The following is the procedure for exporting just-intonation scales from murcchana-s of Ma-grama stored in “-cs.12_scales”.
➡ Read Just intonation: a general framework for explanations.
The scale model
As indicated on page Just intonation: a general framework, just-intonation chromatic scales can be derived from a basic framework made of two cycles of perfect fifths (frequency ratio 3/2).
These produce the 22-shruti framework of Indian musicologists (read Raga intonation) or the series named “Pythagorean” and “1st-order ascending-third” (“LA-1”, “MI-1” etc.) in the approach of western musicologists (see picture on the side).
We found that the “1st-order descending-third cycle” (“LAb+1”, “MIb+1” etc.) in which all notes are higher by a syntonic comma may not be required for the creation of just-intonation chords.
These cycles of fifths are represented graphically (scale “2_cycles_of_fifths” in Csound resource “-cs.tryTunings”):
There are a few differences between this 29-grade division of the octave and the Indian framework, notably the creation of “DO-1” and “FA-1”, two positions lower by one syntonic comma than “DO” (“C” = “Sa” in the Indian convention) and “FA” (“F” = “Ma”). Interestingly, these positions appear in ancient texts under the names “cyuta Sa” and “cyuta Ma”. Other additional positions are “REb-1”, “MIb-1”, “SOLb-1”, “LAb-1” and “SIb-1”.
The rule we follow when producing chromatic scales via transpositions of Ma-grama is that only positions displayed on this graph should be considered valid. When exporting a minor or major chromatic scale from a transposition of Ma-grama, it may occur that a note position is not part of this framework. In all cases of this procedure, the invalid position is one syntonic comma too low. Therefore the exported scale will be “aligned” raising all its positions by one comma.
The term “Pythagorean series” is confusing because any cycle of perfect fifths is Pythagorean by definition. Whether a position in a scale “is” or “is not” Pythagorean depends on the starting note of the series that was announced as “Pythagorean”. In Asselin’s work the starting point of the series in the central column is “FA”. In the Indian system, basic frameworks (Ma-grama and Sa-grama) start from “Sa” (“C” or “do”) and the Pythagorean/harmonic status of a position is determined by factors of its frequency ratio with respect to “Sa”. If a factor “5” is found in the numerator or the denominator, the position is harmonic, or Pythagorean in the reverse case.
Thus, for instance, “DO#” in Asselin’s “Pythagorean” series (two perfect fifths above “SI”) is evaluated as a harmonic position (marked in green) on the Bol Processor graphic and its ratio is 16/15. In reality, “DO#” in Asselin’s series has a frequency ratio 243/128 * 9/16 = 2187/1024 = 1.068 which is very close to 16/15 = 1.067. “DO#-1” in Asselin’s series is two perfect fifths above “SI-1” which yields a frequency ratio 15/8 * 9/16 = 135/128 = 1.054 which is close to 256/243 = 1.053 and marked “Pythagorean” on the Indian scheme. Thus, “DO#” and “DO#-1” have exchanged their properties because each of them is the superposition of two very close positions belonging to different series.
Ignoring schisma differences to take the simplest ratios create this confusion. Therefore, we keep preferring comma indications — e.g. “FA” and “FA-1” — to identify positions, in which the first instance belongs to the series termed “Pythagorean” in Asselin’s work.
This table summarizes a quick procedure for creating all murcchana-s of the Ma-grama chromatic scale and export minor and major chromatic scales therefrom.
Open the “Ma_grama” scale in the “-cs.12_scales” Csound resource and select the Murcchana procedure. To create “Ma01″, move note “F” to note “C” and click TRANSPOSITION.
|F moved to||Murcchana||Minor scale||Raise||Major scale||Identical|
For example, this is the “Ma04″ murcchana obtained by placing “F” (M1 on the Indian scale model) of the moveable wheel on “Eb” (G1 of the outer crown):
The resulting scale “Ma04″ is:
In the last column of the table, “Adjust” indicates the fraction by which the ratios of notes may need to be multiplied so that no position is created outside the Pythagorean and harmonic cycles of fifths according to the Indian system. Practically this is the case when the frequency ratio contains a multiple of 25 in either its numerator or denominator, as this indicates that the position has been constructed by at least two successive major thirds (up or down).
A warning is displayed when this is the case, and a sinple click on ADJUST SCALE fixes positions:
In this example, the warning signals an out-of-range position of “B” (50/27) on the “Ma10″ scale. Also note that “F#” has a multiple of 25 in its numerator.
After clicking ADJUST SCALE, the “Ma10″ scale is finalized with “B” at position 15/8. This has been done by raising all notes by a syntonic comma (81/80) :
This procedure is mentioned in Indian musicology under the name of sadja-sadharana telling that all notes of the scale are raised by one shruti — here, a syntonic comma (Shringy & Sharma 1978). In this model, it is also invoked for scales “Ma11″ and “Ma12″. The result is (as expected) a circular model because “Ma13″ is identical to “Ma01″ as shown by the scale comparator at the bottom of page “-cs.12_scales”.
This circularity is a property of the set of murcchana-s which has no influence on exported minor and major scales because their positions will be aligned in compliance with the basic rule exposed in the first section.
Exporting and aligning minor scales
The “Ma04″ murcchana produces “Cmin” by exporting notes facing marks on the inner wheel.
As explained on page Just intonation: a general framework, the tonic and dominant notes of every minor chord should belong to the “minus-1” position. In this example, “C” and “G” are one comma lower in a “C minor” chord than in a “C major” chord (matching “DO-1” and “SOL-1” on the “2_cycles_of_fifths” scale) , a fact which had been predicted and experimentally checked by Pierre-Yves Asselin (2000 p. 137).
All chromatic minor scales exported from murchana-s of Ma-grama are correctly positioned with respect of enharmonic positions of main notes in just-intonation chords. This can easily be checked comparing ratios with the ones associated with the western series on “2_cycles_of_fifths” (top of this page). This confirms that a tuning system using only two series of perfect fifths is convenient for the construction of a just-intonation framework.
Exporting and aligning major scales
The “Ma04″ murcchana produces “Ebmaj” by exporting notes facing marks on the inner wheel and raising “F”:
According to a rule exposed on page Just intonation: a general framework, the basic note of every major chord should be both in the high position and in the Pythagorean series (blue markings). This is true of “Eb major” chord extracted from the “Ebmaj” chromatic scale, yet not with scales “F#maj”, “Bmaj” and “Emaj” displayed in bold style on the table.
Let us for instance look at “Emaj” exported without precaution from “Ma09″:
Note “E” has a frequency ratio 5/4 which is labeled “MI-1” on scale “2_cycles_of_fifths” (top of this page). Since “MI-1” belongs to a harmonic series, it cannot be taken as a the tonic of a “E major chord”. The Pythagorean “MI” (ratio 81/64) should be used instead.
After its alignment — raising all notes by 1 syntonic comma — the final “Emaj” scale is obtained:
This alignment of exported major scales is done automatically by the Bol Processor when exporting a major chromatic scale.
Asselin, P.-Y. Musique et tempérament. Paris, 1985, republished in 2000: Jobert. Soon available in English.
Shringy, R.K.; Sharma, P.L. Sangita Ratnakara of Sarngadeva: text and translation, vol. 1, 5: 7-9. Banaras, 1978: Motilal Banarsidass. doi:10.2307/2054840. Source in the Web Archive.