# Semantic Daniélou-53

For those of you interested in microtonal composition, here’s a free software developped to use Alain Daniélou’s semantic system, inspired by indian Shrutis.
( Scala files are also available )

Note that there’s also a composition competition.

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Awesome thank you for this! Do you think you will take part in the competition?

You’re welcome !
I think I’ll give it a shot… I saw some interesting exercises in the user manual, I’ll try them and see if something interesting comes out of it.

The “12-note equal temperament” scale, which has become the standard scale for western instruments, evens the scale to identical intervals in each key, at the expense of precision in tuning, since apart from the octave, the other eleven intervals it contains are more or less “out of tune” on an acoustic level.

from the intro page

and i’d love to know : what

more or less “out of tune” on an acoustic level

means ?

Compare and contrast

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:

12√2 = 2 1⁄12 ≈ 1.059463
This interval is divided into 100 cents.

Vs

In music, just intonation (sometimes abbreviated as JI) or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers.

Believe it or not, our ears can hear the difference between large divisors and small divisors, and we generally find the small ratios more pleasant.

So why do we use equal temperament? The answer has to do with the composability and performability of harmony, partially, but it also has to with cultural inertia, as well as the difficulty of instrument design. Acoustic instruments designed for just intonation tend to be complex and unwieldy. It’s only with the advent of electronic instruments that we have easy ways to play with JI tunings.

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well in a very small nutshell they are probably referring to “dissonance” as defined in limit theory. (take the following with a grain of salt, there are plenty of nits to pick.)

so take two frequencies, f1 and f2. say that R is the ratio f1/f2.

now, R may be a rational number, expressible in the form `a/b` where `a` and `b` are integers. these will reduce to some combination of relative primes, like 9/8.

in limit theory (which the writer of this software seems to follow) you would say an interval is “more dissonant” if its rational expression is built from large prime numbers, like 3967 / 4643, and “less dissonant” if its rational expression is composed of small primes, like 9/8 = (33)/(222) (a just-intoned major 2nd) or 4/3 = 22/3 (a just-intoned perfect 4th.) these are both “3-limit” ratios. in just intonation it’s common to restrict intervals to 5-limit - like Pythagoras.

in some respects this maps pretty well to mathematical and psychoacoustic phenomena. this “harmonic limit” describes how often the two waves will come back into phase; tones played together with a simple harmonic relationship tend to be readily perceivable as as single blended tone; and we can more consistently identify the “center pitch” of a cluster of simple ratios.

but this approach does not really lead to consistent results in application. Pythagoras in following his 5-limit rule ended up with ratios like 81/64, which end up having long enough relative periods to sound “rough” and full of beating. it gets worse when you start inverting them. so people started using 7-, 11, 13-limits, and introducing “commas” which are kind of arbitrary ratios that help partition the octave into different invertible regions. it all gets pretty complicated and jargon-heavy. if you go over to the “books” thread and check out the links to Divisions of the Tetrachord i posted, that will give you all the historical tuning theory you could ever want, and more.

and of course all this is rather western-greek-renaissance-centered. other tuning systems in the world place particular value on beating frequencies and “roughness,” the most famous being in Indonesian gamelan which explicitly uses (appromximately) fixed linear frequency relationships between instruments in a pair.

(and yeah, i know danielou was looking at N. indian music. i should really read his book again which i haven’t done in a long time. but he was very directly applying Pythagorean ideas to the analysis of shruthi modes; my knee-jerk reaction is that it misses the point a bit and falls into the same old number-worship that is sometimes directly at odds with sensory reality. but i could stand to be better informed on the specifics of his theories.)

anyways,
bringing it back to the context: in 12tet, only the octave is rational. all other ratios are actually irrational, being powers of the 12th root of 2 - in their mathematically pure form, these frequencies will never come back in phase once started!

so “more or less acoustically dissonant” probably means, more or less far from the nearest just-intoned scale degree (which could have plenty of different definitions.) the scare quotes are deserved because “acoustically dissonant” doesn’t really have an objective definition - the closest might be the psychoacoustic / sensory definition in which “more dissonant” means “more readily perceived as beating.” [this can get a lot more technical but don’t think i can do it justice here/now]

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It seems that Daniélou’s system “thanks to remarkable micro-coincidences” still uses some interesting intervals, other than just mathematical ones, if I get this right (extract from the user manual) :

The 5th harmonic limit proper to the Semantic system uses, among the whole numbers found in its ratios, only products of prime numbers up to 5. In other words, it only uses primes 2, 3, and 5, in accordance with the hypothesis formulated by Alain Daniélou in his book “Sémantique musicale” concerning our perceptions of musical intervals. Furthermore, thanks to remarkable micro-coincidences, the 7th natural overtone (14 occurrences of this interval in the Semantic-53 scale) and the 17th and 19th overtones, to name a few, are naturally present in several ways, notably among indian shrutis, as well as the Semantic system.

As for the “out of tune on an acoustic level” part, I thought this was refering to harmonic overtones sounding “better” together when using just intonation chords, but I’m really not sure about that, I still have to read Daniélou’s book and will surely read “Divisions of the tetrachord”.

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Is this not the same thing as

Which is also along the lines of my own more terse response (which admittedly failed to support its assertions)?

Feels like we are explaining the same thing using different words (which is great!)

thank you for your detailed answer (and taking the time) and thank you for the heads up on the book

so, i happen to be looking at psychoacoustic dissonance right now in a very different context. maybe i can take a stab at it.

first off, here’s William Holder writing in 1731:

From the Premises, it will be easie to comprehend the natural Reason, why the Ear is delighted with those forenamed Concords; and that is, because they all unite in the Motions often, and at the least at every Sixth Course of Vibration, which appears from the Rations by which they are constituted, which are all contained with that Number, and all Rations contained within that Space of Six, make Concords, because the Mixture of their Motions is answerable to the Ration of them, and are made at or before every Sixth Course."

this is pretty much the attitude from the renaissance through the early-mid 20th century, and for most of us it is still the baseline understanding of musical harmonic dissonance, as stated a couple times in this thread in different ways.

but in the 70s we get a more definitive correspondence between harmonics and psychoacoustics in the work of Ernst Terhardt (highly recommended reading btw.) here the discussion gets a little more complicated because actually we have to consider not just fundamental frequencies / pure tones, but harmonic series. assume that all tones in question are harmonic. take two simultaneous tones; each have a number of harmonic partials that are all combining with each other and producing interference tones. if the combining partials are close enough, then their interference frequency is low, and it registers as an amplitude modulation in one critical band on the cochlea’s basilar membrane. this produces a lot of neural activity and contributes to “roughness.”

it’s easy to see from this how pure tones at a perfect 5th apart are completely consonant, while even JI tones a minor 2nd apart (9/8) register as dissonant.

take a more complex example: JI perfect 4th (4/3) with a number of harmonics, played against a tone at the root with a number of harmonics, say 5. the P4th has harmonic numbers 4, 8, 12, 16, 20 (from the numnerator) and 3, 6, 9, 12, 15 (from the denominator.) so this system of interference includes 9/8 and 16/15, both of which are small enough to register as beating in the cochlea (9/8 is pretty fast so its marginal). that’s only 2 combinations out of… uh 25 right? so the JI 4th up to the 5th harmonic sounds “quite consonant.”

if you take a 6/5 minor sixth instead, you get 9/8, 16/15, 24/25, 18/20, and others. if your root is 400hz you get beating frequencies of 50hz and less; the minor 6th has a higher score for “roughness” and it makes sense that went back and forth being considered a “consonant” or “dissonant” interval in medeival / renaissance / baroque / classical theory.

this kind of analysis also makes it clear why just restraining pitch sets to low limits doesn’t really give you consonance as the ear understands it, because it forces you into these high orders of low primes that have complex beating relationships. (plus the music-theory limitations like being severly restrained in key / harmonic movement.)

think about bells, metallophones in general, even low strings on a piano (where the thickness of the string contributes partials.) these are enharmonic sounds - its generally not feasible to quantify their spectral content as a sum of integer ratios. in any pythagorean sense (or the Holder / Rameau sense) these are totally dissonant. its even sort of a mystery and source of debate over how people agree on the “fundamental pitch” for enharmonic sounds (aka residue pitches, aka strike notes.) (tuning bells and metallophones is an amazing art, of which gamelan builders demonstrate unparalleled mastery.)

so while i love working with tunings built on simple ratios, i’m skeptical of approaches that value number theory over sensory and emotional and other contexts. i think the shruti system has a value that goes far beyond how close it can be approximated by 5-limit tunings.

the 7th natural overtone (14 occurrences of this interval in the Semantic-53 scale) and the 17th and 19th overtones, to name a few, are naturally present in several ways, notably among indian shrutis, as well as the Semantic system.

i’ll have to think about what this could mean. makes no sense to me on the face of it - 17 and 19 just can’t be produced by multiplying and dividing the numbers 2, 3, and 5. (they can be arbitrarily approximated, as the greeks knew full well.) i guess the semantic system isn’t 5-limit?

anyway all of the above is just to support the idea that it doesn’t really matter if a ratio is prime, low-limit, or irrational, as far as its level of dissonance to the ear. in my teens and 20s i was a little obsessed with tuning in this “numerological” sense, with pythagoras and rameau. not anymore.

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Thanks for the extra detail! Corrects some errors in my thinking.

trigger warning, zip archive contains 1.8 gigabytes. this is one heavy weight VST plugin!

Thanks a lot for this and the reading recommendations !

Based on what they say about it, I assume the semantic system is based on a 5-limit but not strictly limited to it, to be able to integrate other pitches that feel right, but that’s just my understanding, I did not see it stated clearly…