53-EDO advocacy: (was: Apology on ~17.095-tet confusion)

In-Reply-To: <20000705.121114.-4025147.2.monz@juno.com>
Joe Monzo wrote:

> I haven't been following this thread too closely, but upon
> reading a bit of TD 699.10, I thought that perhaps I'd point
> out, if it hasn't been already, that 53-EDO or -tET is the
> smallest EDO that accurately approximates both 3-limit
> Pythagorean and 5-limit JI tuning.

I suppose you could define "accurately" in such a way as to make this
true. It does have a more precise 5-limit approximation than any smaller
EDO. But it isn't the only 5-limit consistent EDO that is also consistent
with its cycle of fifths. 29 pure fourths are 144.4 steps of 29-EDO, and
approximate to 144 steps.

> This is because 2^(1/53), the 'step-size' of 53-EDO, is
> almost exactly midway between the two relevant commas,
> being ~1.1 cent smaller than the Pythagorean and ~0.8 cent
> larger than the syntonic:
>
> ~21.5 cents - syntonic comma
> ~22.6415 cents - 2^(1/53)
> ~23.46 cents - Pythagorean comma

Well, the optimum 5-limit schismic tuning would be something like -1/8 or
-1/9 schisma, where the comma is flat of both syntonic and Pythagorean
commas. So the precision of 53= doesn't follow from this observation at
all.

If you're only interested in 5-limit approximations, and really really
want 53 notes within the octave, you could try a well temperament based on
-1/9 schisma. You'd have to dispose of about 4 schismas, or 8 cents.

> Therefore, any combinations of tones in 53-EDO will yield
> pitches or intervals that are reasonably close to any of the
> rational ones within a 5-limit system of 53 tones or less.

So you're saying any interval in 53= will be reasonably close to a complex
5-prime-limit interval? Do you think the ear can actually make sense of
this?

> An important point to note is that 53-EDO distinguishes between
> the two different sizes of semitone in both the Pythagorean
> and 5-limit JI tunings; it would thus be inappropriate
> (or at least, radical) for performance of works written with
> a 12-note meantone tuning in mind. An extended meantone is
> perhaps another matter...

This is a general schismic thing. The larger Pythagorean semitone is
confounded with 16/15, so you get three different kinds of semitone. 29=
already distinguishes between them. And makes the distinction very clear.

One thing I found by trying schismic tunings is that the comma in 53= is
too small to be clearly comprehended. It has an annoying "something's not
right" quality. Fatten it up to 41=, and it's heard as a melodic
elaboration.

> If one's interest is to find EDOs which provide good approximations
> to 'Xeno-Gothic' Pythagorean tunings, 53-EDO is the prime candidate.

Hell, no! Xeno-Gothic, as we've been discussing it, uses approximations
up to the 7-prime limit. (Now I've looked at it more closely, the
progression Margo gave as an example acutally shows an anomalous 9-limit
chord resolving onto a 3-limit trine. Interestingly, no factors of 5 are
involved at all.) In this context, considering only EDOs, 41 is far and
away the prime candidate. Dare I say the only one worth bothering with?
No, 29's worth a look as well, as is 17, or even 135 if you can handle
that many notes.

So there you are, I've saved you 12 notes per octave!

Graham

p.s. for those interested, looks like my TX81Z patches are still at
http://x31eq.com/graham/x991203.zip but not linked to as I
haven't updated my website in a looooooong time.

Graham Breed,

I admitted right up front that I wasn't really following this
thread.

Now I'll admit that I haven't really experimented with 53-EDO
and was only speaking from theoretical knowledge. I bow to
your superior expertise in this, and thanks for the corrections
you posted in TD 700.25.

Paul,

The reason I limited myself to a 12-tone meantone was simply
because on first thought it seemed to me that a >12 meantone
would have the two different sizes of semitone, which could
be approximated well with 53-EDO. See my comments to Graham.

Now I'll just run away from this thread with my tail between
my legs. ouch...

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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