Sine Mod ET's

The Tuning List has been rather inactive recently so I
thought I'd post some recent studies of mine. Just before last Xmas,
Brian Thomson posted some examples of sine-modulated equal temperaments.
SMET's are ETs in which the interval width is periodically varied to
improve the consonance of certain intervals, at least in the mode
starting on the note 0. Brian used the formula A*sin(pi*F*N/ET) where F is
the frequency (restricted to even numbers in this case), A is the
amplitude, N is the note index and ET is the number of notes per octave).
The phase is 0 at tone #0.
I wrote a simple 1-D search program BASIC program and minimized
the sum of the abolute values of the errors between the closest SMET
notes and the JI intervals 3/2, 4/3, 5/4, 6/5, 5/3 and 8/5, a somewhat
different set than BT used. The summed absolute error, squared error,
mean deviation, variance and standard deviations were computed in cents
for each SMET. Other evaluation and statistical functions could have beed
used, but these seemed appropriate to me.
Results were obtained for the 12,15,17,19, 22, 24, and 31-tets.
The errors of some unmodulated ET's are small enough that "improvement"
is scarcely necessary and others are so inharmonic there is little point
as even drastic amounts of modulation do not really make much difference.

ET F A STD Un-MOD STD

12 10 0.132579 5.29859 12.05267

15 8 -0.240234 4.69927 13.31584

17 10 0.502893 5.53674 27.30851

19 8 0.1136078 0.48389 5.95502

22 20 0.1699097 2.94625 8.92950

24 10 0.265158 5.29859 12.05267

31 18 0.133739 1.467844 4.58336

A few caveats: the search program was somewhat sensitive to initial
conditions and although I set the acceptable summed error to be less
than 10 exp -7, even in double precision mode, I would not trust the
numbers past the 6th figure. In a few cases, the best solutions had
A's greater than 0.5, which would imply overlapping notes, except
that in some cases two or 3 adjacent intervals varied sufficiently that
no inconsistencies arose. These cases have been omitted from the table
above except for 17. My solution for 19 is very similar to Brian's, the
differences in the amplitude are probably due to my use of a different
interval set and round-off in my computer.

--John

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Bruce Gilson asked:
> my question is how precisely the given ET scale had to match the interval.

The pretty much accepted definition of precise just intonation - at least by
the readers of this list (more on that in the last paragraph) - is that all
notes must exactly match simple whole-number ratios. Simplicity of the ratio,
and accuracy of the match are the ideals, not any particular limit, as with
5-limit Ptolemaic historical tunings.

To me, it seems likely that in the 13th century (if memory serves, that's
roughly when 5-based harmony began to be accepted as harmonious rather than
dissonant) anything but those specific ratios recognized at the time, would have
been considered inaccurate renditions of the accepted ones. So, for example,
I'd almost bet money that even an exactly precise 9:7 third would have been
considered an off 5:4. Time have changed though. Every modern Just
Intonationalist I know of views 9:7 as an ideal in its own right.

Now an absolute, 100% exactly perfect match is almost always impossible, but
therein lies the distinction between inidealities in the performance, and an
accurate rendition of a tempered (nonjust) tuning.

So to present a degenerate example, consider a performance by instruments
tuned to equal-temperament and intended to produce 400-cent thirds, but wherein
the performers consistently produced exactly precise 5:4 thirds. In my book
that would be failed performance in equal-temperament more than a successful
performance in just intonation, although by default, it would be that as well.

My caveat to the "pretty much accepted definition", stems from the rather
puzzling definition presented in the otherwise quite authoritative Harvard
Dictionary of Music. Best I recall, none of us on the list could make head nor
tail of that definition. Feel free to check the archives for more on that. And
a kind ambassador from the list sent (or at least attempted to send) a message
to the Harvard folks. The message read, to summarize down to a single a word,
"huh?!".


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COUL@ezh.nl (Manuel Op de Coul) wrote:


>Your presumption that all of the modes in the list are selections from
>ET scales is wrong. Many of them can indeed be but for most it is
>not a requirement. If I have been vague about it, I apologise.

The notation seemed to imply it. When
you give only a "number of steps in an
octave" and the number of steps between
consecutive notes in the scale [as
2 2 1 2 2 2 1 for a standard major
scale] you have something well-defined
only in ET.

>Messiaen's modes work best in equal temperament. The Indian modes
>aren't necessarily used in just intonation, take the popular harmonium
>for instance, which is tuned more or less equally.
>The Arabic 17-tone modes are to be taken from this scale:
> 256/243 65536/59049 9/8 32/27 8192/6561 81/64 4/3 1024/729 262144/177147 3/2
> 128/81 32768/19683 27/16 16/9 4096/2187 1048576/531441 2/1.

All the scales that are based on ration-
al frequency ratios need such a chart
in addition to the list you provide in
order to be useful.

>The modes of the Indian shruti scale are also not for ET, though the
>theory of shrutis is not universally accepted among Indian
>musicologists and you see different proposals for the frequency
>ratios.

I suppose that what I'm looking for is
something that would enable me to re-
liably reproduce any scale that has a
name. Your chart, for equal-tempered
scales, does that (for the ones you
provide, but obviously any you've
missed can be added). But for non-equal
scales, it still misses the one thing
that makes it useful to my purpose, the
precise ratios that the 12, or 14, or
however many, steps are supposed to
mean.

I like your format, and I'd like a
table such as yours to serve as a base
for what I'm trying to do.
Bruce R. Gilson
brg@netcom.com

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Bruce R. Gilson writes:

> The notation seemed to imply it. When
> you give only a "number of steps in an
> octave" and the number of steps between
> consecutive notes in the scale [as
> 2 2 1 2 2 2 1 for a standard major
> scale] you have something well-defined
> only in ET.

I wasn't aiming at them to be watertight definitions of scales.
Otherwise I could have listed them as such. To stay with your example,
you can have a major scale in many different tunings.
The utility of this notation is that it allows me to use it in
combination with a computer program and for instance

- have a mode-fitting algorithm operate on a given scale and look
if it's similar to a mode (of equal temperament) in the list,
- create a new scale from a mode by entering a different interval for
each amount of steps,
- select a subset from a given scale (not necessarily anyway near equal
temperament) using the mode name.

So rather than a set of exact scales it's meant to be an alternative
notation that can be employed for different purposes. I you have other
ideas I would be very interested to hear them.

Manuel Op de Coul coul@ezh.nl

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