Musical Mathematics

on the art and science of acoustic instruments

 

Table of Contents

 

 

Buy now at  Amazon.com: click New or Used.

 

Also available from the publisher at Chronicle Books, San Francisco.

 

© 2000–2017 Cristiano M.L. Forster
All rights reserved.

 

www.chrysalis-foundation.org

 

 

CHAPTER 10: WESTERN TUNING THEORY AND PRACTICE

 

Part II: Greek classification of ratios, tetrachords, scales, and modes

 

The following section provides background information for subsequent discussions on

Philolaus and Euclid.

 

Section 10.6

         Before the time of the semi-legendary musician Terpander (b. c. 710 b.c.), Greek lyres were built with four strings. Musicians and theorists tuned these strings to tetrachords, or to simple scales that span the interval of a “fourth.”[1] According to Ptolemy (c. a.d. 100 – c. 165), Archytas (fl. c. 400 b.c.) identified three different kinds of tetrachords: the diatonic genus, the chromatic genus, and the enharmonic genus.[2] These three genera also appear in a treatise entitled The Elements of Harmony[3] by Aristoxenus (fl. c. 300 b.c.). The Greeks regarded the first and fourth tones of a given tetrachord as fixed tones, and always tuned them to ratios 1/1 and 4/3, respectively. The second and third tones were considered moveable tones and, therefore, not limited to a specific tuning. Most important, the tuning of the third string, or the interval between the third and fourth tones, characterized the genus of a particular tetrachord. However, for a given genus, musicians did not restrict this interval to only one tuning. For the diatonic genus, the interval was tuned to various “major tones,” for the chromatic genus, to various “minor thirds,” and for the enharmonic genus, to various “major thirds.” Figure 10.1 shows a typical tetrachord for each genus. Here circles indicate fixed tones, dots indicate moveable tones, and the characteristic interval of each tetrachord appears in bold print. We may think of these tetrachords as occurring between E–A, B–E, and A–D in the Western diatonic scale.

 

 

          According to legend, Terpander joined two diatonic tetrachords by a common tone and, thereby, created a heptachord that consists of two conjunct tetrachords:

 

 

Approximately a century later, Pythagoras (c. 570 b.c. c. 500 b.c.) reputedly joined two diatonic tetrachords by inserting a 9/8 “tone” between 4/3 of the lower tetrachord, and 1/1 of the upper tetrachord.[4] The result is an octachord, or a scale comprising of two disjunct tetrachords:

 

 

During the next two hundred years, Greek musicians and theoreticians continued to combine disjunct and conjunct tetrachords in the design of more complicated scales. Two famous scales emerged from this process: the larger scale was called the Greater Perfect System, and the smaller scale, the Lesser Perfect System. Of the Greek texts that survived, the first complete description of the GPS may be found in Euclid’s Division of the Canon.[5]  Figure 10.2 shows that the GPS consists of a 9/8, two conjunct tetrachords, another 9/8, and two more conjunct tetrachords, or fifteen strings that span the interval of a “double-octave.”[6] Figure 10.2 also shows that the LPS consists of a 9/8, and three conjunct tetrachords, or eleven strings that span the interval of an “octave and a fourth.” As in Figure 10.1, circles represent fixed tones, and dots, moveable tones. Finally, note that in the GPS, the standard Greek “octave” occurs between E–E'.

 

 

          The Greeks tuned the tetrachords of the GPS and LPS to either a diatonic, a chromatic, or an enharmonic genus. Figure 10.3 shows three tunings of the GPS based on the tetrachords in Figure 10.1. Since Greek musicians used rational numbers in tuning theory, these notes are only approximations. The rational frequency ratios of the Greek genera should never be confused with the irrational frequency ratios of the conventional 12-tone equal tempered scale. This discrepancy applies especially to the last genus in Figure 10.1. Note a 32/31 “quarter-tone” interval [55.0 ˘] between the first and second tones, and a 31/30 “quarter-tone” interval [56.8 ˘] between the second and third tones of this enharmonic genus. In the last example of Figure 10.3, the notation for the second tone of each tetrachord includes a modern “quarter-tone sharp” sign. In this context, the sign should read an “approximate quarter-tone sharp,” because an equal tempered “quarter-tone” equals 50.0 ˘ exactly. Also in this figure, white notes represent fixed tones, and black notes, moveable tones.

 

 

          Finally, the Greeks also identified seven harmoniai or modes within the GPS. Figure 10.4(a) shows the ancient original names of the modes based on the diatonic genus in Figure 10.1. However, during the Middle Ages, monks gave these modes the medieval ecclesiastical names shown in Figure 10.4(b). Most musicians use these names to this day. In ascending order, the diatonic tetrachord consists of the following three intervals: 1 “semitone” (or 1 “half-tone”), 2 “semitones” (or 1 “whole tone”), and 2 “semitones” (or 1 “whole tone”). In Section 10.21, we will examine the supreme importance of the ancient Lydian Mode in Western music theory. (For a graphic of Ptolemy's Tense Diatonic in the Lydian Mode, please see M.M. Pages > Ibn-Sina, Stifel, and Zarlino > Footnote 33.)

 

 

 

 

CHAPTER: WESTERN TUNING THEORY AND PRACTICE

Part IV: Philolaus, Euclid, Aristoxenus, and Ptolemy

Section 10.10

          Recall from the discussion in Section 10.6 that the first and last tones of a tetrachord are fixed and always sound the interval of a “fourth”; in contrast, the inner two tones are moveable according to the precepts of three different genera. Because the Greeks considered many different kinds of diatonic scales, in the following passage Nicomachus (b. c. a.d. 60) quotes Philolaus (fl. c. 420 b.c.) — the most venerated Pythagorean of antiquity — as an authority on the tuning of the diatonic tetrachord: 

* * * *

The magnitude of harmonia is syllaba and di’oxeian. The di’oxeian is greater than the syllaba in epogdoic ratio. From hypate [E] to mese [A] is a syllaba, from mese [A] to neate [or nete, E'] is a di’oxeian, from neate [E'] to trite [later paramese, B] is a syllaba, and from trite [B] to hypate [E] is a di’oxeian. The interval between trite [B] and mese [A] is epogdoic [9:8], the syllaba is epitritic [4:3], the di’oxeian hemiolic [3:2], and the dia pason is duple [2:1]. Thus harmonia consists of five epogdoics and two dieses; di’oxeian is three epogdoics and a diesis, and syllaba is two epogdoics and a diesis.[7] (Text and ratios in brackets mine.) 

* * * *

Philolaus spoke a Greek dialect called Doric, which explains his unusual vocabulary.[8] In this passage, Philolaus defines three intervals in ratio form:

 

 

Two more interval descriptions are consistent with the latter definitions:

 

 

Refer to Figure 10.1, and note that Philolaus calls the smallest interval of the diatonic genus, ratio 256/243, a diesis.[9] With the exception of Philolaus, all ancient Greek and modern theorists refer to ratio 256/243 as a limma (lit. remainder), or the interval that remains after one subtracts two “whole tones” from a “fourth”:

 

  

 

Finally, Philolaus’ last sentence states

 

 

          Consider now Philolaus’ harmonia in the context of the GPS. Begin by distributing five 9/8 ’s and two 256/243’s according to his note name descriptions. Figure 10.8 shows that Philolaus’ diatonic scale consists of two disjunct tetrachords that span the standard Greek “octave” between E–E'.

 

 

 

According to Figure 10.4, Figure 10.8 shows this scale in the Dorian Mode. For a second perspective, refer to Figure 10.9, which illustrates Philolaus’ diatonic scale distributed over the entire GPS, and in the context of the Dorian and Lydian Modes.

 

 

 

          With respect to the two moveable tones of the lower tetrachord, interval ratio 9/8 that descends from 4/3 requires ancient length ratio 4/3 ÷ 9/8 = 32/27; and interval ratio 9/8 that descends from 32/27 requires ratio 32/27 ÷ 9/8 = 256/243. To calculate these two locations on the previously mentioned canon, substitute 1000.0 mm and these ratios into Equation 3.32, and make two calculations:

 

 

 

For the upper tetrachord, make two similar calculations for 16/9 and 128/81.

          Finally, observe that in the Dorian Mode, Philolaus’ diatonic scale forms a sequence of five descending 3/2’s, and in the Lydian Mode, a sequence of five ascending 3/2’s. If we simplify “octave” equivalents (see Chapter 9, Section 4), the following progressions emerge:

 

  Dorian Mode: 2/1 [E], 4/3 [A], 16/9 [D], 32/27 [G], 128/81 [C], 256/243 [F]

 

Lydian Mode: 1/1 [C], 3/2 [G], 9/8 [D], 27/16 [A], 81/64 [E], 243/128 [B]

 

In Europe, variations of this scale lasted well into the 15th century. Why? Because despite its formidable appearance, musicians can quickly and accurately tune this scale by ear; that is, without the aid of a monochord.

          Before we examine a tuning sequence for the Lydian Mode (or for the standard Western C–C' "octave" range), let us first calculate the frequencies of the complete 8-tone scale. Suppose C4 is tuned to 260.0 cps. According to Equation 3.30, the remaining frequencies are

 

 

 

To achieve these frequencies by ear, first tune F4, G4, and C5 as 4/3, 3/2, and 2/1 above C4, respectively. Now, tune D4, A4, E4, and B4 through a sequence of descending “fourths,” and ascending “fifths,” starting on G4 at 390.0 cps:

 

 

To this day, harpsichord and piano tuners use a similar technique of descending “fourths” and ascending “fifths” in the tuning of their instruments.[10] Undoubtedly, the ancient Greeks also employed such methods in tuning scales on the open strings of lyres, kitharas, and triangular harps.

 

Section 10.11 

          The earliest known mathematical description of a systematic canon tuning is contained in a work entitled Division of the Canon, reputedly written by Euclid (fl. c. 300 b.c.). The last two propositions of this treatise accurately describe the string divisions that produce a 15-tone “double-octave” tuning in the diatonic genus of the GPS. In the following passages, Euclid duplicates the intervals of the lower “octave” between A–A' in the upper “octave” between A'–A''.

 

* * * * 

Proposition 19: To mark out the kanon according to the so-called immutable systema

Let there be a length of the kanon which is also the length AB of the string, and let it be divided into four equal parts, at C, D and E. Therefore BA, being the lowest, will be the bass note. Now this AB is the epitritic [Greek, lit. 1 + 1/3, or 4:3] of CB, so that CB will be concordant [consonant] with AB at the fourth above it. And AB is proslambanomenos [A]: therefore CB is diatonos hypaton [or lichanos hypaton, D]. Again, since AB is double [2:1] BD, BD will be concordant with AB at the octave, and BD will be mese [A']. Again, since AB is quadruple [4:1] EB, EB will be nete hyperbolaion [A''].

     I cut CB in half at F. CB will be double FB, so that CB is concordant with FB at the octave: hence FB is nete synemmenon [should read paranete diezeugmenon, D']. From DB I subtracted DG, a third part of DB. DB will be the hemiolic [Greek, lit. 1 + 1/2, or 3:2] of GB, so that DB will be concordant with GB at the fifth. Therefore GB will be nete diezeugmenon [E'] .

     I then constructed GH, equal to GB, so that HB will be concordant with GB at the octave, making HB hypate meson [E]. From HB I subtracted HK, a third part of HB. HB will be the hemiolic of KB, so that KB is paramese [B']. I marked off LK, equal to KB, and LB will be the lower hypate [B]. Thus we shall have found on the kanon all the fixed notes of the immutable systema

 

Proposition 20: It remains to find the moveable notes

I divided EB into eight parts, and I constructed EM, equal to one of the parts, so that MB is the epogdoic [Greek, lit. 1 + 1/8, or 9:8] of EB. Next, I divided MB into eight parts, and constructed NM, equal to one of these parts. Thus NB is a tone lower than BM, and MB is a tone lower than EB, so that NB will be trite hyperbolaion [F'], and MB will be diatonos hyperbolaion [or paranete hyperbolaion, [G']. I took a third part of NB and constructed NX, so the XB is the epitritic of NB, and is concordant with it at the fourth below: XB is trite diezeugmenon [C']. Again, I took a half of XB and constructed XO, so that OB is concordant at the fifth with XB: therefore OB will be parhypate meson [F]. And I constructed OP, equal to XO, so that PB becomes parhypate hypaton [C]. Finally, I found CR, a fourth part of BC, so that RB becomes diatonos meson [or lichanos meson, G].[11] (Text and ratios in brackets mine.)

* * * * 

 

          Euclid’s text contains several technical difficulties. (1) His reference to nete synemmenon [D'] is incorrect. Since tetrachord synemmenon and tetrachord diezeugmenon do not exist in the same tuning system (see Figure 10.2), and since Euclid duplicates the intervals of the lower “octave” in the upper “octave,” only the tones of tetrachord diezeugmenon are admissible. Hence, we must change nete synemmenon [D'] to read paranete diezeugmenon [D']. (2) In Proposition 19, Euclid gives the impression he is describing fixed tones because in the heading of Proposition 20, he addresses the remaining moveable tones. However, in Proposition 19 he defines lichanos hypaton (CB), ratio 4/3, and paranete diezeugmenon (FB), ratio 8/3, which, in the context of the GPS, are not fixed tones, but moveable tones. (3) Finally, note carefully that Euclid refers to the third tones of tetrachords hypaton, meson, and hyperbolaion not as lichanos hypaton, lichanos meson, and paranete hyperbolaion, respectively, but as diatonos hypaton, ratio 4/3, diatonos meson, ratio 16/9, and diatonos hyperbolaion, ratio 32/9. Compare the diatonic genus in Figure 10.1 to the distribution of tetrachords in Figure 10.10. Observe that the latter three tones — 4/3 [D], 16/9 [G], 32/9 [G'] — produce the characteristic diatonic interval, ratio 9/8, below the fourth tone of their respective tetrachords. Therefore, Euclid substituted the term diatonos to identify these tetrachords as belonging to the diatonic genus, that is

 

 

 

 

 

          In Euclid’s text, fourteen bridge locations depend on previously calculated results, which means he achieved his scale through a series of interdependent divisions. For example, Euclid states that from DB he subtracted DG, or a third part of DB, where DB represents the “octave,” ratio 2/1. Consequently, Figure 10.11(a) shows that on our Sample String, GB has a length of 500 mm (500.0 mm ÷ 3) = 333.3 mm. This example shows the process of shortening a string section by dividing a previously calculated length into three aliquot (exact) parts and subtracting one of those parts; the result is an ascending “fifth,” interval ratio 3/2, from A' to E'. We conclude, therefore, that DB is the hemiolic [1 + 1/2] of GB, where DB has a length of 333.3 mm + (333.3 mm ÷ 2) = 500.0 mm. In contrast, Euclid also states that MB is the epogdoic [1 + 1/8] of EB, where EB represents the “double-octave,” ratio 4/1. Consequently, Figure 10.11(b) shows that MB has a length of 250.0 mm + (250.0 mm ÷ 8) = 281.3 mm. This example shows the process of lengthening a string section by dividing a previously calculated length into eight aliquot parts and adding one of those parts; the result is a descending “whole tone,” interval ratio 9/8, from A'' to G'. Euclid then states that XB is the epitritic [1 + 1/3] of NB, where NB represents the “minor sixth,” ratio 256/81, in the upper “octave.” Consequently, Figure 10.11(c) shows that XB has a length of 316.4 + (361.4 mm ÷ 3) = 421.9 mm. Again, this example demonstrates the process of lengthening a string section; the result is a descending “fourth,” interval ratio 4/3, from F' to C'.

  

 

Figure 10.12 illustrates the string divisions in the order described by Euclid. Here, dots with arrows point to previously calculated divisions. Fourteen proportions in the left column show that Euclid’s scale requires only five different length ratios, namely: 4/1, 2/1, 3/2, 4/3, and 9/8.

 

 

          With respect to length MB, ratio 32/9 in the upper “octave,” the utilization of length ratio 9/8 is especially noteworthy. Euclid could have produced G' by first tuning D' — a “fifth,” ratio 3/2 — down from 4/1, and then tuning G' — a “fourth,” or ratio 4/3 — up from 8/3:

 

 

However, such a procedure would have violated the practice of defining the genus of a given tetrachord by tuning the characteristic interval as an interval that descends from the fourth tone to the third tone of the tetrachord. In this case, length ratio 9/8 that descends from 4/1 to 32/9 defines the diatonic genus:

 

 

Table 10.4 gives a detailed analysis of the calculations of the ratios below the bridges in Figure 10.12. As described in Chapter 3, Section 7, I use arrows in this table to indicate the inverse proportionality between modern length ratio x/n and frequency ratio n/x, or l.r. x/n f.r. n/x. This technique provides a method of analysis that is universally applicable to all tuning descriptions based on string length divisions. For this reason, tables such as this also appear in Chapter 11 on world tunings. Finally, as discussed in Chapter 3, Section 20, since ancient length ratio n/x is indistinguishable from frequency ratio n/x, Table 10.4 represents an attempt to simplify the language of string division and ratio calculation. Provided we remember that in the writing of most creative theorist-musicians, (1) frequency ratios are a function of length ratios, and that in the context of interval divisions, (2) we must carefully distinguish between ancient length ratio n/x and frequency ratio n/x, simplicity of language greatly enhances comprehension.

 

 


 

[1]Chalmers, J.H., Jr. (1993). Divisions of the Tetrachord. Frog Peak Music, Hanover, New Hampshire.

 

     Chalmers cites literally hundreds of tetrachord divisions, mostly by Western theorists.

 

 

[2]Barker, A., Translator (1989). Greek Musical Writings, Volume 2, pp. 303–304. Cambridge University Press, Cambridge, England.

 

 

[3]Macran, H.S., Translator (1902). The Harmonics of Aristoxenus, p. 198. Georg Olms Verlag, Hildesheim, Germany, 1990.

 

 

[4]See Section 10.7 for a detailed discussion on the early development of the heptachord and octachord.

 

 

[5]Barbera, A., Translator (1991). The Euclidean Division of the Canon, p. 187. University of Nebraska Press, Lincoln, Nebraska.

 

 

[6]West, M.L. (1992). Ancient Greek Music, pp. 219–222. The Clarendon Press, Oxford, England, 1994.

 

     The Greek names for the tones in the GPS describe strings and their position on the lyre. For example, lichanos refers to the forefinger string.

 

 

[7](a) Greek Musical Writings, Volume 2, pp. 36–38.

 

     Barker identifies this quotation as Fragment 6.

 

 

(b) Levin, F.R., Translator (1994). The Manual of Harmonics, of Nicomachus the Pythagorean, p. 125. Phanes Press, Grand Rapids, Michigan.

 

 

[8]Burkert, W. (1962). Lore and Science in Ancient Pythagoreanism, p. 394. Translated by E.L. Minar, Jr. Harvard University Press, Cambridge, Massachusetts, 1972.

 

     “These considerations, along with the archaic terminology, allow us to regard Philolaus’ Fragment 6 as one of the oldest pieces of evidence for Greek music.”

 

 

[9]Modern theorists refer to the discrepancy between an “octave” and four “minor thirds” as a large diesis (lit. large separation). Four “minor thirds” = 6/5 × 6/5 × 6/5 × 6/5 = 1296/625. Large Diesis = 1296/625 ÷ 2/1 = 648/625 = 62.6 ˘. And they refer to the discrepancy between an “octave” and three “major thirds” as a small dieses. Three “major thirds” = 5/4 × 5/4 × 5/4 = 125/64. Small Diesis = 2/1 ÷ 125/64 = 128/125 = 41.1 ˘. The large diesis is a discrepancy by which four “minor thirds” exceed an “octave,” and a small dieses is a discrepancy by which an “octave” exceeds three “major thirds.”

 

 

[10]White, W.B. (1917). Piano Tuning and Allied Arts, 5th ed., pp. 86–87. Tuners Supply Company, Boston, Massachusetts, 1972.

 

 

[11](a) Greek Musical Writings, Volume 2, pp. 205–207.

 

(b) The Euclidean Division of the Canon, pp. 179–185.