Wanted: A mathematical definition for consonant - see Euler -supplemetary information
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From: JEMebius on 6 Jun 2010 16:47 JEMebius wrote: > 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. > > Supplementary information: (A) Euler's term "gradus" reads in full "gradus suavitatis", which literally means "degree of sweetness". (B) Online resources on Euler: http://www.eulersociety.org/ http://www.eulerarchive.org/ Johan E. Mebius
From: rabid_fan on 6 Jun 2010 17:07
On Sun, 06 Jun 2010 21:47:37 +0100, JEMebius wrote: >> >> This treatise presumably contains mathematical definitions of >> "consonant", "dissonant" and "measure of consonance". >> All such attempts at a mathematical definition can be traced back to the mystical cult of Pythagoras, who had first investigated the numerical ratios of pure intervals. But all these folks have been barking up the wrong tree. Look up "critical bandwidth of the basilar membrane." Consonance is directly related to this critical bandwidth. Consonance is a physiological manifestation. Also, when dealing with real musical instruments, the concept of harmonics is inapplicable. Non-linear effects of sound production result in "harmonics" that are not part of an integral series. Thus the term "partials" is used to describe the overtones produced by real musical instruments. |