“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):

**From 4/3 up:**FA, DO, SOL, RE, LA, MI, SI, FA#, DO#, SOL#, RE#, LA#**From 4/3 down:**FA, SIb, MIb, LAb, REb, SOLb**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**From 320/243 down:**FA-1, SIb-1, MIb-1, LAb-1, REb-1, SOLb-1**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**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 3^{7} * 4^{5} = 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 2^{11} = 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.

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.

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 16 ^{th} 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.