Just intonation: a general framework

“Just into­na­tion” (into­na­tion pure in French) is a word employed by com­posers, musi­cians and musi­col­o­gists to des­ig­nate var­i­ous aspects of com­po­si­tion, per­for­mance and instru­ment tun­ing. These point at the same goal of “playing/singing in tune” — what­ev­er it means. Implementing a gener­ic abstract mod­el of just into­na­tion in the Bol Processor is a chal­lenge beyond our cur­rent com­pe­tence… We address it in a prag­mat­ic man­ner, look­ing at a few musi­cal tra­di­tions fol­low­ing the same goal with the aid of reli­able the­o­ret­i­cal models.

A com­plete and con­sis­tent frame­work for con­struct­ing just-intonation scales — or “tun­ing sys­tems” — was the grama-murcchana mod­el elab­o­rat­ed in ancient India. This the­o­ry has been exten­sive­ly com­ment­ed and (mis)interpreted by Indian and Western schol­ars: read a detailed sur­vey in 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 Bol Processor + Csound.

This page is a con­tin­u­a­tion of Microtonality but it 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 for tun­ing musi­cal instru­ments have been doc­u­ment­ed for more than 2000 years in sev­er­al parts of the World. For prac­ti­cal and per­son­al rea­sons, we will focus on work in Europe and the Indian subcontinent.

Systems described as “just into­na­tion” are attempts to set up a tun­ing in which all tonal inter­vals would be con­so­nant. A large body of the­o­ret­i­cal work on just into­na­tion is avail­able — check Wikipedia for links and abstracts.

Models are amenable to Hermann von Helmholtz’s notion of con­so­nance deal­ing with the per­cep­tion of the pure sine wave com­po­nents of com­plex sounds con­tain­ing mul­ti­ple tones. According to the the­o­ry of har­mo­ny, 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 hit or plucked in a gen­tle way. However, this har­monic­i­ty is miss­ing in many wind instru­ments, notably reed instru­ments such as the sax­o­phone or the shehnai in India, and indeed in per­cus­sion instru­ments or bells which com­bine mul­ti­ple modes of vibration.

If just into­na­tion is invoked for tun­ing a musi­cal instru­ment, it must there­fore be anal­o­gous to a zither, a swara man­dal, a harp­si­chord, a piano or a pipe organ, includ­ing elec­tron­ic devices pro­duc­ing sim­i­lar sounds.

Maybe because of their late “dis­cov­ery” of cal­cu­lus — actu­al­ly “bor­rowed” from Indian, Persian and Arabic sources — Europeans cul­ti­vat­ed a fas­ci­na­tion for 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 designed pro­ce­dures to tune their instru­ments, lis­ten­ing to inter­vals and pick­ing up the ones that made sense to their ears — read page The two-vina exper­i­ment. Following the devel­op­ment of music acoustics, attempts were done to inter­pret these pro­ce­dures in terms of fre­quen­cy ratios. However, this was a risky ven­ture because the dream of per­fec­tion lead to the sim­plis­tic pro­mo­tion of “per­fect ratios”.

Seeking the kind of per­fec­tion embod­ied by num­bers is the best way to pro­duce taste­less music. While just into­na­tion — inter­vals with no beats — is nowa­days achiev­able on elec­tron­ic instru­ments, it is based on a nar­row con­cept of tonal­i­ty. This can be ver­i­fied by lis­ten­ing to ancient west­ern music played with var­i­ous types of tem­pera­ment — read page Comparing tem­pera­ments — and indeed to Indian clas­si­cal music — read page Raga into­na­tion.

The “Greek” approach

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

The prac­tice of poly­phon­ic music on fixed-tuning instru­ments showed that this per­fect con­so­nance is nev­er achieved with 12 grades in an octave — the con­ven­tion­al chro­mat­ic scale. In west­ern clas­si­cal har­mo­ny, it would demand a retun­ing of the instru­ment accord­ing to the musi­cal genre, the musi­cal piece and the har­mon­ic con­text of every note sequence or chord.

Imperfect tonal inter­vals gen­er­ate unde­sired 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 reveals that this is inher­ent to arith­metics and not a defect of instru­ment design. Imagine the tun­ing of ascend­ing fifths (ratio 3/2) by suc­ces­sive steps on a harp with a shift of octaves to main­tain the result­ing note with­in the orig­i­nal octave. Frequency ratios would be 3/2, 9/4, 27/16, 81/64 etc. At this stage, the note seems to be locat­ed at a major third although its actu­al ratio (81/64 = 1.265) is high­er than 5/4 (1.25). The 81/64 inter­val is named a 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 a syn­ton­ic com­ma.

Whoever designed the so-called “Pythagorean tun­ing” went fur­ther in their inten­tion to describe all musi­cal notes via cycles of fifths. Moving 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 sound­ing of some of the result­ing inter­vals, things turn bad if one hopes to ter­mi­nate the cycle on the ini­tial note. If the series start­ed on “C” it does end on “C” (or “B#”), yet with a ratio 531441/524288 = 1.01364 slight­ly high­er than 1. This gap is called the Pythagorean com­ma, con­cep­tu­al­ly dis­tinct from the syn­ton­ic com­ma (1.0125) although their sizes are almost iden­ti­cal. This para­dox is a mat­ter of sim­ple arith­metics: pow­ers of 2 (octave inter­vals) nev­er match pow­ers of 3.

Needless to say, attribut­ing this sys­tem to “ancient Greeks” is pure fan­ta­sy because (unlike Egyptians) they did not know the usage of fractions!

Despite the com­ma prob­lem, tun­ing instru­ments by series of per­fect fifths was com­mon prac­tice in Middle-Aged Europe, fol­low­ing the organum which con­sist­ed in singing/playing par­al­lel fifths or fourths to enhance a melody. One of the old­est trea­tis­es on “Pythagorean tun­ing” was pub­lished by Henri Arnault de Zwolle cir­ca 1450 (Asselin 2000 p. 139). With this tun­ing, major “Pythagorean” thirds sound­ed harsh, which explains that in those days the major third was rat­ed a dis­so­nant interval.

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

The Indian approach

The grama-murcchana mod­el was described in Natya Shastra, a Sanskrit trea­tise on the per­form­ing arts com­piled in India about twen­ty cen­turies ago. Chapter 28 con­tains a dis­cus­sion of the “har­mon­ic scale” based on a divi­sion of the octave in 22 shru­ti-s, where­as it enlists only sev­en swaras (notes) used by musi­cians: “Sa”, “Re”, “Ga”, “Ma”, “Pa”, “Dha”, “Ni”. These may be mapped to west­ern con­ven­tion­al 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 may be extend­ed to a 12-grade (chro­mat­ic) scale thanks to diesis and flat alter­ations respec­tive­ly rais­ing or low­er­ing a note by a semi­tone. Altered notes in the Indian sys­tem are com­mon­ly named “komal Re”, “komal Ga”, “Ma tivra”, “komal Dha” and “komal Ni”. The word “komal” may be trans­lat­ed as flat and “tivra” as diesis.

The focus of 20^th cen­tu­ry research in Indian musi­col­o­gy has been to “quan­ti­fy” shru­ti-s in a sys­tem­at­ic man­ner and 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 num­ber ratios, be it fre­quen­cies or lengths of vibrat­ing strings. This point has been over­looked by “colo­nial musi­col­o­gists” due to their lack of insight into Indian math­e­mat­ics and their fas­ci­na­tion for a mys­tique of num­bers inher­it­ed from Neopythagoreanism.

Bharata Muni, the author(s) of Natya Shastra, may have heard about “Pythagoras tun­ing”, a the­o­ry which they could expand con­sid­er­ably because of their advance in the usage of cal­cu­lus. Despite this, the whole chap­ter on musi­cal scales does not cite a sin­gle num­ber. This para­dox is dis­cussed on 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 that should be for­mal­ized with alge­bra rather than a finite set of inte­ger ratios.

How (and why) should one divide the octave in 22 micro-intervals when only 5 to 12 notes are named in most Indo-European musi­cal sys­tems? A few inno­cent eth­no­mu­si­col­o­gists claimed that Bharata’s mod­el must be a vari­ant of the Arabic “quar­ter­tone sys­tem” or even a 22-grade tem­pered scale… If so, why not 24 shru­ti-s? Or any arbi­trary num­ber? The two-vina exper­i­ment yields shru­ti-s unequal in size. No sum of micro­tones mea­sur­ing 54.5 cents in a 22-grade tem­pered scale would pro­vide an inter­val close to 702 cents — the per­fect fifth giv­ing 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 to low­er all the notes of one instru­ment by “one shru­ti” and he gives a list of notes that will match between the two instru­ments. The oper­a­tion is repeat­ed three more times until all match­es have been made explic­it. This yields a sys­tem of equa­tions (and inequa­tions) for the 22 unknown vari­ables. Additional equa­tions can be inferred from a pre­lim­i­nary state­ment that the octave and the major fifth are “con­so­nant” (sam­va­di), there­by fix­ing ratios close to 2/1 and 3/2. (Read the detailed process on page The Two-vina Experiment and the maths in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra.)

Still, one more equa­tion is required which Bharata’s mod­el does not pro­vide. Interestingly, in Natya Shastra the major third is clas­si­fied “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 16^th cen­tu­ry — as fixed-pitch key­board instru­ments had become pop­u­lar (Asselin 2000 p. 139) — which many musi­col­o­gists take for grant­ed in their inter­pre­ta­tion of the Indian mod­el. […] thirds were con­sid­ered inter­est­ing and dynam­ic con­so­nances along with their invers­es the 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 the flex­i­bil­i­ty of into­na­tion schemes in Indian music — read 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. Compliance with the two-vina exper­i­ment only implies that the com­ma takes its val­ue between 0 and 56.8 cents (Bel 1988a). 

➡ 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) mea­sures 1200 cents, and each semi­tone is approx­i­mate­ly 100 cents.

The con­struc­tion and assess­ment of raga scale types based on this flex­i­ble mod­el is demon­strat­ed on page Raga into­na­tion.

Extending the Indian model to western harmony

An incen­tive to apply­ing the Indian frame­work to west­ern clas­si­cal music is that both tra­di­tions have giv­en a pri­or impor­tance to the con­so­nance of per­fect fifths asso­ci­at­ed with a 3/2 fre­quen­cy ratio. In addi­tion, let us agree with fix­ing the har­mon­ic major third to inter­val 5/4 (384 cents), there­by lead­ing to 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 = approx. 90 cents) and the minor semi­tone (25/24 = approx. 70 cents).

These inter­vals were well-known to west­ern musi­col­o­gists who tried to fig­ure out just-intonation scales playable on (12-grade per octave) key­board instru­ments. 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”:

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

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

Positions R1, R2 etc. are labelled with abbre­vi­a­tions of names Sa, Re, Ga, Ma, Pa, Dha, Ni. For instance, Ga (“E” in English) may 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.

Notes of the chro­mat­ic scale have been labeled using the Italian/Spanish/French con­ven­tion “do”, “re”, “mi”, “fa”… rather than English 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 telling how each posi­tion may be derived from the base note (Sa). For instance, the pic­togram near N2 (“B flat” = “si bemol”) shows 2 ascend­ing per­fect fifths and 1 descend­ing major third.

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

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

The frame­work imple­ment­ed in Bol Processor deals with inte­ger ratios allow­ing high accu­ra­cy. Nonetheless, it delib­er­ate­ly wipes out schis­ma dif­fer­ences. This is done by adjust­ing cer­tain ratios, for instance 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). In case 2187/2048 and fur­ther com­plex ratios of the same series were deemed unprac­ti­cal, the Indian mathematician/physicist (fol­low­ing the Āryabhaţīya) would replace them all with “16/15 āsan­na (near val­ue)”… This is exem­plary of Indian math­e­mat­ics designed for cal­cu­lus rather than proof con­struc­tion. In the Western Platonicist approach, math­e­mat­ics tar­get­ed “exact val­ues” as a sign of per­fec­tion, which led its pro­po­nents to fac­ing seri­ous prob­lems with “irra­tional” num­bers and even the log­ic under­ly­ing for­mal proof-making pro­ce­dures (Raju 2007 p. 387-389).

This dia­gram and the move­able gra­ma wheel that will be next intro­duced could be built with any size of the syn­ton­ic com­ma in 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 of the flex­i­ble mod­el, just 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 when opt­ing for a Pythagorean inter­val (e.g. G4, ratio 81/64) 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 named wolf fifth as its use in melod­ic phras­es or chords is held to sound “out of tune”, with a negative/devilish mag­i­cal conotation.

No posi­tion of 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 on sev­er­al steps — here, max­i­mum 5 or 6 up and down. However, tun­ing a har­mon­ic major third requires a lit­tle more atten­tion. Therefore, imag­in­ing an accu­rate tun­ing pro­ce­dure based on a suc­ces­sion of major thirds would be unre­al­is­tic — even though, indeed, this can be achieved with the help of elec­tron­ic devices.

On top of the dia­gram (posi­tion “fa#”) we notice that none of the two cycles of fifths clos­es on itself because of 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 par­tic­u­lar­i­ty at the top of the pic­ture is the appar­ent dis­rup­tion of sequence L-C-M. However, remem­ber­ing that L = M + C indi­cates that the reg­u­lar­i­ty is restored by opt­ing between P1/M3 and P2/M4.

Approximations have no impli­ca­tion on the sound­ing of musi­cal inter­vals because no human ear would appre­ci­ate a schis­ma dif­fer­ence (2 cents). However, oth­er dif­fer­ences need to remain explic­it since inte­ger ratios denote the tun­ing pro­ce­dure by which the scale may be con­struct­ed. Thus, the replace­ment inte­ger ratio may turn out more com­plex than the “schis­mat­ic” one, as is the case with 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 doc­u­ment­ed in great detail by 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-grade scales via tem­pera­ment — com­pro­mis­ing the pure inter­vals of just into­na­tion to meet oth­er require­ments. Techniques of tem­pera­ment applic­a­ble to Bol Processor are dis­cussed on pages Microtonality and Comparing tem­pera­ments.

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

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

Extending series of per­fect fifths beyond the sixth step cre­ates com­pli­cat­ed ratios that may be approx­i­mat­ed (with a schis­ma dif­fer­ence) to the ones pro­duced by har­mon­ic major thirds (ratio 5/4). Positions on the right (major third upward, first order) are one syn­ton­ic com­ma low­er than their equiv­a­lents in the cen­tral series, and posi­tions on the left (major third down­ward, first order) 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 to the left (“DOb#+2”, “SOLb#+2” etc.) but these second-order series are only used for the con­struc­tion of tem­pera­ments — read page Microtonality.

This mod­el pro­duces 3 to 4 posi­tions for each note, a 41-grade scale that would 41 keys (or strings) per octave on a mechan­i­cal instru­ment! This is a rea­son for tem­per­ing 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 along 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 pro­duced 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 suf­fi­cient to deter­mine the 3 or 4 posi­tions of each note, giv­en that sev­er­al ones may reach the same posi­tion at a schis­ma dis­tance. For instance, “REb” is at the same posi­tion as “DO#-1”. The IMAGE link dis­plays this scale with (sim­pli­fied) fre­quen­cy ratios:

Compared with the mod­el advo­cat­ed by Arnold (1974, see pic­ture on top), this sys­tem accepts har­mon­ic posi­tions on both sides of Pythagorean posi­tions, imply­ing that Sa (“C” or “do”) may take three dif­fer­ent posi­tions just like all unal­tered notes. In Indian music, Sa is unique because it is the base note of every clas­si­cal per­for­mance of raga, fixed by the drone (tan­pu­ra) and tuned at the con­ve­nience of singers or instru­ment play­ers. Nonetheless, 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 suit­able grid for con­struct­ing basic chords in just into­na­tion. For instance, a “C major” chord is made of its ton­ic “DO”, its dom­i­nant “SOL” a per­fect fifth high­er and “MI-1” at a har­mon­ic major third above “DO”. The first two notes may 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 that will be made explic­it later.

This does not entire­ly solve the prob­lem of play­ing tonal music in just into­na­tion. Sequences of chords must be prop­er­ly aligned. For instance, should one take the same “E” in “C major” and in “E major”? The answer is “no” but the rule needs to be made explicit.

How is it pos­si­ble to select the prop­er one among the 3⁷ * 4⁵ = 2 239 488 chro­mat­ic scales dis­played on this graph?

In the approach fol­lowed by Pierre-Yves Asselin (2000) — inspired by the work of Conrad Letendre in Canada — rules have been 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 assess­ment by practitioners.

The grama framework

Using Bharata’s mod­el — read page The two-vina exper­i­ment — we can con­struct chro­mat­ic (12-grade) scales in which each tonal posi­tion (among 11) has two options: har­mon­ic or Pythagorean. This is a rea­son for say­ing that the frame­work is based on 22 shru­ti-s. In Indian musi­co­log­i­cal lit­er­a­ture, the term shru­ti is ambigu­ous since it either des­ig­nates a tonal posi­tion or an interval.

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

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

Two options for each of the 11 notes yields a set of 2¹¹ = 2048 chro­mat­ic scales. Among these, only 12 are “opti­mal­ly con­so­nant”, i.e. con­tain­ing 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 describ­ing a basic 12-tone “opti­mal” scale named “Ma-grama”. This scale is named “Ma_grama” in Csound resource “-cs.12_scales”:

Clicking link IMAGE on the “Ma_grama” page yields a graph­ic rep­re­sen­ta­tion of this scale:

On this image, per­fect fifths are blue lines and the (unique) wolf fifth between C and G is a red line. Note 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 in which 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). However, this sim­ple rule is bro­ken when com­plex ratios have been replaced with sim­ple equiv­a­lents at a dis­tance of a schis­ma. Therefore, the blue and green marks on Bol Processor images are main­ly for facil­i­tat­ing the iden­ti­fi­ca­tion of a posi­tion: a note appear­ing near a blue mark­ing might as well belong to the har­mon­ic series with a more com­plex ratio bring­ing it near the Pythagorean position.

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

This Ma-grama is the start­ing point for gen­er­at­ing all “opti­mal­ly con­so­nant” chro­mat­ic scales. This is done by trans­po­si­tions of per­fect fifths (up or down). Visualizing trans­po­si­tions becomes clear if the base scale is drawn on a cir­cu­lar wheel allowed to move inside the out­er crown shown above. The fol­low­ing is Arnold’s com­plete mod­el show­ing Ma-Grama in the basic posi­tion pro­duc­ing the “Ma01″ scale:

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

Intervals are shown on the graph. For instance, 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:

C l Db c+m D c+l Eb c+m E c+l F c+m F# c+l G l Ab 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

This con­struc­tion of the “A minor” scale is com­pli­ant with the west­ern scheme for pro­duc­ing just-intonation chords: the basic note “A” (ratio 5/3) is “LA-1” on the “3_cycles_of_fifths” scale), locat­ed in the “major third upward” series as well as its dom­i­nant “MI-1”, where­as “C” (ratio 1/1) belongs to the series termed “Pythagorean”.

At first view, the scale con­struct­ed by this M1 trans­po­si­tion also resem­bles a “C major” scale, yet 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 using an alter­nate basic scale named “Sa-Grama” in which P4 replaces P3. 

P3 is named “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 inter­vals of the chro­mat­ic scale are list­ed with sig­nif­i­cant inter­vals high­light­ed in col­or. The wolf fifth is col­ored in red. Remember that if the scale is opti­mal­ly con­so­nant only one cell will be col­ored in red.

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

We may use “Ma01″ as a 23-grade micro­ton­al scale in Bol Processor pro­duc­tions because all notes rel­e­vant to the chro­mat­ic scale have been labelled. However it is more prac­ti­cal to extract a 12-grade scale with only labelled notes. This can be done on the “Ma01″ page. The image shows the expor­ta­tion of “Cmaj” scale con­tain­ing 12 grades 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 man­ner, a 12-grade “Amin” can be export­ed with­out rais­ing “D”.

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

In all 12-grade 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, giv­en that the machine rec­og­nizes both options.

Producing the 12 chromatic scales

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

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

For instance, pro­duce “Ma02″ by trans­pos­ing “Ma01″ of a per­fect fourth “C to F” (see pic­ture). Nothing else needs to be done. All trans­po­si­tions have been 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 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 instance mov­ing from “C major” to “C minor”.

To get the “C minor” scale, we need to cre­ate “Ma04″ via four suc­ces­sive descend­ing fifths (or ascend­ing fourths). Be care­ful that writ­ing “C to F” on the form will not always pro­duce a per­fect fourth trans­po­si­tion because the “F to C” inter­val might be a wolf fifth! This hap­pens when mov­ing from “Ma03″ to “Ma04″. In this case, select for instance “D to G”.

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

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

This shift of the base note can be made visu­al by mov­ing the inner wheel. After 4 trans­po­si­tions, posi­tion M1 of the inner wheel will match posi­tion G1 of the out­er wheel, yield­ing the fol­low­ing configuration:

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 shifts of the ton­ic on minor chords.

Pierre-Yves him­self men­tions a one-comma low­er­ing of “C” and “G” in the “C minor” chord. However, this was one among two options pre­dict­ed by his the­o­ret­i­cal mod­el. He checked it play­ing the Cantor elec­tron­ic organ at the University, report­ing musi­cians rat­ed this option as 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”:

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

This choice is based on per­cep­tu­al expe­ri­ence, name­ly “pratyakṣa pramāṇa” in Indian epis­te­mol­o­gy — read page The two-vina exper­i­ment. We fol­low an empir­i­cal approach rather than search­ing for an “axiomat­ic proof”. The ques­tion is not which of the two options shall be true, but which one pro­duces music that sounds cor­rect.

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 five options sug­gest­ed by his the­o­ret­i­cal mod­el, the author select­ed the one pre­ferred by all musi­cians. They even spon­ta­neous­ly choose this into­na­tion when singing with­out any spe­cif­ic instruc­tion. In addi­tion, this ver­sion is com­pli­ant with Zarlino’s “nat­ur­al scale”.

In the pre­ferred option, 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”.

On the pic­ture, tri­an­gles whose sum­mit points to the right are major chords, and the one point­ing to the left is the “D minor” chord.

Asselin’s con­clu­sion (2000 p. 137) is 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 in full agree­ment with the mod­el con­struct­ed by grama-murcchana. Since minor chro­mat­ic scales are export­ed from trans­po­si­tions of Ma-grama with all its grades 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 2^nd-order series of fifths in the right­most col­umn involv­ing two con­sec­u­tive ascend­ing major thirds result­ing in a low­er­ing of 2 syn­ton­ic com­mas. This process is fur­ther explained on 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 will hear the sequence of chords in equal-tempered into­na­tion, then in just-intonation.

Identity of the last occur­rence with Asselin’s favorite choice is marked by fre­quen­cies in the Csound score: “D4” in the third chord (D minor) is low­er by one com­ma than “D4” in the fourth chord (G major), where­as all oth­er notes (for instance “F4”) have the same fre­quen­cies in the four chords.

Show Csound score

To sum­ma­rize, the ton­ic and dom­i­nant notes of every minor chord belongs to the “low­er” har­mon­ic series of per­fect fifths appear­ing in the right col­umn of Asselin’s draw­ing repro­duced above. Conversely, the ton­ic and dom­i­nant notes of every major chord belongs to the “Pythagorean” series of per­fect fifths in the cen­tral column.

Checking note sequences

Rules set­ting the rel­a­tive posi­tions of major and minor modes (see above) only deal with the three notes defin­ing 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-grade) scales. Therefore, they also set up the enhar­mon­ic posi­tions of all notes that would be played in this har­mon­ic con­text.

Do these com­ply with just into­na­tion? In the­o­ry they do, 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 did exper­i­ments to check this the­o­ret­i­cal mod­el with my Shruti Harmonium pro­duc­ing pro­grammed inter­vals with 1-cent accu­ra­cy. Pierre-Yves Asselin played clas­si­cal pieces while Jim was manip­u­lat­ing 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 one sev­er­al dif­fer­ent scales to ren­der just into­na­tion. To this effect, vari­ables point­ing at 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.

Scale comparison

At the bot­tom of 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 floating-point fre­quen­cy ratios otherwise.

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

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

In a sim­i­lar way, trans­po­si­tions “Ma13″, “Ma14″ etc. are iden­ti­cal to “Ma01″, “Ma02″ etc. The series of chro­mat­ic scales is (as expect­ed) cir­cu­lar because “Ma13″ returns to “Ma01″.

More details about fre­quen­cies, block keys etc. may be found on page Microtonality.

Is this perfect?

This whole page is ded­i­cat­ed to tonal sys­tems defined in terms of inte­ger ratios (i.e. ratio­nal num­bers) mea­sur­ing tonal inter­vals. There exist­ed at least two strong incen­tives sup­port­ing the idea that every “pure” tonal inter­val should be worked as the 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.

Music his­to­ry (in the West) dates back to ideas attrib­uted to greek philoso­pher “Pythagoras” (read above) believ­ing that all things were made of [ratio­nal] num­bers. This approach stum­bled on the impos­si­bil­i­ty of match­ing the octave to a suc­ces­sion 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 much more advanced than the Greeks with respect to 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 design a tun­ing sys­tem meet­ing the require­ments of (approx­i­mate­ly) tune­ful har­mo­ny and trans­po­si­tion for the sup­port of oth­er instru­ments and human voic­es. This had made it log­i­cal to aban­don a great vari­ety of tun­ing sys­tems — notably the ones based on tem­pera­ment — and adopt equal tem­pera­ment as the stan­dard. At this stage, com­posers no longer explored the sub­tleties of melodic/harmonic tonal inter­vals; har­mo­ny involv­ing groups of singers and/or orches­tra paved the way to musi­cal innovation.

When 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 refut­ed by a sys­tem­at­ic analy­sis of this cor­pus of pre­ludes and fugues with an instru­ment using all tun­ing pro­ce­dures en vogue dur­ing the Baroque peri­od — read page The Well-tempered Clavier.

Composers and instru­ment design­ers did not tune “by num­bers”, as tun­ing pro­ce­dures were not doc­u­ment­ed on that mod­el (read Asselin P-Y., 2000). They rather tuned “by ear” in order to achieve a per­ceived reg­u­lar­i­ty of sets of inter­vals: tem­pera­ment in gen­er­al. This was indeed a breach 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. Likewise, the twelve inter­vals of the equal-tempered scale are expressed by irra­tional fre­quen­cy ratios.

Overture

The goal of just into­na­tion is to pro­duce “opti­mal­ly con­so­nant” chords and note sequences, a legit­i­mate approach when con­so­nance is the touch­stone of the high­est achieve­ment in art music. This was indeed the case of sacred music aim­ing at a “divine per­fec­tion” ensured by the absence of “wolf tones” and oth­er odd­i­ties. However, from a broad­er view­point, music is also the field of both expec­ta­tion and sur­prise. In an artis­tic process, this may imply devi­a­tions from “rules” — the same way poet­ry requires a breach of seman­tic and syn­tac­tic rules of a language…

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

When Greek-French com­pos­er Iannis Xenakis — well-known for his for­mal­ized approach of tonal­i­ty — lis­tened to Bach’s First pre­lude for Well-Tempered Clavier played in just into­na­tion on the Shruti Harmonium, he told 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

References

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, 71^e (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.

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