Just intonation: a general framework

Just into­na­tion (into­na­tion pure in French) is a word used by com­posers, musi­cians and musi­col­o­gists to describe var­i­ous aspects of com­po­si­tion, per­for­mance and instru­ment tun­ing. They all point to the same goal of “playing/singing in tune” — what­ev­er that means. Implementing a gener­ic abstract mod­el of just into­na­tion in the Bol Processor is a chal­lenge beyond our cur­rent com­pe­tence… We approach it prag­mat­i­cal­ly by look­ing at some musi­cal tra­di­tions that pur­sue the same goal with the help of reli­able the­o­ret­i­cal models.

A com­plete and con­sis­tent frame­work for the con­struc­tion of just-intonation scales - or “tun­ing sys­tems” - was the grama-murcchana mod­el elab­o­rat­ed in ancient India. This the­o­ry has been exten­sive­ly com­ment­ed on and (mis)interpreted by Indian and Western schol­ars: for a detailed review see Rao & van der Meer 2010. We will show that an arguably accept­able inter­pre­ta­tion yields a frame­work of chro­mat­ic scales that can be extend­ed to Western clas­si­cal har­mo­ny and eas­i­ly han­dled by the Bol Processor + Csound.

This page is a con­tin­u­a­tion of Microtonality but can be read independently.

All exam­ples shown on this page are avail­able in the sam­ple set bp3-ctests-main.zip shared on GitHub. Follow instruc­tions on Bol Processor ‘BP3’ and its PHP inter­face to install BP3 and learn its basic oper­a­tion. Download and install Csound from its dis­tri­b­u­tion page.

Historical background

Methods of tun­ing musi­cal instru­ments have been doc­u­ment­ed in var­i­ous parts of the world for over 2000 years. For prac­ti­cal and per­son­al rea­sons we will con­cen­trate on work in Europe and the Indian subcontinent.

Systems described as “just into­na­tion” are attempts to cre­ate a tun­ing in which all tonal inter­vals are con­so­nant. There is a large body of the­o­ret­i­cal work on just into­na­tion - see Wikipedia for links and abstracts.

Models are amenable to Hermann von Helmholtz’s notion of con­so­nance which deals with the per­cep­tion of the pure sinu­soidal com­po­nents of com­plex sounds con­tain­ing mul­ti­ple tones. According to the the­o­ry of con­so­nance, the fre­quen­cies of these upper par­tials are inte­ger mul­ti­ples of the fun­da­men­tal fre­quen­cy of the vibra­tion. In mechan­i­cal musi­cal instru­ments, this is close to real­i­ty when long strings are gen­tly struck or plucked. However, this har­mo­ny is lack­ing in many wind instru­ments, espe­cial­ly reed instru­ments such as the sax­o­phone or the Indian shehnai, and even in per­cus­sion instru­ments or bells which com­bine sev­er­al modes of vibration.

Therefore, if just into­na­tion is invoked to. tune a musi­cal instru­ment, it must be anal­o­gous to a zither, a swara man­dal, a harp­si­chord, a piano or a pipe organ, includ­ing elec­tron­ic devices that pro­duce sim­i­lar sounds.

Perhaps because of their late “dis­cov­ery” of cal­cu­lus — actu­al­ly “bor­rowed” from Indian, Persian and Arabic sources — Europeans cul­ti­vat­ed a fas­ci­na­tion with num­bers strong­ly advo­cat­ed by priests as an image of “God’s per­fec­tion”. We may recall Descarte’s claim that the length of a curve is “beyond human under­stand­ing” — because π can­not be writ­ten as an inte­ger ratio…

In real life, musi­cians devel­oped pro­ce­dures for tun­ing their instru­ments by lis­ten­ing to inter­vals and pick­ing out the ones that made sense to their ears — see The two-vina exper­i­ment page. After the devel­op­ment of musi­cal acoustics, attempts were made to inter­pret these pro­ce­dures in terms of fre­quen­cy ratios. This was a risky ven­ture, how­ev­er, because the dream of per­fec­tion led to the sim­plis­tic pro­mo­tion of “per­fect ratios”.

Seeking the kind of per­fec­tion embod­ied in num­bers is the best way to pro­duce bland music. Although just into­na­tion — inter­vals with­out beats — is now pos­si­ble on elec­tron­ic instru­ments, it is based on a nar­row con­cept of tonal­i­ty. This can be ver­i­fied by lis­ten­ing to ancient Western music played in dif­fer­ent tem­pera­ments — see page Comparing tem­pera­ments — and even to Indian clas­si­cal music — see page Raga into­na­tion.

The “Greek” approach

Greek women play­ing ancient Harp, Cithara and Lyre musi­cal instru­ments (source)

Models of vibrat­ing strings attrib­uted to the “ancient Greeks” sug­gest that fre­quen­cy ratios of 2/1 (the octave), 3/2 (the major fifth) and 5/4 (the major third) pro­duce con­so­nant inter­vals, while oth­er ratios pro­duce a cer­tain degree of dis­so­nance.

The prac­tice of poly­phon­ic music on fixed-tuned instru­ments has shown that this per­fect con­so­nance is nev­er achieved with 12 notes in an octave — the con­ven­tion­al chro­mat­ic scale. In Western clas­si­cal har­mo­ny, it would require retun­ing the instru­ment accord­ing to the musi­cal genre, the piece of music and the har­mon­ic con­text of each melod­ic phrase or chord.

Imperfect tonal inter­vals pro­duce unwant­ed beats because their fre­quen­cy ratio can­not be reduced to sim­ple 2, 3, 4, 5 frac­tions. A sim­ple thought exper­i­ment, myth­i­cal­ly attrib­uted to Pythagoras of Samos, shows that this is inher­ent in arith­metic and not a defect in instru­ment design. Imagine the tun­ing of ascend­ing fifths (ratio 3/2) by suc­ces­sive steps on a harp with an octave shift to keep the result­ing note with­in the orig­i­nal octave. The fre­quen­cy ratios would be 3/2, 9/4, 27/16, 81/64 and so on. At this stage, the note appears to be a major third although its actu­al ratio (81/64 = 1.265) is high­er than 5/4 (1.25). The 81/64 inter­val is called the Pythagorean major third, which may sound “out of tune” in a con­ven­tion­al har­mon­ic con­text. The fre­quen­cy ratio (81/80 = 1.0125) between the Pythagorean and har­mon­ic major thirds is called the syn­ton­ic com­ma.

Whoever devised the so-called “Pythagorean tun­ing” went fur­ther in their inten­tion to describe all musi­cal notes by cycles of fifths. Going fur­ther up, 243/128, 729/512… etc. effec­tive­ly pro­duces a full chro­mat­ic scale: C - G - D - A - E - B - F♯ - C♯ - G♯… etc. But in addi­tion to the harsh sound of some of the result­ing inter­vals, things get bad if one hopes to end the cycle on the ini­tial note. If the series start­ed on ‘C’, it will end on ‘C’ (or ‘B#’), but with a ratio of 531441/524288 = 1.01364, slight­ly high­er than 1. This gap is called the Pythagorean com­ma, which is con­cep­tu­al­ly dif­fer­ent from the syn­ton­ic com­ma (1.0125), although their sizes are almost iden­ti­cal. This para­dox is a mat­ter of sim­ple arith­metic: pow­ers of 2 (octave inter­vals) nev­er equal pow­ers of 3.

The attri­bu­tion of this sys­tem to the “ancient Greeks” is, of course, pure fan­ta­sy, since they (unlike the Egyptians) did­n’t have any use for fractions!

A 19-key per octave (from “A” to “a”) key­board designed by Gioseffo Zarlino (1517-1590) (source)

Despite the com­ma prob­lem, tun­ing instru­ments by series of per­fect fifths was com­mon prac­tice in medieval Europe, fol­low­ing the organum which con­sist­ed of singing/playing par­al­lel fifths or fourths to enhance a melody. One of the old­est trea­tis­es on “Pythagorean tun­ing” was pub­lished around 1450 by Henri Arnault de Zwolle (Asselin 2000 p. 139). In this tun­ing, major “Pythagorean” thirds sound­ed harsh, which explains why the major third was con­sid­ered a dis­so­nant inter­val at the time.

Because of these lim­i­ta­tions, Western fixed-pitch instru­ments using chro­mat­ic (12-tone) scales nev­er achieve the pitch accu­ra­cy dic­tat­ed by just into­na­tion. For this rea­son just into­na­tion is described in the lit­er­a­ture as “incom­plete” (Asselin 2000 p. 66). Multiple divi­sions (more than 12 per octave) are required to pro­duce all “pure” ratios. This has been unsuc­cess­ful­ly attempt­ed on key­board instru­ments, although it remains pos­si­ble on a computer.

The Indian approach

Bharata Muni’s “Natya Shastra”

The grama-murcchana mod­el was described in the Natya Shastra, a Sanskrit trea­tise on the per­form­ing arts writ­ten in India some twen­ty cen­turies ago. Chapter 28 con­tains a dis­cus­sion of the “har­mon­ic scale”, which is based on a divi­sion of the octave into 22 shru­ti-s, while only sev­en swara-s (notes) are used by musi­cians: “Sa”, “Re”, “Ga”, “Ma”, “Pa”, “Dha”, “Ni”. These can be mapped onto con­ven­tion­al Western music nota­tion “C”, “D”, “E”, “F”, “G”, “A”, “B” in English, or “do”, “re”, “mi”, “fa”, “sol”, “la”, “si” in Italian/Spanish/French.

This 7-swara scale can be extend­ed to a 12-degree (chro­mat­ic) scale by means of diesis and flat alter­ations, which raise or low­er a note by a semi­tone. Altered notes in the Indian sys­tem are com­mon­ly called “komal Re”, “komal Ga”, “Ma tivra”, “komal Dha” and “komal Ni”. The word “komal” can be trans­lat­ed as “flat” and “tivra” as “diesis”.

The focus of 20th cen­tu­ry research in Indian musi­col­o­gy has been to ‘quan­ti­fy’ shruti-s in a sys­tem­at­ic way and to assess the rel­e­vance of this quan­tifi­ca­tion to the per­for­mance of clas­si­cal raga.

A strik­ing point in the ancient Indian the­o­ry of musi­cal scales is that it does not rely on numer­i­cal ratios, be they fre­quen­cies or lengths of vibrat­ing strings. This point was over­looked by ‘colo­nial musi­col­o­gists’ because of their lack of insight into Indian math­e­mat­ics and their fas­ci­na­tion with a mys­ti­cism of num­bers inher­it­ed from Neopythagoreanism.

As report­ed by Jonathan Barlow (per­son­al com­mu­ni­ca­tion, 3/9/2013, links my own):

The ustads in India from way back con­sid­ered that they fol­lowed Pythagoras, but ear­ly on they made the dis­cov­ery that try­ing to tune by num­bers was a los­ing game, and Ibn Sina (Avicenna) (980-1037 AD), who was their great philoso­pher of aes­thet­ics, said in plain terms that it was wis­er to rely on the ears of the experts. Ahobala tried to do the num­bers thing (and Kamilkhani) but they are rel­e­gat­ed to a foot­note of 17th C musicology.

Bharata Muni, the author(s) of the Natya Shastra, may have heard of “Pythagorean tun­ing”, a the­o­ry that Indian sci­en­tists could have expand­ed con­sid­er­ably, giv­en their matu­ri­ty in the use of cal­cu­lus.. Despite this, not a sin­gle num­ber is quot­ed in the entire chap­ter on musi­cal scales. This para­dox is dis­cussed on my page The Two-vina exper­i­ment. In A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra, I showed a min­i­mal rea­son: Bharata’s descrip­tion leads to an infi­nite set of solu­tions, which should be for­malised with some alge­bra, rather than a set of inte­ger ratios.

How (and why) should the octave be divid­ed into 22 micro-intervals when most Indo-European musi­cal sys­tems only name 5 to 12 notes? Some naive eth­no­mu­si­col­o­gists have claimed that Bharata’s mod­el must be a vari­ant of the Arabic “quar­ter­tone sys­tem”, or even a tem­pered scale with 22 inter­vals… If so, why not 24 shruti-s? Or any arbi­trary num­ber? The two-vina exper­i­ment pro­duces shruti-s of unequal sizes. No sum of micro­tones of 54.5 cents in a 22-degree tem­pered scale would pro­duce an inter­val close to 702 cents — the per­fect fifth that gives con­so­nance (sam­va­di) to musi­cal scales.

In the (thought?) exper­i­ment described in Natya Shastra (chap­ter 28), two vina-s — stringed instru­ments sim­i­lar to zithers — are tuned iden­ti­cal­ly. The author sug­gests low­er­ing all the notes of one instru­ment by “one shru­ti” and he gives a list of notes that will match between the two instru­ments. The process is repeat­ed three more times until all the match­es have been made explic­it. This gives a sys­tem of equa­tions (and inequa­tions) for the 22 unknown vari­ables. Additional equa­tions can be derived from a pre­lim­i­nary state­ment that the octave and the major fifth are “con­so­nant” (sam­va­di), thus fix­ing ratios close to 2/1 and 3/2. (Read the detailed pro­ce­dure on my page The Two-vina Experiment and the math­e­mat­ics in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra.)

However, a new equa­tion is need­ed, which Bharata’s mod­el does not pro­vide. Interestingly, in Natya Shastra the major third is clas­si­fied as “asso­nant” (anu­va­di). Setting its fre­quen­cy ratio to 5/4 is there­fore a reduc­tion of this mod­el. In fact, it is a dis­cov­ery of European musi­cians in the ear­ly 16th cen­tu­ry — when fixed-pitch key­board instru­ments had become pop­u­lar (Asselin 2000 p. 139) — that many musi­col­o­gists take for grant­ed in their inter­pre­ta­tion of the Indian mod­el. […] thirds were con­sid­ered inter­est­ing and dynam­ic con­so­nances along with their inverse, sixths, but in medieval times they were con­sid­ered dis­so­nances unus­able in a sta­ble final sonor­i­ty (Wikipedia).

The reduc­tion of Bharata’s mod­el does not fit with the flex­i­bil­i­ty of into­na­tion schemes in Indian music — see page Raga into­na­tion. Experimental work on musi­cal prac­tice is not “in tune” with this inter­pre­ta­tion of the the­o­ry. The shru­ti sys­tem should be inter­pret­ed as a “flex­i­ble” frame­work in which the vari­able para­me­ter is the syn­ton­ic com­ma, name­ly the dif­fer­ence between a Pythagorean major third and a har­mon­ic major third. Adherence to the two-vina exper­i­ment only implies that the com­ma takes its val­ue between 0 and 56.8 cents (Bel 1988a).

Building tona­grams on the Apple II from tonal data col­lect­ed by Bel’s MMA (1982)

➡ Measurements in “cents” refer to a log­a­rith­mic scale. Given a fre­quen­cy ratio ‘r’, its cent val­ue is 1200 x log(r) / log(2). The octave (ratio 2/1) is 1200 cents, and each semi­tone is about 100 cents.

The con­struc­tion and eval­u­a­tion of raga scale types based on this flex­i­ble mod­el is explained on my page Raga into­na­tion.

Extending the Indian model to Western harmony

One incen­tive for apply­ing the Indian frame­work to Western clas­si­cal music is that both tra­di­tions have giv­en pri­or­i­ty to the con­so­nance of per­fect fifths asso­ci­at­ed with a 3/2 fre­quen­cy ratio. In addi­tion, let us agree to fix the har­mon­ic major third to the inter­val 5/4 (384 cents), result­ing in a syn­ton­ic com­ma of 81/80 (close to 21.5 cents). The sys­tem of equa­tions derived from the two-vina exper­i­ment is com­plete, and it yields two addi­tion­al sizes of shruti-s: the Pythagorean lim­ma (256/243 = about 90 cents) and the minor semi­tone (25/24 = about 70 cents).

These inter­vals were known to Western musi­col­o­gists who were try­ing to find just into­na­tion scales that could be played on key­board instru­ments (12 degrees per octave). Gioseffo Zarlino (1517-1590) is a well-known con­trib­u­tor to this the­o­ret­i­cal work. His “nat­ur­al scale” was an arrange­ment of the three nat­ur­al inter­vals yield­ing the fol­low­ing chro­mat­ic scale — named “just into­na­tion” in “-cs.tryScales”:

A “just into­na­tion” chro­mat­ic scale derived from Zarlino’s mod­el of “nat­ur­al scale”
(Image cre­at­ed by Bol Processor)

This should not be con­fused with Zarlino’s mean­tone tem­pera­ment (image), read pages Microtonality and Comparing tem­pera­ments.

In 1974, E. James Arnold, inspired by the French musi­col­o­gist Jacques Dudon, designed a cir­cu­lar mod­el to illus­trate the trans­po­si­tion of scales (mur­ccha­na) in Bharata’s mod­el. Below is the sequence of inter­vals (L, C, M…) over an octave as derived from the two-vina exper­i­ment.

The out­er crown of Arnold’s mod­el for his inter­pre­ta­tion of the grama-murcchana sys­tem. The Pythagorean series of per­fect fifths is drawn in red and the har­mon­ic series of per­fect fifths is drawn in green. The dot­ted blue line is a har­mon­ic major third.

Positions R1, R2 etc. are labelled with abbre­vi­a­tions of names Sa, Re, Ga, Ma, Pa, Dha, Ni. For exam­ple, Ga (“E” in English) can have four posi­tions, G1 and G2 being enhar­mon­ic vari­ants of komal Ga (“E flat” = “mi bemol”), while G3 and G4 are the har­mon­ic and Pythagorean posi­tions of shud­dha Ga (“E”= “mi”) respectively.

The notes of the chro­mat­ic scale have been labelled using the Italian/Spanish/French con­ven­tion “do”, “re”, “mi”, “fa”… rather than the English con­ven­tion to avoid con­fu­sion: “D” is asso­ci­at­ed with Dha (“A” in English, “la” in Italian/French) and not with the English “D” (“re” in Italian/French).

Frequency ratios are illus­trat­ed by pic­tograms show­ing how each posi­tion can be derived from the base note (Sa). For exam­ple, the pic­togram near N2 (“B flat” = “si bémol”) shows 2 ascend­ing per­fect fifths and 1 descend­ing major third.

Cycles of per­fect fifths have been marked with red and green seg­ments. The red series is gen­er­al­ly called “Pythagorean” — con­tain­ing G4 (81/84) — and the green one “har­mon­ic” — con­tain­ing G3 (5/4). The blue arrow shows a har­mon­ic major third going from S (“C” = “do”) to G3. Both cycles are iden­ti­cal, with the har­mon­ic and Pythagorean posi­tions dif­fer­ing by 1 syn­ton­ic comma.

Theoretically, the har­mon­ic series could also be con­struct­ed in a “Pythagorean” way, by extend­ing the cycles of per­fect fifths. Thus, after 8 descend­ing fifths, G3 (“E” = “mi”) would be 8192/6561 (1.248) instead of 5/4 (1.25). The dif­fer­ence is a schis­ma (ratio 1.001129), an inter­val beyond human per­cep­tion. It is there­fore more con­ve­nient to show sim­ple ratios.

The frame­work imple­ment­ed in Bol Processor deals with inte­ger ratios, which allows for high accu­ra­cy. Nevertheless, it delib­er­ate­ly eras­es schis­ma dif­fer­ences. This is the result of approx­i­mat­ing cer­tain ratios, e.g. replac­ing 2187/2048 with 16/15.

Note that there is no trace of schis­ma in the clas­si­cal Indian the­o­ry of musi­cal scales; there would­n’t be even if Bharata’s con­tem­po­raries had con­struct­ed them via series of ratio­nal num­bers, because of their deci­sion to dis­re­gard infin­i­tes­i­mals as “non-representable” enti­ties (cf. Nāgārjuna’s śūniyavā­da phi­los­o­phy, Raju 2007 p. 400). If 2187/2048 and oth­er com­plex ratios of the same series were deemed imprac­ti­cal, the Indian mathematician/physicist (fol­low­ing the Āryabhaţīya) would replace them all with “16/15 āsan­na (near val­ue)”… This is an exam­ple of Indian math­e­mat­ics designed for cal­cu­la­tion rather than proof con­struc­tion. In the Western Platonic approach, math­e­mat­ics aimed at “exact val­ues” as a sign of per­fec­tion, which led its pro­po­nents to face seri­ous prob­lems with “irra­tional” num­bers and even with the log­ic under­ly­ing for­mal proof pro­ce­dures (Raju 2007 p. 387-389).

This dia­gram, and the mov­ing gra­ma wheel that will be intro­duced next, could be built with any size of the syn­ton­ic com­ma in the range 0 to 56.8 cents (Bel 1988a). The two-vina exper­i­ment implies that L = M + C. Thus, the syn­ton­ic com­ma is also the dif­fer­ence between a lim­ma and a minor semi­tone. To build a frame­work for the flex­i­ble mod­el, sim­ply allow all har­mon­ic posi­tions to move by the same amount in the direc­tion of their Pythagorean enhar­mon­ic variants.

While major thirds would be 1 com­ma larg­er if a Pythagorean inter­val (e.g. G4, ratio 81/64) were cho­sen instead of a har­mon­ic one (G3, ratio 5/4), major fifths also dif­fer by 1 com­ma, but the Pythagorean fifth (P4, ratio 3/2) is larg­er than the har­mon­ic one (P3, ratio 40/27). The lat­ter has been called the wolf fifth because its use in melod­ic phras­es or chords is said to sound “out of tune”, with a negative/evil mag­i­cal connotation.

No posi­tion on this mod­el requires more than 1 ascend­ing or descend­ing major third. This makes sense to instru­ment tuners who know that tun­ing per­fect fifths by ear is an easy task that can be repeat­ed in sev­er­al steps — here a max­i­mum of 5 or 6 up and down. Tuning a har­mon­ic major third, how­ev­er, requires a lit­tle more atten­tion. It would there­fore be unre­al­is­tic to imag­ine a pre­cise tun­ing pro­ce­dure based on a sequence of major thirds —although this can be achieved with the aid of elec­tron­ic devices.

At the top of the pic­ture (posi­tion “fa#”) we notice that nei­ther of the two cycles of fifths clos­es on itself due to the pres­ence of a Pythagorean com­ma. The tiny dif­fer­ence (schis­ma, ratio 1.001129) between Pythagorean and syn­ton­ic com­mas is illus­trat­ed by two pairs of posi­tions: P1/M3 and P2/M4.

Another pecu­liar­i­ty at the top of the pic­ture is the appar­ent dis­rup­tion of the sequence L-C-M. However, remem­ber­ing that L = M + C, the reg­u­lar­i­ty is restored by choos­ing between P1/M3 and P2/M4.

Approximations have no effect on the sound of musi­cal inter­vals, since no human ear would appre­ci­ate a schis­ma dif­fer­ence (2 cents). However, oth­er dif­fer­ences must remain explic­it, since inte­ger ratios indi­cate the tun­ing pro­ce­dure by which the scale can be con­struct­ed. Thus the replace­ment inte­ger ratio may be more com­plex than the “schis­mat­ic” one, as in the case of R1, ratio 256/243 instead of 135/128, because the lat­ter is built with a sim­ple major third above D4 instead of belong­ing to the Pythagorean series.

Tuning Western instruments

The prob­lem of tun­ing fixed pitch instru­ments (harp­si­chord, pipe organ, pianoforte…) has been well doc­u­ment­ed by the organ/harpsichord play­er, builder and musi­col­o­gist Pierre-Yves Asselin (Asselin 2000). In his prac­ti­cal approach, just into­na­tion is a back­ground mod­el that can only be approx­i­mat­ed on 12-degree scales by tem­pera­mentcom­pro­mis­ing the pure inter­vals of just into­na­tion to meet oth­er require­ments. Temperament tech­niques applied to the Bol Processor are dis­cussed on the Microtonality and Comparing tem­pera­ments pages.

Source: Pierre-Yves Asselin (2000 p. 61)

The col­umn at the cen­tre of this pic­ture, with notes with­in ellipses, is a series of per­fect fifths which Asselin called “Pythagorean”.

Series of fifths are infi­nite. Selecting sev­en of them (in the mid­dle col­umn) cre­ates a scale called the “glob­al dia­ton­ic frame­work” (milieu dia­tonique glob­al, see Asselin 2000 p. 59). In this exam­ple, the frame­works are those of “C” and “G” (“do” and “sol” in French).

Extending series of per­fect fifths beyond the sixth step pro­duces com­pli­cat­ed ratios that can be approx­i­mat­ed (with a schis­ma dif­fer­ence) to those pro­duced by har­mon­ic major thirds (ratio 5/4). Positions on the right (major third up, first order) are one syn­ton­ic com­ma low­er than their equiv­a­lents in the mid­dle series, and posi­tions on the left (major third down, first order) are one syn­ton­ic com­ma higher.

It is pos­si­ble to cre­ate more columns on the right (“DO#-2”, “SOL#-2” etc.) for posi­tions cre­at­ed by 2 suc­ces­sive jumps of a har­mon­ic major third, and in the same way on the left (“DOb#+2”, “SOLb#+2” etc.), but these second-order series are only used for the con­struc­tion of tem­pera­ments — see page Microtonality.

This mod­el pro­duces 3 to 4 posi­tions for each note, a 41-degree scale, which would require 41 keys (or strings) per octave on a mechan­i­cal instru­ment! This is one rea­son for the tem­per­ing of inter­vals on mechan­i­cal instru­ments, which amounts to select­ing the most appro­pri­ate 12 posi­tions for a giv­en musi­cal repertoire.

This tun­ing scheme is dis­played on scale “3_cycles_of_fifths” in the “-cs.tryTunings” Csound resource of Bol Processor.

Series of names have been entered, togeth­er with the frac­tion of the start­ing posi­tion, to pro­duce cycles of per­fect fifths in the scale. Following Asselin’s nota­tion, the fol­low­ing series have been cre­at­ed (trace gen­er­at­ed by the Bol Processor):

  1. From 4/3 up: FA, DO, SOL, RE, LA, MI, SI, FA#, DO#, SOL#, RE#, LA#
  2. From 4/3 down: FA, SIb, MIb, LAb, REb, SOLb
  3. From 320/243 up: FA-1, DO-1, SOL-1, RE-1, LA-1, MI-1, SI-1, FA#-1, DO#-1, SOL#-1, RE#-1, LA#-1
  4. From 320/243 down: FA-1, SIb-1, MIb-1, LAb-1, REb-1, SOLb-1
  5. From 27/20 up: FA+1, DO+1, SOL+1, RE+1, LA+1, MI+1, SI+1, FA#+1, DO#+1, SOL#+1, RE#+1, LA#+1
  6. From 27/20 down: FA+1, SIb+1, MIb+1, LAb+1, REb+1, SOLb+1

This was more than enough to deter­mine the 3 or 4 posi­tions of each note, since sev­er­al notes can reach the same posi­tion at a schis­ma dis­tance. For exam­ple, “REb” is in the same posi­tion as “DO#-1”. The IMAGE link shows this scale with (sim­pli­fied) fre­quen­cy relationships:

The “3_cycles_of_fifths” scale: a graph­i­cal rep­re­sen­ta­tion of three series of per­fect fifths used to per­form Western music in “just into­na­tion” accord­ing to Asselin (2000).
(Image cre­at­ed by Bol Processor)

Compared to the mod­el advo­cat­ed by Arnold (1974, see fig­ure above), this sys­tem accepts har­mon­ic posi­tions on either side of the Pythagorean posi­tions, which means that Sa (“C” or “do”), like all unal­tered notes, can take three dif­fer­ent posi­tions. In Indian music, Sa is unique because it is the fun­da­men­tal note of every clas­si­cal per­for­mance of a raga, fixed by the drone (tan­pu­ra) and tuned to suit the singer or instru­men­tal­ist. However, we will see that trans­po­si­tions (murcchana-s) of the basic Indian scale(s) pro­duce some of these addi­tion­al positions.

A tun­ing scheme based on three (or more) cycles of per­fect fifths is a good grid for con­struct­ing basic chords in just into­na­tion. For exam­ple, a “C major” chord is made up of its ton­ic “DO”, its dom­i­nant “SOL” a per­fect fifth high­er, and “MI-1” a major har­mon­ic third above “DO”. The first two notes can belong to a Pythagorean series (blue marks on the graph) and the last one to a har­mon­ic series (green marks on the graph). Minor chords are con­struct­ed in a sim­i­lar way, which will be explained later.

This does not com­plete­ly solve the prob­lem of play­ing tonal music with just into­na­tion. Sequences of chords must be cor­rect­ly aligned. For exam­ple, should one use the same “E” in “C major” and in “E major”? The answer is “no”, but the rule must be made explicit.

How is it pos­si­ble to choose the right one among the 37 * 45 = 2 239 488 chro­mat­ic scales shown in this graph?

In the approach of Pierre-Yves Asselin (2000) — inspired by the work of Conrad Letendre in Canada — rules were derived from options val­i­dat­ed by lis­ten­ers and musi­cians. Conversely, the gra­ma frame­work exposed below is a “top-down” approach — from a the­o­ret­i­cal mod­el to its eval­u­a­tion by practitioners.

The grama framework

Using Bharata’s mod­el — see page The two-vina exper­i­ment — we can con­struct chro­mat­ic (12-degree) scales in which each tonal posi­tion (out of 11) has two options: har­mon­ic or Pythagorean. This is one rea­son to say that the frame­work is based on 22 shru­ti-s. In Indian musi­co­log­i­cal lit­er­a­ture, the term shru­ti is ambigu­ous, as it can mean either a tonal posi­tion or an interval.

In Bol Processor BP3 this “gra­ma” frame­work is edit­ed as fol­lows in “-cs.12_scales”:

The 22-shruti frame­work as per Bharata’s mod­el with a syn­ton­ic com­ma of 22 cents (see full image)

We use lower-case labels for R1, R2 etc. and append a ‘_’ after labels to dis­tin­guish enhar­mon­ic posi­tions from octave num­bers. So, “g3_4” means G3 in the fourth octave.

Two options for each of the 11 notes yields a set of 211 = 2048 chro­mat­ic scales. Of these, only 12 are “opti­mal­ly con­so­nant”, i.e. they con­tain only one wolf fifth (small­er by 1 syn­ton­ic com­ma). These 12 scales are the ones used in har­mon­ic or modal music to expe­ri­ence max­i­mum con­so­nance. The author(s) of Naya Shastra had this inten­tion in mind when they described a basic 12-tone “opti­mal” scale called “Ma-grama”. This scale is called “Ma_grama” in Csound resource “-cs.12_scales”:

The “Ma-grama” basic chro­mat­ic scale built on the 22-shruti frame­work (see full image)

Click on the IMAGE link on the “Ma_grama” page to obtain a graph­i­cal rep­re­sen­ta­tion of this scale:

The Ma-grama chro­mat­ic scale, Bol Processor graph­ic display

In this pic­ture the per­fect fifths are blue lines and the (unique) wolf fifth between C and G is a red line. Note that posi­tions marked in blue (“Db”, “Eb” etc.) are Pythagorean and har­mon­ic posi­tions (“D”, “E” etc.) appear in green. Normally, a “Pythagorean” posi­tion on this frame­work is one where nei­ther the numer­a­tor nor the denom­i­na­tor of the frac­tion is a mul­ti­ple of 5. Multiples of 5 indi­cate jumps of har­mon­ic major thirds (ratio 5/4 or 4/5). This sim­ple rule is bro­ken, how­ev­er, when com­plex ratios are replaced by sim­ple equiv­a­lents at a dis­tance of one schis­ma. Therefore, the blue and green mark­ings on the Bol proces­sor images are main­ly used to facil­i­tate the iden­ti­fi­ca­tion of a posi­tion: a note appear­ing near a blue mark­ing could as well belong to the har­mon­ic series with a more com­plex ratio, bring­ing it close to the Pythagorean position.

It will be impor­tant to remem­ber that all the notes of the Ma-grama scale are in their low­est enhar­mon­ic posi­tions. Other scales are cre­at­ed by rais­ing a few notes by a comma.

This Ma-grama is the start­ing point for the gen­er­a­tion of all “opti­mal­ly con­so­nant” chro­mat­ic scales. This is done by trans­pos­ing per­fect fifths (upwards or down­wards). The visu­al­i­sa­tion of trans­po­si­tions becomes clear when the basic scale is drawn on a cir­cu­lar wheel which is allowed to move with­in the out­er crown shown above. The fol­low­ing is Arnold’s com­plete mod­el, show­ing the Ma-Grama in the basic posi­tion, pro­duc­ing the “Ma01″ scale:

The fixed (out­er) and mov­able (inner) shru­ti wheels in posi­tion for the “M1” trans­po­si­tion of Ma-grama, which pro­duces the “Ma01” scale

This posi­tion­ing of the inner wheel on top of the out­er wheel is called a “trans­po­si­tion” (mur­ccha­na).

Intervals are shown on the graph. For exam­ple, R3 (“D” = “re”) is a per­fect fifth to D3 (“A” = “la”).

The “Ma01″ scale pro­duced by this M1 trans­po­si­tion pro­duces the “A minor” chro­mat­ic scale with the fol­low­ing intervals:

CDb c+m D c+l Eb c+m E c+l F c+m F# c+l GAb c+m A c+l Bb c+m B c+l C

  • m = minor semi­tone = 70 cents
  • l = lim­ma = 90 cents
  • c = com­ma = 22 cents
The “A minor” chro­mat­ic scale pro­duced by the M1 trans­po­si­tion of Ma-grama (i.e. “Ma01”)

This con­struc­tion of the “A minor” scale cor­re­sponds to the Western scheme for the pro­duc­tion of just into­nat­ed chords: the fun­da­men­tal “A” (ratio 5/3) is “LA-1” on the “3_cycles_of_fifths” scale, which is in the “major third upwards” series as well as its dom­i­nant “MI-1”, while “C” (ratio 1/1) belongs to the “Pythagorean” series.

At first sight, the scale con­struct­ed by this M1 trans­po­si­tion also resem­bles a “C major” scale, but with a dif­fer­ent choice of R3 (har­mon­ic “D” ratio 10/9) instead of R4 (Pythagorean “D” ratio 9/8). To pro­duce the “C major” scale, “D” should be raised to its Pythagorean posi­tion, which amounts to R4 replac­ing R3 on Bharata’s mod­el. This is done by using an alter­na­tive root scale called “Sa-Grama” in which P4 replaces P3.

P3 is called “cyu­ta Pa” mean­ing “Pa low­ered by one shru­ti” — here a syn­ton­ic com­ma. The wheel rep­re­sen­ta­tion sug­gests that oth­er low­ered posi­tions may lat­er be high­light­ed by the trans­po­si­tion process, name­ly cyu­ta Ma and cyu­ta Sa.

At the bot­tom of the “Ma01″ page on “-cs.12_scales”, all the inter­vals of the chro­mat­ic scale are list­ed, with the sig­nif­i­cant inter­vals high­light­ed in colour. The wolf fifth is coloured red. Note that when the scale is opti­mal­ly con­so­nant, only one cell is coloured red.

Harmonic struc­ture of the “Ma01” trans­po­si­tion of Ma-grama, as dis­played by the Bol Processor
Ma01 tun­ing scheme, dis­played by the Bol Processor

A tun­ing scheme is sug­gest­ed at the bot­tom of page “Ma01″. It is based on the (pure­ly mechan­i­cal) assump­tion that per­fect fifths are tuned first with­in the lim­it of 6 steps. Then har­mon­ic major thirds and minor sixths are high­light­ed, and final­ly Pythagorean thirds and minor sixths can also be tak­en into account.

Exporting a major chro­mat­ic scale with the sen­si­tive note raised by 1 comma

We can use “Ma01″ as a 23-degree micro­ton­al scale in Bol Processor pro­duc­tions because all the notes rel­e­vant to the chro­mat­ic scale have been labelled. However it is more prac­ti­cal to extract a 12-degree scale with only labelled notes. This can be done on the “Ma01″ page. The image shows the expor­tat of the “Cmaj” scale with 12 degrees and a raised posi­tion of D.

Using “Cmaj” for the name makes it easy to declare this scale in its spe­cif­ic har­mon­ic con­text. In the same way, a 12-degree “Amin” can be export­ed with­out rais­ing the “D”.

“D” (“re”) is there­fore the sen­si­tive note when switch­ing between the “C major” scale and its rel­a­tive “A minor”.

In all 12-degree export­ed scales it is easy to change the note con­ven­tion — English, Italian/Spanish/French, Indian or key num­bers. It is also pos­si­ble to select diesis in replace­ment of flat and vice ver­sa, as the machine recog­nis­es both options.

Producing the 12 chromatic scales

A PowerPoint ver­sion of Arnold’s mod­el can be down­loaded here and used to check the trans­po­si­tions pro­duced by the Bol Processor BP3.

Creating “Ma02” as a trans­po­si­tion of “Ma01”

To cre­ate suc­ces­sive “opti­mal­ly con­so­nant” chro­mat­ic scales, the Ma-grama should be trans­posed by descend­ing or ascend­ing per­fect fifths.

For exam­ple, cre­ate “Ma02″ by trans­pos­ing “Ma01″ from a per­fect fourth “C to F” (see pic­ture). Nothing else needs to be done. All the trans­po­si­tions are stored in Csound resource “-cs.12_scales”. Each of these scales can then be used to export a minor and a major chro­mat­ic scale. This pro­ce­dure is explained in detail on the page Creation of just-intonation scales.

Enharmonic shift of the tonic

An inter­est­ing point raised by James Arnold in our paper L’intonation juste dans la théorie anci­enne de l’Inde : les appli­ca­tions aux musiques modale et har­monique (1985) is the com­par­i­son of minor and major scales of the same ton­ic, for exam­ple mov­ing from “C major” to “C minor”.

To get the “C minor” scale, we need to cre­ate “Ma04″ by using four suc­ces­sive descend­ing fifths (or ascend­ing fourths). Note that writ­ing “C to F” on the form does not always pro­duce a per­fect fourth trans­po­si­tion because the “F to C” inter­val may be a wolf fifth! This hap­pens when going from “Ma03″ to “Ma04″. In this case, select, for exam­ple, “D to G”.

From “Ma04″ we export “Cmin”. Here comes a surprise:

(See full image)
The “C minor” scale derived from the “Ma04” trans­po­si­tion of Ma-grama

The inter­vals are those pre­dict­ed (see “A minor” above), but the posi­tions of “G”, “F” and “C” have been low­ered by one com­ma. This was expect­ed for “G” because of the replace­ment of P4 by P3. The bizarre sit­u­a­tion is that both ‘C’ and ‘F’ are one com­ma low­er than what seemed to be their low­est (or only) posi­tion in the 22-shruti mod­el. The authors of Natya Shastra had antic­i­pat­ed a sim­i­lar process when they invent­ed the terms “cyu­ta Ma” and “cyu­ta Sa”

This shift­ing of the base note can be seen by mov­ing the inner wheel. After 4 trans­po­si­tions, the posi­tion M1 of the inner wheel will cor­re­spond to the posi­tion G1 of the out­er wheel, giv­ing the fol­low­ing configuration:

The “Ma04” trans­po­si­tion of Ma-grama show­ing low­ered C, F and G

This shift of the ton­ic was pre­sent­ed as a chal­leng­ing find­ing in our paper (Arnold & Bel 1985). Jim Arnold had done exper­i­ments with Pierre-Yves Asselin play­ing Bach’s music on the Shruti Harmonium and both liked the shift of the ton­ic on minor chords.

Two options for tun­ing a “C minor” chord. Source: Asselin (2000)

Pierre-Yves him­self men­tions a one-comma low­er­ing of “C” and “G” in the “C minor” chord. However, this was one of two options pre­dict­ed by his the­o­ret­i­cal mod­el. He test­ed it by play­ing the Cantor elec­tron­ic organ at the University, and reports that musi­cians found this option to be more pun­gent“déchi­rant” — (Asselin 2000 p. 135-137).

The oth­er option (red on the pic­ture) was that each scale be “aligned” in ref­er­ence to its base note “C” (“DO”). This align­ment (one-comma rais­ing) can be done click­ing but­ton “ALIGN SCALE” on scale pages wher­ev­er the basic note (“C”) is not at posi­tion 1/1. Let us lis­ten to the “C major”/ “C minor” / “C major” sequence, first “non-aligned” then “aligned”:

“C major”/ “C minor” / “C major” sequence, first non-aligned then aligned

Clearly, the “non-aligned” ver­sion is more pun­gent than the “aligned” one.

This choice is based on per­cep­tu­al expe­ri­ence, or “pratyakṣa pramāṇa” in Indian epis­te­mol­o­gy — see The two-vina exper­i­ment. We take an empir­i­cal approach rather than seek­ing an “axiomat­ic proof”. The ques­tion is not which of the two options is true, but which one pro­duces music that sounds right.

Checking the tuning system

Checking a chord sequence

The con­struc­tion of just into­na­tion using the grama-murcchana pro­ce­dure needs to be checked in typ­i­cal chord sequences such as the “I-IV-II-V-I” series dis­cussed by Pierre-Yves Asselin (2000 p. 131-135):

After try­ing out five options sug­gest­ed by his the­o­ret­i­cal mod­el, the author chose the one pre­ferred by all the musi­cians. This is the into­na­tion they spon­ta­neous­ly choose when singing, with­out any spe­cial instruc­tion. This ver­sion also cor­re­sponds to Zarlino’s “nat­ur­al scale”.

The best option for a just-intonation ren­der­ing of the
“I-IV-II-V-I” har­mon­ic series (Asselin 2000 p. 134)

In the pre­ferred option, the ton­ics “C”, “F” and “G” belong to the Pythagorean series of per­fect fifths, except “D” in the “D minor” chord which is one com­ma low­er than in “G major”.

In the pic­ture, the tri­an­gles with the top point­ing to the right are major chords, and the one point­ing to the left is the “D minor” chord.

Asselin (2000 p. 137) con­cludes that the minor mode is one syn­ton­ic com­ma low­er than the major mode. Conversely, the major mode should be one syn­ton­ic com­ma high­er than the minor mode.

This is ful­ly con­sis­tent with the mod­el con­struct­ed by grama-murcchana. Since minor chro­mat­ic scales are export­ed from trans­po­si­tions of Ma-grama with all its degrees in the low­est posi­tion, their base notes are also dri­ven to the low­est posi­tions. However this requires a scale “adjust­ment” in the cas­es of “Ma10″, “Ma11″ and “Ma12″ so that no posi­tion is cre­at­ed out­side the basic Pythagorean/harmonic scheme of the Indian sys­tem. Looking at Asselin’s draw­ing (above), this means that no posi­tion would be picked up in the 2nd-order series of fifths in the right­most col­umn with two suc­ces­sive ascend­ing major thirds result­ing in a low­er­ing of 2 syn­ton­ic com­mas. This process is explained in more detail on the page Creation of just-intonation scales.

Let us lis­ten to the pro­duc­tion of the “-gr.tryTunings” gram­mar:

S --> Temp - Just
Temp --> Cmaj Fmaj Dmin Gmaj Cmaj
Just --> _scale(Cmaj,0) Cmaj _scale(Fmaj,0) Fmaj _scale(Dmin,0) Dmin _scale(Gmaj,0) Gmaj _scale(Cmaj,0) Cmaj
Cmaj --> {C3,C4,E4,G4}
Fmaj --> {F3,C4,F4,A4}
Dmin --> {D3,D4,F4,A4}
Gmaj --> {G3,B3,D4,G4}

First we hear the sequence of chords in equal tem­pera­ment, then in just intonation.

The “I-IV-II-V-I” har­mon­ic series in equal-tempered and just-intonation

The iden­ti­ty of the last occur­rence with Asselin’s favourite choice is marked by fre­quen­cies in the C-sound score: “D4” in the third chord (D minor) is one com­ma low­er than “D4” in the fourth chord (G major), while all oth­er notes (e.g. “F4”) have the same fre­quen­cies in the four chords.

; I - Cmaj
i1 6.000 1.000 130.815 90.000 90.000 0.000 0.000 0.000 0.000 ; C3
i1 6.000 1.000 261.630 90.000 90.000 0.000 0.000 0.000 0.000 ; C4
i1 6.000 1.000 327.038 90.000 90.000 0.000 0.000 0.000 0.000 ; E4
i1 6.000 1.000 392.445 90.000 90.000 0.000 0.000 0.000 0.000 ; G4

; IV - Fmaj
i1 7.000 1.000 174.420 90.000 90.000 0.000 0.000 0.000 0.000 ; F3
i1 7.000 1.000 261.630 90.000 90.000 0.000 0.000 0.000 0.000 ; C4
i1 7.000 1.000 348.840 90.000 90.000 0.000 0.000 0.000 0.000 ; F4
i1 7.000 1.000 436.050 90.000 90.000 0.000 0.000 0.000 0.000 ; A4

; II - Dmin
i1 8.000 1.000 145.350 90.000 90.000 0.000 0.000 0.000 0.000 ; D3
i1 8.000 1.000 290.700 90.000 90.000 0.000 0.000 0.000 0.000 ; D4
i1 8.000 1.000 348.840 90.000 90.000 0.000 0.000 0.000 0.000 ; F4
i1 8.000 1.000 436.050 90.000 90.000 0.000 0.000 0.000 0.000 ; A4

; V - Gmaj
i1 9.000 1.000 196.222 90.000 90.000 0.000 0.000 0.000 0.000 ; G3
i1 9.000 1.000 245.278 90.000 90.000 0.000 0.000 0.000 0.000 ; B3
i1 9.000 1.000 294.334 90.000 90.000 0.000 0.000 0.000 0.000 ; D4
i1 9.000 1.000 392.445 90.000 90.000 0.000 0.000 0.000 0.000 ; G4

; I - Cmaj
i1 10.000 1.000 130.815 90.000 90.000 0.000 0.000 0.000 0.000 ; C3
i1 10.000 1.000 261.630 90.000 90.000 0.000 0.000 0.000 0.000 ; C4
i1 10.000 1.000 327.038 90.000 90.000 0.000 0.000 0.000 0.000 ; E4
i1 10.000 1.000 392.445 90.000 90.000 0.000 0.000 0.000 0.000 ; G4

To sum­marise, the ton­ic and dom­i­nant of each minor chord belong to the “low­er” har­mon­ic series of per­fect fifths appear­ing in the right-hand col­umn of Asselin’s draw­ing repro­duced above. Conversely, the ton­ic and dom­i­nant of each major chord belong to the “Pythagorean” series of per­fect fifths in the mid­dle column.

Checking note sequences

Switches for (pro­gram­ma­ble) enhar­mon­ic adjust­ments on Bel’s Shruti Harmonium (1980)

The rules for deter­min­ing the rel­a­tive posi­tions of major and minor modes (see above) deal only with the three notes that define a major or minor chord. Transpositions (murcchana-s) of the Ma-grama pro­duce basic notes in the same posi­tions, but these are also chro­mat­ic (12-degree) scales. Therefore, they also estab­lish the enhar­mon­ic posi­tions of all the notes that would be played in that har­mon­ic context.

Do these com­ply with just into­na­tion? In the­o­ry, yes, because the 12 chro­mat­ic scales obtained by these trans­po­si­tions are “opti­mal­ly con­so­nant”: each of them con­tains no more than a wolf fifth.

In 1980, James Arnold con­duct­ed exper­i­ments to ver­i­fy this the­o­ret­i­cal mod­el using my Shruti har­mo­ni­um, which pro­duced pro­grammed inter­vals to an accu­ra­cy of 1 cent. Pierre-Yves Asselin played clas­si­cal pieces while Jim manip­u­lat­ed switch­es on the instru­ment to select enhar­mon­ic variants.

Listen to three ver­sions of an impro­vi­sa­tion based on Mozart’s musi­cal dice game. The first one is equal-tempered, the sec­ond one uses Serge Cordier’s equal-tempered scale with an extend­ed octave (1204 cents, see Microtonality) and the third uses sev­er­al dif­fer­ent scales to repro­duce a just into­na­tion. To this end, vari­ables point­ing to scales based on the har­mon­ic con­text have been insert­ed in the first gram­mar rules:

S --> _vel(80) Ajust Bjust
Ajust --> Cmaj A1 A2 Gmaj A3 Cmaj A4 Dmaj A5 Cmaj A6 Gmaj A7 A8 Cmaj A1 A2 Gmaj A3 Cmaj A4 Dmaj A5 Cmaj A6 Gmaj A7 A’8
Bjust --> Gmaj B1 Cmaj B2 Dmaj B3 Cmaj B4 Fmaj B5 B6 Gmaj B7 Cmaj B8 Gmaj B1 Cmaj B2 Dmaj B3 Cmaj B4 Fmaj B5 B6 Gmaj B7 Cmaj B8
Cmaj --> _scale(Cmaj,0)
Dmaj --> _scale(Dmaj,0)
Fmaj --> _scale(Fmaj,0)
Gmaj --> _scale(Gmaj,0)
… etc.

An exam­ple of Mozart’s musi­cal dice game, equal-tempered
The same exam­ple, equal-tempered scale with octave stretched at 1204 cents
The same exam­ple in just intonation

Scale comparison

At the bot­tom of the pages “-cs.12_scales” and “-cs.Mozart”, all scales are com­pared for their inter­val­ic con­tent. The com­par­i­son is based on frac­tions where these have been declared, or on floating-point fre­quen­cy ratios otherwise.

The com­par­i­son con­firms that the “Amin” chro­mat­ic scale is iden­ti­cal to “Fmaj”.

By rais­ing “D” in “Ma01″ we have cre­at­ed “Sa01″, the first trans­po­si­tion of the Sa-grama scale. From “Sa01″ we can make “Sa02″ etc. by suc­ces­sive trans­po­si­tions (one fourth upwards). But the com­para­tor shows that “Sa02″ is iden­ti­cal to “Ma01″.

Similarly, the trans­po­si­tions “Ma13″, “Ma14″ etc. are iden­ti­cal to “Ma01″, “Ma02″ etc. The series of chro­mat­ic scales is (as expect­ed) cir­cu­lar, since “Ma13″ returns to “Ma01″.

Comparison of scales stored in “-cs.12_scales”

For more details on fre­quen­cies, block keys, etc., see the Microtonality page.

Is this perfect?

This entire page is devot­ed to tonal sys­tems defined in terms of whole-numbered ratios (i.e. ratio­nal num­bers) mea­sur­ing tonal inter­vals. There were at least two strong incen­tives for the idea that any “pure” tonal inter­val should be treat­ed as a ratio of two whole num­bers, such as 2/1 for the octave, 3/2 for a “per­fect” fifth, 5/4 for a “har­mon­ic” major third, etc.

The his­to­ry of music (in the West) goes back to ideas attrib­uted to the Greek philoso­pher “Pythagoras” (see above) , who believed that all things were made of [ratio­nal] num­bers. This approach stum­bled upon the impos­si­bil­i­ty of mak­ing the octave cor­re­spond to a series of “per­fect fifths”…

As we found out — read above and The Two-vina exper­i­ment — this approach was not fol­lowed in India despite the fact that Indian sci­en­tists were sig­nif­i­cant­ly more advanced than the Greeks in the field of cal­cu­lus (Raju C.K., 2007).

Another incen­tive to the use of ratio­nal num­bers was Hermann von Helmholtz’s notion of con­so­nance (1877) which became pop­u­lar after the peri­od of Baroque music in Europe, fol­low­ing the ini­tial claim of a “nat­ur­al tonal sys­tem” by Jean-Philippe Rameau in his Traité de l’har­monie réduite à ses principes naturels (1722). The devel­op­ment of key­board stringed instru­ments such as the pipe organ and the pianoforte had made it nec­es­sary to devel­op a tun­ing sys­tem that met the require­ments of (approx­i­mate­ly) tune­ful har­mo­ny and trans­po­si­tion to sup­port oth­er instru­ments and the human voice. It was there­fore log­i­cal to aban­don a wide vari­ety of tun­ing sys­tems, espe­cial­ly those based on tem­pera­ment, and to adopt equal tem­pera­ment as the stan­dard. By this time, com­posers were no longer explor­ing the sub­tleties of melodic/harmonic inter­vals; har­mo­ny involv­ing groups of singers and/or orches­tra paved the way for musi­cal innovation.

Looking back to the Baroque peri­od, many musi­col­o­gists tend to believe that the tun­ing sys­tem advo­cat­ed by J.S. Bach in The Well-tempered Clavier must have been equal tem­pera­ment… This belief can be dis­proved by a sys­tem­at­ic analy­sis of this cor­pus of pre­ludes and fugues on an instru­ment using all the tun­ing sys­tems en vogue dur­ing the Baroque peri­od — read the page The Well-tempered Clavier.

Composers and instru­ment mak­ers did not tune “by num­bers”, as tun­ing pro­ce­dures were not doc­u­ment­ed (see Asselin P-Y., 2000). Rather, they tuned “by ear” in order to achieve a per­ceived reg­u­lar­i­ty of sets of inter­vals: the tem­pera­ment in gen­er­al. This was indeed a break with the “Pythagorean” mys­tique, because these tem­pera­ments can­not be reduced to fre­quen­cy inter­vals based on inte­ger ratios.

For instance, Zarlino’s mean­tone tem­pera­ment — read this page — is made of 12 fifths start­ing from “E♭” (“mi♭”) up to “G#” (“sol#”) dimin­ished by 2/7 of a syn­ton­ic com­ma (ratio 81/80). The fre­quen­cy ratio of each fifth is therefore

\[\ \frac{3}{2}\left(\frac{80}{81}\right)^{\frac{2}{7}}=\ 1.5\ x\ 0.99645\dots\ =\ 1.4946\dots\ \left(or\ 695.81\dots\ cents\right)\]

which can­not be reduced to an inte­ger ratio. In the same way, the twelve inter­vals of the equal tem­pera­ment scale are expressed in terms of irra­tional fre­quen­cy ratios.


The goal of just into­na­tion is to pro­duce “opti­mal­ly con­so­nant” chords and sequences of notes, a legit­i­mate approach when con­so­nance is the touch­stone of the high­est achieve­ment in art music. This was indeed the case in sacred music, which aimed for a “divine per­fec­tion” guar­an­teed by the absence of “wolf tones” and oth­er odd­i­ties. In a broad­er sense, how­ev­er, music is also the field of expec­ta­tion and sur­prise. In an artis­tic process, this can mean depart­ing from “rules” — just as poet­ry requires break­ing the seman­tic and syn­tac­tic rules of a language…

Even when chords are per­fect­ly con­so­nant and con­form to the rules of har­mo­ny (as per­ceived by the com­pos­er), note sequences may devi­ate from their the­o­ret­i­cal posi­tions in order to cre­ate a cer­tain degree of ten­sion or to make a bet­ter tran­si­tion to the next chord.

When the Greek-French com­pos­er Iannis Xenakis - known for his for­malised approach to tonal­i­ty - heard Bach’s First Prelude for Well-Tempered Clavier played on the Shruti Harmonium in just into­na­tion, he declared his pref­er­ence for the equal-tempered ver­sion! This made sense for a com­pos­er whose music had been praised by Tom Service for its “deep, pri­mal root­ed­ness in rich­er and old­er phe­nom­e­na even than musi­cal his­to­ry: the physics and pat­tern­ing of the nat­ur­al world, of the stars, of gas mol­e­cules, and the pro­lif­er­at­ing pos­si­bil­i­ties of math­e­mat­i­cal prin­ci­ples” (Service T, 2013).

Bernard Bel — Dec. 2020 / Jan. 2021


Arnold, E.J.; Bel, B. L’intonation juste dans la théorie anci­enne de l’Inde : ses appli­ca­tions aux musiques modale et har­monique. Revue de musi­colo­gie, JSTOR, 1985, 71e (1-2), p.11-38.

Asselin, P.-Y. Musique et tem­péra­ment. Paris, 1985, repub­lished in 2000: Jobert. Soon avail­able in English.

Bel, B. A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra. Note interne, Groupe Représentation et Traitement des Connaissances (CNRS), March 1988a.

Bel, B. Raga : approches con­ceptuelles et expéri­men­tales. Actes du col­loque “Structures Musicales et Assistance Informatique”, Marseille 1988b.

Rao, S.; Van der Meer, W. The Construction, Reconstruction, and Deconstruction of Shruti. Hindustani music: thir­teenth to twen­ti­eth cen­turies (J. Bor). New Delhi, 2010: Manohar.

Raju, C. K. Cultural foun­da­tions of math­e­mat­ics : the nature of math­e­mat­i­cal proof and the trans­mis­sion of the cal­cu­lus from India to Europe in the 16th c. CE. Delhi, 2007: Pearson Longman: Project of History of Indian Science, Philosophy and Culture : Centre for Studies in Civilizations.

Service, T. A guide to Iannis Xenakis’s music. The Guardian, 23 April 2013.

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