Well-Temperament Comparator Excelsior!

http://lumma.org/music/theory/WellTemperamentComparator.zip

With my third attempt in the last, um, year, I think I
finally have something that's worth something. It may still
need refinement/debugging, but I finally think I've got the
right approach. Comments/corrections greatly appreciated.

This spreadsheet, like its forebears, shows the error from
JI in each key of 24 different circulating temperaments. It
offers a choice of 3 different error measures in both the
5- and 7-limits to do so:

1. pairwise RMS error
In 'Schwartz et al - Statistical Structure of Human Speech
Sounds' a large corpus of spoken sounds from a variety of
languages was analyzed for its spectral content. Harmonics
2-6 were strongest, harmonics 1 and 7 were about half as
strong, and all other harmonics were very weak. So a choice
between 5- and 7-limit JI seems supported by this study; I
measure the RMS average of the errors of the dyads in an
approximate 2:3:4:5:6 chord for the 5-limit, or a 2:3:4:5:6:7
chord for the 7-limit.

2. Tenney-weighted error
Tenney Harmonic distance has been found to agree with
psychoacoustic ratings of dyadic dissonance for simple
ratios, and penalizing error more strongly when it effects
more consonant ratios is something that may make sense. I
use the sum of the absolute values of the Tenney-weighted
errors of the primes 2 3 5 for the 5-limit, and of 2 3 5 7
for the 7-limit.

3. harmonic entropy
Using data supplied by Paul Erlich for s=1.0 ('harmonic
entropy for acute listeners'), I sum the entropy of the same
dyads used in the RMS calculation (rounded to the nearest
cent) and subtract the entropy of their justly-tuned versions.
This should give the most accurate picture of the dissonance
of the chord of any of these measures (whether or not that
chord reflects what we hear when playing music in a 'key' of
a well temperament).

From an implementation point of view, this spreadsheet is
far better than previous versions. It uses explicit absolute
and relative references, 3-D references, and smarter formulas
to keep the fields isomorphic. To say that in English, you
can now add a new well-temperament to the sheet without having
to do a ton of work.

What conclusions can we reach from the data is this sheet?
Well, I haven't looked closely at it yet, but these jumped
out at me:

. Harmonic entropy says that if you're going to playing in
all keys, your average experience of dissonance will be less
in equal temperament than in any well temperament. The same
isn't true for the other error measures. As long as you don't
have what George Secor calls "harmonic waste" you can do just
as well on average as in ET. You can even beat ET in some
cases my using tempered octaves.

. Flat octaves are better if you believe in RMS, or Tenney
weighting in the 7-limit. They're worse if you believe in
harmonic entropy or 5-limit Tenney weighting.

Note: many of the temperaments on this sheet are specially
designed to do something with beat rates, or to allow
historically-accurate performances. The sheet doesn't
reflect these considerations directly.

-Carl

Also, the key of F is better than the key of C in most
historical temperaments. I never knew that. I always
thought C was supposed to be the best key!

With my well temperaments I try to favor keys like
D, A, and E because they're easier to play (at least,
for me) on the piano. It isn't always possible
given the pattern of key contrast though.

-Carl

[I wrote...]
> http://lumma.org/music/theory/WellTemperamentComparator.zip
//
> What conclusions can we reach from the data is this sheet?
> Well, I haven't looked closely at it yet, but these jumped
> out at me:
>
> . Harmonic entropy says that if you're going to playing in
> all keys, your average experience of dissonance will be less
> in equal temperament than in any well temperament. The same
> isn't true for the other error measures. As long as you don't
> have what George Secor calls "harmonic waste" you can do just
> as well on average as in ET. You can even beat ET in some
> cases my using tempered octaves.
>
> . Flat octaves are better if you believe in RMS, or Tenney
> weighting in the 7-limit. They're worse if you believe in
> harmonic entropy or 5-limit Tenney weighting.
>
> Note: many of the temperaments on this sheet are specially
> designed to do something with beat rates, or to allow
> historically-accurate performances. The sheet doesn't
> reflect these considerations directly.