Wanted: A mathematical definition for consonant - see Euler
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From: JEMebius on 6 Jun 2010 16:29 Androcles wrote: > "Jay Bala" <jay1bala(a)gmail.com> wrote in message > news:2abb25ee-7925-4d66-a5e4-62dce7d8293d(a)d37g2000yqm.googlegroups.com... > | Who can give me a mathematical definition for consonant, possibly tied > | to even order harmonics. I don't words, I want an equation. > | > | I have something, but I think its not the best. I wanted to see whats > | out there before I put anymore time into it. > | > | Thanks, > | Jay Bala. > | > http://www.merriam-webster.com/dictionary/consonant > 1 : being in agreement or harmony : free from elements making for discord > 2 : marked by musical consonances > 3 : having similar sounds <consonant words> > 4 : relating to or exhibiting consonance : resonant > > > consonant = integer ratio > > Where a = x/y and a belongs to the set of natural numbers {1,2,3,...}, x and > y are consonant. > > In music, F# = 1480 Hz and G = 1568 Hz are dissonant (and sound it), > 1480/1568 = 0.943877551 > which is not an integer. > However, for F# = 185 Hz and G = 18,130 Hz (185*98) can be mathematically > consonant if G is taken from a different and much higher octave; but 18,130 > Hz > is a higher frequency than most people can hear and we don't have 98 * 12 = > 1176 > keys on a piano, most have only 88 keys or less. > Therefore we can in practice limit the set of consonant natural numbers > to those we can hear which have the ratio x/y that belong to the set > {1,2,3,4,5,6,7} in music, but not in mathematics. There is no limitation > to even or odd numbers being consonant in music or mathematics. > > Google "measure of consonance" yields about 285 hits, among which... http://www.huygens-fokker.org/bpsite/consonance.html http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.111.2789&rep=rep1&type=pdf The latter paper mentions Leonhard Euler's Gradus function. Another paper on measures of consonance: http://staff.science.uva.nl/~ahoningh/publicaties/measures_of_consonance.pdf This paper mentions Euler's treatise on music: Euler, L. (1739). Tentamen novae theoriae musicae. (*) In E. Bernoulli et al. (Ed.), Opera Omnia, Volume 1 of III, Stuttgart. Teubner (1926). This treatise presumably contains mathematical definitions of "consonant", "dissonant" and "measure of consonance". Happy tuning - happy singing: Johan E. Mebius --------------------------------------------- (*) Essay on a new theory of music. |