source file: m1396.txt Date: Sun, 26 Apr 1998 01:20:42 EDT Subject: Re: TUNING Resolution From: Ascend11 =09I recently joined the tuning forum and have followed the recent discus= sion=0Aabout tuning resolution with interest. I have worked fairly exten= sively with=0AFourier additive analysis and pitch extraction algorithms (= which I received=0Afrom the University of Illinois=92 CERL music lab and = further developed), and=0Ahave analyzed a number of digitally recorded vo= ice and acoustic instrument=0Asounds (16 bit signed integer samples at a = 40 KHz sampling rate). In this=0Awork I was impressed by the closeness o= f the frequencies of the partials of=0Amany, but not all instrument sound= s to being exact integer multiples of the=0Abasis frequency as determined= by pitch extraction (as a frequency vs time=0Atrajectory which I usually= computed at points roughly a millisecond apart over=0Athe course of the = sound). Frequently the deviations of time averaged partial=0Afrequencies= from n times the time averaged basis frequencies (n an integer)=0Awere l= ess than one cent. =09Here are some results which I obtained: For three recorded soprano no= tes of=0Aroughly one second duration with vowel sounds =93ah=94 and =93ee= =94 at e4 and f4, I=0Aaveraged the =93global=94 frequency of the sound ov= er the main portion of the note=0A(appreciable sound volume) and then ave= raged the frequencies of the first 20=0Apartials over this same portion o= f the note. For three notes, the average=0Aabsolute values for the devia= tions of the average partial frequencies from n=0Atimes the average =93gl= obal=94 frequency were: 0.44, 1.04, and 0.48 cents. For=0Athree recorded= baritone =93ah=94 sounds at a2, g3, and c4, having durations of=0Aroughl= y 2 seconds, these deviations in cents (average for first 20 partials)=0A= were 0.57, 0.55, and 0.41. For a trombone f3 sound, the average absolute= =0Avalue of the deviation of a partial=92s average frequency from n times= the=0Asound=92s global frequency was .48 cents (first 20 partials). =09The averages of the instantaneous deviations of the frequencies of par= tials=0Afrom n times the global frequency over a set of time points space= d a=0Amillisecond apart over the duration of the sounds appeared to be co= nsiderably=0Agreater than this. =09In the case of a flute d4 note a little under a second in duration, th= e=0Adeviations in the average partial frequencies from n times the global= =0Afrequency of the sound amounted to 4.55 cents, roughly ten times great= er than=0Athese deviations were for the sung notes and the trombone note. =09I analyzed a piano a2 sound of roughly 2 seconds=92 duration and here = will give=0Athe cent deviations of the average partial frequencies from n= times the=0Aaverage global frequency over the early portion of the sound= just after the=0Apeak of the attack for the first 20 partials: Partial = 1: -11.4 cents; par. 2:=0A-6.7 cents; par. 3: -7.3; par. 4: -6.7; par. 5:= -6.5; par. 6: -5.8; par. 7:=0A-4.8; par. 8: -3.5; par. 9: -1.1; par. 10:= +1.0; par. 11: +5.1; par. 12: +5.0;=0Apar. 13: +7.9; par. 14: 10.4; par.= 15: +8.9; par. 16: +16.9; par. 17: +20.1;=0Apar. 18: +23.0; par. 19: +24= 6; par. 20: +29.1. The results for a c4 piano=0Anote I analyzed were si= milar but not identical to those for the a2. =09I've gone into some specific detail as I believe this is necessary in = order=0Ato give a meaningful picture of the results of this work. =09I have the impression that seemingly small shifts in the frequencies o= f=0Amusical sounds can have surprisingly large effects on the cumulative= =0Aimpression which the music creates. =09 =09Note: I am doing non-real-time additive synthesis of musical notes on = a=0AMacintosh 8500 computer for purposes of developing demonstrations and= also=0Alistening tests for use in research in musical aesthetics. Recen= tly I've had=0Aa piano retuned to quarter comma mean tone temperament and= have been exploring=0Aits harmonies and have prepared a few demonstratio= ns of pieces of music played=0Aside by side in equal temperament and in q= uarter comma mean tone temperament.=0AMany have found the difference betw= een the effects of these different tuning=0Asystems to be striking. =09Dave Hill, La Mesa, CA=0A