Subject: TUNING Resolution

Gary Morrison wrote: =93For three notes, the average absolute values for=
the=0Adeviations of the average partial frequencies from n times the ave=
rage global=0Afrequency were: 0.44, 1.04, and 0.48 cents.

=93Thanks Dave, for the details on your results.

=93Do you have any way of characterizing your process' margin of error=
?
That is, could these sounds be absolutely exactly harmonic (short of nois=
e
components and human/random fluctuations in the entire tone of course), a=
nd
your process could only detect their pitch within that accuracy (that of
course being quite impressive accuracy)? Or is it more likely, in your
view, that these pitch deviations from exact harmonic frequencies are ver=
y
definitely present in the overtone structures of the tones?=94

=09Good question.

=09I had been away from the additive analysis system I=92d worked with fo=
r several=0Ayears and I have been rapidly reviewing the work done with th=
at system and=0Areacquainting myself with the various problems concerning=
validity and=0Aaccuracy which I had wrestled with when developing and te=
sting it.

=09While I have not made a formal rigorous analysis of margins of error f=
or the=0Amany data values generated by my additive analysis machinery, w=
hich I=92ve=0Anamed the Cro-Magnon System, the following line of reasonin=
g gives me reason=0Ato believe that the AVERAGE figures for harmonicity o=
f a sound=92s partials are=0Agenerally very accurate. For each of a pitc=
hed sound=92s partials the method=0Atracks that partial=92s phase over a =
succession of small time windows covering=0Athe entire analyzable portion=
of the sound. My recollection is that as long=0Aas the tracking analysi=
s indicated no breaks, any individual partial=92s phase=0Acould be accura=
tely tracked over many many cycles - to something on the order=0Aof a hun=
dredth of a cycle or better. This would correspond to an accuracy of=0Ao=
ne part in 100,000 for a 1000 Hz partial being tracked over one second.

=09When I first did analyses and resyntheses, although I logged the phase=
of a=0Asound=92s partials at roughly every millisecond, in the syntheses=
I did based on=0Athe analyses, I used a partial frequency value only to =
resynthesize the sound.=0AThe resynthesized sounds generated this way wer=
e very nearly indistinguishable=0Afrom originals, but the fact that throu=
gh rounding errors, etc. the phases of=0Ahigh partials in the resynthesiz=
ed sounds might still drift appreciably from=0Athe phase values for the a=
nalyzed sound troubled me. I modified my original=0Aresynthesis algorith=
m to force the partial phases for the resynthesized sound=0Ato match thos=
e found by analysis of the original recorded sound. When I=0Alistened to=
two resyntheses, one using phase tracking and one without phase=0Atracki=
ng, I found it difficult to tell any difference between them and in any=
=0Acase my perceptions might be influenced by what I knew about the metho=
ds I=92d=0Aused. I asked a musician with excellent hearing in the high f=
requency region=0Ato compare the sounds without telling him anything abou=
t the methods used to=0Agenerate them. He very quickly responded that on=
e of the sounds seemed a much=0Amore natural musical sound than the other=
, in which he detected something=0Awhich didn=92t sound right - a kind of=
clashing in the high frequency region.=0AThe sound which he found to be =
more natural sounding was the one in which I=92d=0Aused phase tracking. =
This result suggested that the analytical method was=0Aindeed tracking th=
e phases of a sound=92s partials.

=09The method also yielded normal appearing inharmonicity for the partial=
s of=0Atwo piano sounds I analyzed using it.

=09I reviewed in greater depth the results for the frequency of partials =
of one=0Aof the soprano sounds mentioned in my last post. The results I =
gave there=0Awere for analyses which included very early and late parts =
of the sounds where=0Athe amplitude was low and where there is less stead=
iness in the sounds. I=0Alooked at the frequency behavior of the partial=
s over the main part of the=0Asound, and here I found that the average pa=
rtial frequencies (total delta=0Aphase divided by duration of that portio=
n of the sound) were still closer to=0Apure harmonic than they were for t=
he sound with attack and decay portions=0Aincluded. For a .7 sec. stretc=
h over the main part of a sung e4 =93ah=94, I=0Aobtained deviations in ce=
nts for the first 10 partials of: -.07, .02, -.01,=0A-.05, -.04, -.17, .0=
8, .07, .02, and -.01 respectively.

=09However I did find that the average INSTANTANEOUS deviation from harmo=
nic for=0Athe partials was much greater than the averaged deviation - i.e=
over portions=0Aof the sound a partial=92s phase would be advancing fas=
ter than n times that of=0Athe overall sound while for other portions of =
the sound its phase would be=0Aadvancing more slowly, with there being an=
approximate balance between the=0Atimes of more rapid advance and slower=
advance. For the first 10 partials I=0Aobtained over that .7 sec. time =
stretch cent values for the RMS instantaneous=0Adeviation from harmonic o=
f: 3.20, 1.92, 1.08, 1.12, 1.72, 3.25, 4.08, 3.17,=0A2.58, and 1.95 respe=
ctively. I plotted the deviations from harmonic for some=0Aof the partia=
ls over this .7 sec. stretch and found that peaks in positive=0Afrequency=
deviation for the first three partials corresponded to peaks in the=0Afr=
equency vibrato, while troughs in the frequency deviation for the 4th=0Ap=
artial corresponded to these peaks in frequency vibrato. I do not presen=
tly=0Ahave an explanation for these results.

=09I=92ll add that the analysis of a single one second sound yields a dat=
a file of=0Apartial amplitude, frequency, and phase trajectories having s=
everal hundred=0Athousand individual data points. I=92ve usually analyze=
d partials up to=0Afrequencies well above 10 KHz (frequently over 100 par=
tials), as inclusion of=0Athe high partials is necessary to resynthesize =
a sound accurately resembling=0Athe original. I=92ve only studied a few =
of the many sound characteristics which=0Aare described in intriguing det=
ail by this wealth of data. This system is not=0Aready to give to a user=
not skilled in programming (not thoroughly documented,=0Aetc.), but it c=
ould be made so with some work and I=92d like to make it=0Aavailable for =
use to others who could use it.

=09Dave Hill, La Mesa, CA=0A

On Tue, 28 Apr 1998, A440A wrote:

> The popularity of ET in Western music is due, (IMHO) to the loss of
> interest in tonality. With renewed interest in tonality today, the
> alternatives to ET are becoming more prevelant. Computer-equipped keyboard
> tuners, such as myself, now are having a lot easier time offering customers a
> choice in temperament and the results are encouraging.
> Regards,
> Ed Foote
> Precision Piano Works
> Nashville, Tn.
> http://www.airtime.co.uk/forte/history/edfoote.html

Actually it seems to me that the fascination with polyphony along with
12ET's somewhat universal simplicity, has caused the predominance of
12TET-- it has taken 250-300 years to somewhat exhaust the rich endowment
of this tuning.

Bruce Kanzelmeyer