Wendy Carlos and optimum diatonic meantone

Paul Erlich wrote:

>Wendy Carlos does not appear to agree with Manuel Op de Coul, Dave Keenan,
>and Carl Lumma's perceptions of the relative dissonance of errors in the
>perfect fifth versus errors in the thirds. Here are some comments she makes
>in her article, "Tuning at the Crossroads":
>
>[Many Carlos quotes omitted]
>"Since low partials are involved, the resulting flat
>fifths beat much slower than if some higher ratio were mistuned by the same
>5.3766 cents -- if you will, the fifth can "stand it" better."
>[More Carlos quotes omitted]

Paul, I think you are reading too much into Wendy's self-described "capsule
explanation". She was only trying to establish that 1/4 comma meantone (or
_any_ meantone) distributes the syntonic comma better than equal tempered.
She wasn't distinguishing between any characteristic meantones. Beats were
an easily explained motivation. Relative increase in sensory dissonance
would be just as good a motivation for this but much harder to explain.

I don't think anyone would find an error of 30 cents in fifths as
acceptable as the errors of 14 and 16 cents that folks routinely put up
with in major and minor thirds. And certainly no one finds errors of 60
cents in an octave or unison to be as benign. I thought we had agreed that
the error weighting function for these simple intervals is somewhere
between equal weighted and weight proportional to 1/limit, and certainly
not proportional to the limit.

However !!!!, I can conceive that when the differences between coinciding
partials are so small as to give rise primarily to a slow beat (say <5Hz),
rather than roughness, even in the upper registers, then weight according
to beating _might_ become a more important criterion than relative increase
in sensory dissonance.

So which inversion of which triad do we make minimum beating, 4:5:6, 3:4:5,
10:12:15? Do we use RMS or Max_Abs? I'll assume 4:5:6 and Max_Abs.

interval frequency of coinciding partials (relative, arbitrary units)
---------------------------------------------
fifth 6
major third 10
minor third 15

Let x be the flatness of the fifth, measured in commas (a positive fraction).
The flatness of the major third is 4x - 1
The flatness of the minor third is 1 - 3x

For such small deviations we can ignore the logarithmic relationship and we
get relative beats of

interval frequency of beat (relative, arbitrary units)
---------------------------------------------
fifth 6x
major third 40x - 10
minor third 15 - 45x

The minimimum Max-Abs beat rate will occur when the (absolute value of the)
rates are the same for major and minor third.

40x - 10 = 15 - 45x
=> 85x = 25
=> x = 25/85 = 5/17 ~= 0.294

I see now that 5/17 comma meantone gives equal beats for all three
intervals in the 4:5:6 chord.

How come this number hasn't come up before? What did I screw up?

Using RMS we want the minimum of
36x^2 + 1600x^2 - 800x + 100 + 2025x^2 - 675x + 225
= 3661x^2 - 1475x + 325

This will occur when the first derivative is zero i.e. when
2 * 3661x - 1475 = 0

so x = 1475/7322 ~= 0.201

i.e. around 1/5 comma.

How come this is so small and so far from what I got above (equal beating)?
What did I screw up this time?

[Paul Erlich:]
>Tonight I finally got a hold of copies of Mandelbaum's dissertation and
Jorgenson's tome. The first few pages of both contain information on the
proposed tuning system of Robert Smith. According to Mandelbaum, Smith's
ideal fifth in his
>1759 _Harmonics_ is flat by, you guessed it, 5/18 of a comma. Mandelbaum's
description makes it seem as if Smith might have used the same derivation
as me. According to Jorgenson, however, Smith's ideal fifth derived from
equal-beating
>considerations and satisfies the equation 3x^3+4x=16. That implies a fifth
0.0027 cents smaller than that of 5/18-comma meantone. Though this is
really splitting paramecial hairs, anyone know who's right about Smith?

Can you explain how 3x^3+4x=16 is derived from equal-beating
considerations? I guess that x here represents the fifth-size as a
frequency ratio rather than its logarithmic error in commas.

Regards,
-- Dave Keenan
http://dkeenan.com

>Paul, I think you are reading too much into >Wendy's self-described "capsule
>explanation". She was only trying to establish >that 1/4 comma meantone (or
>_any_ meantone) distributes the syntonic comma >better than equal tempered.

But she also stated that a comma-off-third is _more_ than four times worse than a 1/4-comma-off-fifth. This could be a reference to the squared error she likes to use, but she just finished saying that the fifths can take mistuning better than thirds.

>I don't think anyone would find an error of 30 >cents in fifths as
>acceptable as the errors of 14 and 16 cents that >folks routinely put up
>with in major and minor thirds.

30 cents, no. 25 cents, as suggested by inverse limit-weighting, would still be problematic from a purely melodic point of view. Let's stick to the deviations in 12-tET and meantone. You have to remember how very engrained 12-tET is in most people's musical consciousness, and that in many parts of the world other non-JI intervals are equally engrained. The quality of JI or meantone thirds is really quite different from that of 12-tET, and to a musician the former is hard to put up with at first, until a good exposure period, after which a return to 12-tET is _really_ hard to put up with.

>I see now that 5/17 comma meantone gives equal >beats for all three
>intervals in the 4:5:6 chord.

>How come this number hasn't come up before? What >did I screw up?

I think Wilson's meta-meantone had the same equal-beating goal. Was that 5/17-comma? Jorgenson's book is loaded with equal-beating tunings -- equal beating seems some kind of holy grail in Jorgenson's mind. Maybe his bias led him to distort Smith's views.

>Can you explain how 3x^3+4x=16 is derived from >equal-beating
>considerations? I guess that x here represents >the fifth-size as a
>frequency ratio rather than its logarithmic error >in commas.

Yes. According to Jorgenson, Smith wanted a fifth and a major sixth to beat at the same rate when their lower note was the same.

Paul writes:
>Jorgenson's book is loaded with equal-beating tunings -- equal beating
>seems some kind of holy grail in Jorgenson's mind. Maybe his bias led him
>to distort Smith's views.

Greetings,
I don't think it is a "Holy Grail" so much as being a widely documented
method for aural tuning in the past. The use of equal beating comparisons is
a way of measurement which makes tuning easier to quantify and we know that
the easiest way has always been the favored method by the average tuner. The
equal beating temperaments have their strengths and weakness, but for a
practical matter, I think their ease was responsible for their popularity.
Regards,
Ed Foote