RE: [tuning] Good news! Paul Erlich tuning experiment

Joseph wrote,

>Well, the "cat's out of the bag" since Paul mistakenly posted one of the
>.mp3 files to the Tuning List, but he just sent me some terrific and
>varied examples of his 12-22-tET tuning system!

Actually, I sent Joseph two chords in 12-tET, three chords in 22-tET, and
three chords in JI. The JI chords were 4:5:6:7, 5:6:7:9, and
1/7:1/6:1/5:1/4, and the others were the nearest approximations to those in
22-tET and 12-tET.

>They are fascinating .mp3s, since they differ so markedly from one to
>another. I'm not certain how Paul was able to do this, or what
>equipment he was using (he will probably explain subsequently), but it
>certainly does work.

As I pointed out, the reason you may not have noticed much difference with
your timbres is because they only contained partials 1 2 4 8 and 16, so had
no opportunities for first-order beating to arise. My timbres have all
partials, so lots of beating is there. In particular, I used a
rectified-sine waveform for each tone, created .wav files in Matlab, and
converted the .wavs to .mp3s using CoolEdit. I also kept the average (log)
pitch of each chord constant, so that one wouldn't have to worry about
whether a net pitch shift up or down would impact the comparison.

By the way, the relative amplitudes of the partials in a rectified sine wave
are:

Fundamental -- 1
2nd partial -- 1/5
3rd partial -- 3/35
4th partial -- 1/21
5th partial -- 1/33
6th partial -- 3/143
7th partial -- 1/65
8th partial -- 1/85
9th partial -- 3/323
10th partial -- 1/133
nth partial -- 3/((2n-1)(2n+1))

So you would have heard even more difference between the tunings had I used
a parabolic wave (amplitude of nth partial ~ 1/(n^2)) or especially a
sawtooth wave (amplitude of nth partial ~ 1/n)). And with pure sine waves,
or a timbre like the one you used with only octave partials, you might not
have heard any difference at all, especially in the utonal chords (in the
otonal chords difference tones can help you out even with sine waves).

I wrote,

>By the way, the relative amplitudes of the partials in a rectified sine
wave are:

>Fundamental -- 1
>2nd partial -- 1/5
>3rd partial -- 3/35
>4th partial -- 1/21
>5th partial -- 1/33
>6th partial -- 3/143
>7th partial -- 1/65
>8th partial -- 1/85
>9th partial -- 3/323
>10th partial -- 1/133
>nth partial -- 3/((2n-1)(2n+1))

I'm getting ready to make .wav files for all those tetrads, and I think I'd
like to use a timbre richer in harmonics than the rectified sine wave. For a
waveform which follows the sine curve from peak to crest, but then
immediately jumps up to the peak again, I find the following relative
amplitudes:

Fundamental -- 1
2nd partial -- 2/5
3rd partial -- 9/35
4th partial -- 4/21
5th partial -- 5/33
6th partial -- 18/143
7th partial -- 7/65
8th partial -- 8/85
9th partial -- 27/323
10th partial -- 10/133
nth partial -- 3n/((2n-1)(2n+1))

That sounds really buzzy, like a sawtooth wave. So I decided to keep
investigating. For a waveform which follows the sine curve from 45 degrees
to 315 degrees, but then immediately jumps up to 45 degrees again, I find
the following relative amplitudes:

Fundamental -- 1
2nd partial -- 14/55
3rd partial -- 7/45
4th partial -- 28/247
5th partial -- 35/391
6th partial -- 2/27
7th partial -- 49/775
8th partial -- 8/145
9th partial -- 7/143
10th partial -- 70/1591
nth partial -- 7n/((4n-3)(4n+3))

For a waveform which follows the sine curve from 30 degrees to 330 degrees,
but then immediately jumps up to 30 degrees again, I find the following
relative amplitudes:

Fundamental -- 1
2nd partial -- 22/119
3rd partial -- 33/299
4th partial -- 44/551
5th partial -- 11/175
6th partial -- 66/1271
7th partial -- 77/1739
8th partial -- 88/2279
9th partial -- 99/2891
10th partial -- 2/65
nth partial -- 11n/((6n-5)(6n+5))

This one still has the "buzz", just a little attenuated. So it seems that
the buzz is associated with the amplitudes going as 1/n for high enough
partials. I think I'll continue using the rectified sine wave, then, unless
anyone has any other suggestions.

P.S.
In general, if the waveform follows the sine curve from 180/x degrees to
360-(180/x) degrees, the relative amplitude of the nth partial will be

(2x-1)n/((xn-(x-1))(xn+(x-1)))

(does someone care to prove this using calculus?)

If x gets really close to 1 (say 1+d, where d approaches zero), you
essentially have a sawtooth wave, and taking the limit, you get n/n^2 = 1/n,
as you would expect. If x gets really large, you essentially have a sine
wave, and taking the limit, you get 1 for n=1, and 0 for n>1, again as you
would expect.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> I wrote,
>
> >By the way, the relative amplitudes of the partials in a rectified
sine
> wave are... <snip>

I'm a little confused... I thought that the sine wave doesn't
have ANY upper partials... or is this different since it's
"rectified." I'm sorry, but I guess I don't know what this means...

Any help would be appreciated!

___________ _____ ___ __ _
Joseph Pehrson

A sine wave "theoretically" does not contain overtones - but this can
be an abstraction in synthesis, as many synths have "dirty sines"
that are not exactly pure tones. I'm guessing the closest that one
may come to a pure tone, may be ones generated by a computer. You
know the old Moog analogs have sine wave occillators, but if you look
at the wave in a spectrum analysizer, you see that this isn't exactly
so.

Bye,

Jacky

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:
> I'm a little confused... I thought that the sine wave doesn't
> have ANY upper partials... or is this different since it's
> "rectified." I'm sorry, but I guess I don't know what this means...
>
> Any help would be appreciated!
>
> ___________ _____ ___ __ _
> Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:

> I'm a little confused... I thought that the sine wave doesn't
> have ANY upper partials... or is this different since it's
> "rectified." I'm sorry, but I guess I don't know what this means...

It's the latter. "Rectified" means, basically, that you take the
absolute value of the sine function. So it's a completely different
shape. That's the timbre I used for each voice in the .mp3s I sent
you so far (the JI, 22-tET, and 12-tET ones).