For more than twenty centuries, musicians, instrument makers and musicologists figured out scale models and tuning procedures for the creation of music embodying the concept of “consonance”. There was a shared notion of the octave and the major fifth (interval “C” to “G”) being the building blocks of these models, and the harmonic major third (interval “C” to “E”) lately played a significant role in European baroque and classical music.
Computer-controlled electronic instruments open new avenues for the implementation of microtonality including just-intonation frameworks dividing the octave in more than 12 grades (https://bolprocessor.org/microtonality/). Throughout centuries, Indian art music claimed its adherence to a division of 22 intervals (the ṣruti-swara system) theorized in Nāṭyaśāstra, a Sanskrit treatise dating back between 400 BCE to 200 CE. Since consonance (saṃvādī) is the basis of both ancient Indian and European tonal systems, we felt the urge for a theoretical framework encompassing all models.
Unfortunately, the topic of “just intonation” is exposed in an altogether confusing and reductive manner (read Wikipedia) due to musicologists’ focus on integer ratios aimed at reflecting the distribution of higher partials in periodical sounds. While these speculative models of intonation may respond to beliefs in the mystical properties of natural numbers — as claimed by Pythagoreanists — they were rarely checked against non-directed musical practice. Instrument tuners mostly rely on their own auditory perception of intervals rather than resorting to numbers, despite the availability of “electronic tuners”…
Interestingly, the ancient Indian theory of natural scales does not rely on arithmetics. This should be surprising given that in Vedic times mathematicians/philosophers had laid out the foundations of calculus and infinitesimals which much later were exported from Kerala to Europe and borrowed/appropriated by European scholars — read C.K. Raju’s Cultural Foundations of Mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE. This epistemological paradox was an incentive to decrypt the model depicted by the author(s) of Nāṭyaśāstra via a thought experiment: the two-vina experiment (https://bolprocessor.org/two-vina-experiment).
Earlier interpretations of this model, mimicking the Western habit of dealing with intervals as frequency ratios, failed to explain the intervalic structure of ragas in Hindustani classical music. In fact, the implicit model of Nāṭyaśāstra is a “flexible” one because the size of the major third (or equivalently the pramāņa ṣruti) is not stated in advance. Read Raga intonation (https://bolprocessor.org/raga-intonation) and listen to examples to grasp the articulation between the theory and practice of intonation in this context.
In Europe, the harmonic major third was finally accepted as a “celestial interval” after the Council of Trent (1545-1563) putting an end to the banishment of polyphonic singing in religious gatherings. Along with the development of fixed-pitch keyboard instruments, this gave way to the elaboration of theoretical models and tuning procedures attempting to include this interval in “pure intonation”. In theory, this is not feasible on a chromatic (12-grade scale) but it can be figured out and applied to Western harmony if more grades (29 to 41) are permitted. However, choosing enharmonic positions suitable for a harmonic context remains an uncertain venture.
Once again, the Indian model came to the rescue because it can be extended to produce a consistent series of twelve “optimally consonant” chromatic scales in compliance with chord intervals in Western harmony. Each scale contains 12 grades, which is more than the notes of chords it is applicable for. Sound examples are provided to illustrate this process (https://bolprocessor.org/just-intonation-framework/).
Tuning mechanical keyboard instruments (church organ, harpsichord, pianoforte) for 12-grade scales made it necessary to distribute unwanted dissonance (the syntonic comma) in an acceptable manner over series of fifths and fourths. Many tempered tuning procedures were designed, during the 16th to 19th centuries, with emphasis on either “perfect fifths” or “pure major thirds”, in response to the constraints of specific musical repertoires.
These techniques have been documented in detail by organ player and instrument designer Pierre-Yves Asselin, along with methods for achieving the tuning on a mechanical instrument such as the harpsichord. His book Musique et tempérament (soon available in English) was a guideline for implementing a similar approach in the Bol Processor (https://bolprocessor.org/microtonality/). This framework will make it possible to listen to baroque and classical works, using Csound instruments, with the very tunings their composers had been favoring — according to historical sources.