythagorean Tuning 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 ': 9:8

1:1 2187:2048

19683:1638481:64

1.0

1.068

1.125

1.20

1.27

c'

c♯'

d'

e♭'

e'

0

114

204

318

90

114

204

114

90

204

0

114

204

318

c'

c♯'

d'

e♭'

9:8

19683:16384

204

1.0 2187:2048

408

4:3

729:512

3.2

1.33

729:512

3:2

f'

90

f♯'

114

498 204

612 204

408

498

612

e'

f'

f♯'

180

81:64 4:3

Figure 9:

729:512

90

g'

702

4:3

27:16 6561:4096 243:128

4:3

27:16

816

906

g♯' 114

204

a'

90

16:9↓

243:128

1020

1110

b♭'

114

2:1

204

2.0

b'

90

c ''

90

1200

204

702

816

906

1020

1110

1200

g'

g♯'

a'

b♭'

b'

c ''

3.2

4:3

204

204

27:16 6561:4096 243:128

2:1

step-by-step tuning c' - c''

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 e♭'-g♯' rather than a 678-cent Wolf fifth g♯-e♭'. 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.

4.3.2 Medieval modes in Pythagorean tuning 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. 29/09/15

25

Pythagorean Tuning 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 B♭ 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 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 fc' 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

d

e

0

204

32:27

4:3

f

g

90

294

204

498

204

3.2

27:16

a

b

702

204

906

90

16:9

2:1

c'

d'

996

204

1200 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

90

294

204

204

498

204

702

90

792

204

996

1200 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

90

29/09/15

204

204

204

90

204

996

1200 204

26

Pythagorean Tuning Mode IV (Hypophrygian) F

1:1

256:243

B

c

0 90

90

32:27

4:3

d

e

204

294

204

498

90

1024:729

128:81

f

g

588

204

792

204

16:9

2:1

a

b

294

204

498

Mode V (Lydian) F

1:1

9.8

f

g

0

204

204

81:64

729:512

a

204

408

b

204

612

3:2

27:16

702

906

c' 90

243:128

d'

204

2:1

e'

204

1100

f' 1200 90

Mode VI (Hypolydian) 1:1

9.8

c

d 204

0

F

4:3

81:64

e 204

3:2

f

408

204

498

27:16

g

204

a

702

90

906

2:1

243:128

b

204

c'

1100

204

1200 90

Mode VII (Mixolydian) F

1:1

9.8

g

a 204

0

4:3

81:64

b 204

3:2

c'

408

204

498

27:16

d' 204

e'

702

90

906

2:1

16:9

f'

90

g'

996

204

1200 204

Mode VIII (Hypomixolydian) 1:1

9.8

g

a 204

0

F

32:27

4:3

b

c'

90

294

204

498

3:2

27:16

d' 204

e'

906

702

204

2:1

16:9

f'

90

g'

996

204

1200 204

Mode IX (Aeolian) F

1:1

9.8

a

b 204

0

32:27

4:3

c'

d'

90

294

204

498

204

3:2

128:81

e'

f'

702

204

792

204

16:9

2:1

g'

a'

996

90

1200 204

Mode X (Hypoaeolian) F

1:1

256:243

e

f 90

0

32:27

4:3

g

a

204

294

90

498

204

3:2

128:81

b

c'

702

204

792

204

16:9

2:1

d'

e'

996

90

1200 204

Mode XI (Ionian) F

1:1

9.8

c'

d' 90

0 204

29/09/15

81.64

4:3

e'

f'

204

294

498 90

204

3:2

27.16

g'

a'

702

906 204

243:128

2:1

b'

c''

204

1100

1200 90

27

Pythagorean Tuning Mode XII (Hypoionian) F

1:1

9.8

81:64

4:3

3:2

27:16

16:9

2:1

c'

d'

e'

f'

g'

a'

b'

c''

0

204

408

498

702

906

90

204

204

204

90

204

996

1200 204

Figure ??: Medieval modes

29/09/15

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