Musical Mathematics

on the art and science of acoustic instruments


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Part VI: Just Intonation


Section 10.36


In the West, the modern history of just intonation begins with Bartolomeo Ramis (c. 1440 – c. 1500), who published a treatise in 1482 entitled Musica practica. Ramis proposed a 12-tone chromatic scale, which, for the first time in European music history,[1] included six 5-limit ratios. Although Ramis spent the most productive time of his life in Italy, he was born in a small town called Baéza in southern Spain.[2] After the Moslems invaded Spain in 711, they referred to this region as al-Andalus, a word that is etymologically “. . . connected with the name of the Vandals, who had occupied the land before the Arabs.[3] The Moslems controlled various areas of Andalusia until 1492. During this eight hundred year reign, Arabian artists, scientists, and scholars created magnificent cultural centers in the cities of Cordova, Granada, and Seville. Henry George Farmer (1882–1965), an eminent historian of Arabian music, recounts


* * * *


Both Adelard of Bath and Roger Bacon had advocated that European students should go to the Arabic fountain head in the schools of Muslim Spain. That some did go is evident from a statement of Ibn al-Hijari (12th century) who said that ‘students from all parts of the world flocked to Cordova’ to sit at the feet of Arabic scholars.[4]

* * * *


As discussed in Chapter 11, Part IV, Al-Farabi, Ibn Sina, and Safi Al-Din all wrote monumental music treatises in which they classified ratios 5/4 and 6/5 as bona fide consonances. To most music historians the fact that Ramis lived the first thirty-two years of his life in Andalusia before he left for Italy c. 1472,[5] and the fact that he was the first European music theorist to advocate the systematic implementation of 5-limit ratios, is a pure coincidence not worthy of investigation. However, the highly likely possibility exists that Ramis not only learned a great deal of musical mathematics from his Arabian counterparts, but that his famous 5-limit chromatic scale has its theoretical origins in Safi Al-Din’s 17-tone ‘ud tuning.


          Although Ramis’ original monochord in A includes a traditional Pythagorean “major third” between A–C#, ratio 81/64, Ramis broke with European orthodoxy when he specified


            5/4’s between Bb–D, C–E, F–A', and G–B',


6/5’s between A–C, D–F, and E–G,


 5/3’s between C–A', F–D' and G–E',


            8/5’s between D–Bb', E–C', A–F, and B–G.


Recall from the discussion in Section 10.26 that Ramis rejected the 3-limit Pythagorean scale because “. . . although it is useful and pleasing to theorists, it is laborious and difficult for singers . . .”[6] This statement clearly defines Ramis’ intent. At the above-mentioned locations, Ramis substituted simple 5/4’s for complex 81/64’s, simple 6/5’s for complex 32/27’s, simple 5/3’s for complex 27/16’s, and simple 8/5’s for very complex 128/81’s. Ramis included these 5-limit consonances because he observed and experienced the natural tendencies of singers, and decided to reject conventional but impractical Pythagorean music theory that excludes all prime numbers except 2 and 3. Again, remember that this breach occurred approximately two hundred years before the discovery of the harmonic series, which means that Ramis did not have the benefit of science to corroborate his theory. Franchino Gaffurio (1451–1522), and other contemporary music theorists, cited ancient texts to demonstrate that Ramis’ 5-limit chromatic scale subverted venerated Pythagorean traditions.[7] However, these objections soon evaporated, and as a consequence, singing as an art became a more enjoyable experience.


          Ramis’ monochord consists of two separate parts. He begins by describing the first 16 length ratios of his 12-tone “double-octave” scale:


Due to Arial font limitations, in the following translation and discussion,

 Ħ represents Ramis' "square" symbol, and the modern "natural" symbol.

 See Figures 10.30 and 10.31, and Table 10.21.




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     Accordingly, let a string of any length be used which may be extended over a somewhat concave piece of wood. Let the end to which the string is tied be shown by point a; let the other place, at the opposite end, to which the string is drawn and tightened, be shown by point q. Then let the distance q a, that is, the length of the entire string, be divided into two equal parts, and let the mid-point be written with the letter h. Again we will divide the length of the string h a in half and put d in the middle of the division. The length h d again will be divided and f is placed in the middle of the division.


     Note that the same is also to be done with the other half of the string, namely, h q, for in the first division letter p will be written in the middle; in the division of h p the letter l will be placed equidistantly, and keeping the same intervallic rule let n be placed between l and p. Then when we have divided f n in half we will write the letter i.


     We shall not go further through this median division to the smaller segments until we have made other divisions, but now we will divide the entire string a q into three parts, and measuring from q, at the [one-] third part m is placed and e at two-thirds of the string. Then let e q again be divided into three parts, and going from q to e let the sign square Ħ be placed at two-thirds of the distance; where the length of square Ħ and q is doubled let b be written.


     Then we will again divide m h in half and at the middle point we will put the letter k. Now we duplicate the length of k q and place c at the end of the duplication; but equidistant between e and square Ħ let letter g be placed. Then if we divide g into two equal parts letter o will be put in the middle; thus the entire monochord has been divided in a legitimate partition . . .[8] (Text in brackets mine.)


* * * *


          Figure 10.30 shows these divisions in the order described by Ramis; it also includes the ancient length ratios or modern frequency ratios of this scale, and the modern note names as well. Note carefully that almost all bridge locations depend on previously calculated results. Because of this, a single error near the beginning of Ramis’ description will cause many subsequent mistakes. To help simplify the illustration, dots with arrows point to previously calculated divisions; furthermore, brackets that indicate the locations of tones labeled e, b, and c represent a doubling of the string lengths that indicate the locations of tones labeled m, Ħ, and k, respectively.




          In the second part, Ramis describes 9 more length ratios, which are all accidentals. In Ramis’ time, the natural sign Ħ represented a tone raised by a “semitone.” Therefore, with respect to the “seventh,” Ħ signifies B Ħ, the “major seventh,” as opposed to Bb, the “minor seventh.” Also, observe that in the following passage, Ramis does not include ratio 32/15, or Bb' of the upper “octave,” because he defined it in the first part.


* * * *


     And then we will form the conjunctions of soft b in this way: having duplicated the length of q i we will mark the first conjunction of soft b [Bb], and so the first will be between a and b. Then the length of the string i and the first b is divided in the middle and is marked as the second b [Eb], which will be between d e. But if we have divided the median length of the second b q we will mark the fourth b [Eb'] between l m. By dividing in half the length of the fourth b and the second b we will mark the third b [Ab]. But if the length of the third b q is so divided the fifth b [Ab'] will be marked. Thus from the division we will have five conjunctions of soft b resulting from a correct division.


     But if we wish to obtain square Ħ conjunctions we will divide b q into three parts, and coming from letter q to b we will put forth [the fourth] Ħ [F#'] at the end of the [one-] third part, namely, between n o, and the second square Ħ [F#] at two-thirds of the length, which will fall between f g. If the length of q to the second square Ħ is divided into three parts, coming from letter q to the second square Ħ we will place the third square Ħ [C#'] at two-thirds of the length, and it will be between k l ; if the length of it and of q is doubled, accordingly the first Ħ [C#] between c d will result. Thus from this division we will have four square Ħ conjunctions arising from a correct division . . .[9] (Text in brackets mine.)


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          Figure 10.31(a) shows the locations of five flats described in the first paragraph, and Figure 10.31(b), of four sharps described in the second paragraph.





          Now, refer to Table 10.21, which gives the modern length ratios and inversions that result in the frequency ratios of the complete “double-octave” scale. 








          Figure 10.32 shows Ramis’ original monochord in A, and a modal version in C.





Table 10.22 unfolds the inner structures of these two scales.





Here Columns 1–3 show that the original scale consists of an ascending spiral of four “fifths” above 1/1, and of a descending spiral of seven “fifths” below 1/1. The ascending progression represents the ubiquitous pentatonic scale known the world over as ratios: 1/1, 9/8, 81/64, 3/2, 27/16, [2/1]. In Column 3, note the “flat fifth,” interval ratio 40/27 [680.45 ¢], which indicates the transition from D to G, or from 3-limit ratio 4/3 to 5-limit ratio 9/5: 4/3 ÷ 9/5 = 40/27. Also, in Column 3, a “schismatic fifth,” interval ratio 16384/10935 [700.00 ¢], indicates the transition from Ab (i.e., G#) to C#, or from 5-limit ratio 256/135 to 3-limit ratio 81/64: 256/135 ÷ 81/64 = 16384/10935. The latter constitutes an extremely close rational approximation of the irrational “tempered fifth” of 12-tone equal temperament.


          To understand the origin of the “schismatic fifth,” we must first calculate the discrepancy called a schisma (lit. split), or the imperceptible interval between the ditonic comma and the syntonic comma:



Note that a schisma and a 1/12 ditonic comma are extremely similar:



Consequently, a “just fifth,” ratio 3/2, reduced by a schisma very closely approximates a “just fifth” reduced by a 1/12 ditonic comma:



          In Table 10.22, Columns 4 and 5, observe that Ramis’ scale in A includes a total number of six 3-limit ratios tuned one syntonic comma sharp. Examine the illustration below to see that in a descending spiral of “fifths,” which begins on A, the seventh “fifth” is Ab, or 3-limit ratio 4096/2187:





If we increase this ratio by one syntonic comma, then the result equals a 5-limit ratio: Ab (+ 4/4 comma) = 4096/2187 × 81/80 = 256/135, the scale’s “major seventh.” Similarly, Eb (+ 4/4 comma) = 1024/729 × 81/80 = 64/45; Bb (+ 4/4 comma) = 256/243 × 81/80 = 16/15; F (+ 4/4 comma) = 128/81 × 81/80 = 8/5; C (+ 4/4 comma) = 32/27 × 81/80 = 6/5; G (+ 4/4 comma) = 16/9 × 81/80 = 9/5.


          Now, turn to Table 10.22, Columns 7 and 8, which indicate that Ramis’ modal version in C includes a total number of six 3-limit ratios tuned one syntonic comma flat. For example, in an ascending spiral of “fifths” that begins on C, the second “fifth” is D, or 3-limit ratio 9/8. If we decrease this ratio by one syntonic comma, then the result equals a 5-limit ratio: D (– 4/4 comma) = 9/8 ÷ 81/80 = 10/9, the scale’s “small whole tone.” Similarly, A (– 4/4 comma) = 27/16 ÷ 81/80 = 5/3; E (– 4/4 comma) = 81/64 ÷ 81/80 = 5/4; B (– 4/4 comma) = 243/128 ÷ 81/80 = 15/8; F# (– 4/4 comma) = 729/512 ÷ 81/80 = 45/32; C# (– 4/4 comma) = 2187/2048 ÷ 81/80 = 135/128. Columns 10 and 11 give this 12-tone scale in chromatic order.


          Next, refer to Columns 12 and 13, which show 12 selected tones from an ‘ud tuning described by Safi Al-Din (d. 1294) in a treatise entitled Risalat al-Sharafiya fi’l-nisab al-ta’lifiya (The Sharafian treatise on musical conformities in composition).[10] Safi Al-Din’s First ‘Ud Tuning is based on an ascending spiral of four “fifths” above 1/1, and on a descending spiral of fourteen “fifths” below 1/1. [See Chapter 11, Table 40(a).] Consequently, his “double-octave” consists of a theoretical 19-tone scale that produces two slightly different 17-tone ‘ud tunings. [See Chapter 11, Table 40(b).] However, note carefully that both scales include all the ratios in Column 13. If we increase six 3-limit ratios in Column 13 by one schisma, we obtain six 5-limit ratios in Column 11. For example, C# (+ 4/4 comma) = 256/243 × 32805/32768 = 135/128; D (+ 4/4 comma) = 65536/59049 × 32805/32768 = 10/9; . . . ; etc. Since the finest ear cannot distinguish between two frequencies unless they are at least 2 cents apart, Ramis’ scale in Column 11 and Safi Al-Din’s scale in Column 13 are aurally indistinguishable, and for all practical purposes identical. As discussed in Chapter 11, Section 67, we may interpret the latter six 3-limit ratios as schisma variants, because we tend to perceive such complex 3-limit ratios as simple 5-limit ratios.


          Safi Al-Din was fully aware of the historic origins and the musical derivations of the schisma variants. As discussed in Chapter 11, Sections 68–69, more than three hundred years before Safi Al-Din, Al-Farabi described how musicians tuned these complex 3-limit ratios on the tunbur of Khurasan by ear! Furthermore, turn to Chapter 11, Table 38, and notice that one derives the Arabian schisma variants either by extending the ascending spiral to the 7th, 8th, 9th, and 10th “fifth” above 1/1, or by extending the descending spiral to the 7th, 8th, 9th, and 10th “fifth” below 1/1. Safi Al-Din’s uninterrupted spiral of fourteen descending “fifths” below 1/1 indicates that he was fully aware of the musical and mathematical derivations of the schisma variants. However, since Ramis offered no musical or mathematical explanation for the presence of interval ratio 16384/10935 — the sole “schismatic fifth” that sounds between C#–Ab — it is highly likely that he was unaware of its existence! We conclude, therefore, that Ramis probably modeled his 5-limit scale after Safi Al-Din’s 3-limit scale; that is, Ramis substituted his six 5-limit ratios for Safi Al-Din’s six schisma variants. Consequently, the “schismatic fifth” simply occurred by default.


          As described by Al-Farabi and Safi Al-Din, the utilization of schisma variants originated in the ancient Persian tunbur tuning tradition. Safi Al-Din, who was arguably the greatest music theorist of the Arabian Renaissance, was also famous throughout the Arabian empire for his organization and identification of 84 Melodic Modes. (See Chapter 11, Sections 74–75.) In Andalusia, musicians played these melodic modes for two hundred years before Ramis began his formal education. Many modes identified by Safi Al-Din are in use to this day. (See Chapter 11, Section 83.)


          Scales that require flat consecutive 3/2’s are very reminiscent of tempered tunings. So, it would not be entirely incorrect to describe the scales of Safi Al-Din and Ramis as “just temperaments.” Finally, Ramis’ insistence on just 5/4’s has undoubtedly played a crucial role in later developments of meantone temperament.



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[1] (a) Dupont, W. (1935). Geschichte der musikalischen Temperatur, pp. 20–21. C.H. Beck’sche Buchdruckerei, Nördlingen, Germany.


     In the Erlangen University Library, Dupont discovered an anonymous treatise from the second half of the 15th century entitled Pro clavichordiis faciendis, which describes a monochord division intended as a clavichord tuning. Dupont cites the original Old German text and includes a modern synopsis of this passage. To many scholars the following tuning is the earliest known European 12-tone scale that includes 5-limit ratios:



In this scale, D and A are tuned as “major thirds,” ratio 5/4, down from Gb and Db, respectively. Consequently, D is a schisma equivalent of 9/8, and A is a schisma equivalent of 27/16. According to Dupont, the anonymous author is unaware of these schisma “. . . comma differences.” Finally, B is tuned as a 5/4 up from G.


(b) Adkins, C.D. (1963). The Theory and Practice of the Monochord, pp. 238–241. Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.


     Adkins includes a detailed diagram of this division.


(c) Barbour, J.M. (1951). Tuning and Temperament, pp. 92–93. Da Capo Press, Inc., New York, 1972.



[2] Miller, C.A., Translator (1993). Musica practica, by Bartolomeo Ramis de Pareia, p. 15. Hänssler-Verlag, Neuhausen-Stuttgart, Germany.


     Hence, his Spanish name Bartolomé Ramos.



[3] Hitti, P.K. (1937). History of the Arabs, p. 498. Macmillan and Co. Ltd., London, England, 1956.



[4] Farmer, H.G. (1965). The Sources of Arabian Music, p. xxi. E.J. Brill, Leiden, Netherlands.



[5] Musica practica, p. 17.



[6] Ibid., p. 46.



[7] Palisca, C.V. (1985). Humanism in Italian Renaissance Musical Thought, pp. 232–234. Yale University Press, New Haven, Connecticut.


     On pp. 233–234, Palisca gives excerpts of Gaffurio’s attack on Ramis.



[8] Musica practica, pp. 47–48.



[9] Ibid., p. 85.



[10] D’Erlanger, R., Bakkouch, ‘A.‘A., and Al-Sanusi, M., Translators (Vol. 1, 1930; Vol. 2, 1935; Vol. 3, 1938; Vol. 4, 1939; Vol. 5, 1949; Vol. 6, 1959). La Musique Arabe, Volume 3. Librairie Orientaliste Paul Geuthner, Paris, France.