et 5ths

This is a reply to an off-topic thread from MMM.

--- In MakeMicroMusic@yahoogroups.com, "a_sparschuh"
<a_sparschuh@y...> wrote:

> I.m.o. any et below 665 remains incompetent amateur work.

And you are probably the only one on the planet who thinks so. Other
people are either opposed to any approximation at all, or willing to
accept that getting to within 0.00001 cents of the JI interval is not
actually necessary. This is far more accuracy than the MIDI Tuning
Standard, for instance, even allows. It is of a degree of accuracy
that doing acoustical experiments becomes difficult, and something
probably best left to something like Csound. A single beat of a C-G
interval, with the C at 256 Hz, would take 11 hours to complete. By
comparison, 306 would take 13 minutes, and 53 66 seconds; the last is
still slow enough that it already would give problems. The MIDI tuning
standard, by the way, can in theory make the beat 40 minutes long, but
no longer.

> > Take 53, and make the circle close exactly, and you are in
> > business.

> simply tune in practice a few (2^(31/53))^n artificial created 5ths
> and hear yourself how the errors accumulate:

But the *only* intervals you can test with your ears are fourths,
fifths and their octave equivalents, or things which *approximate* to
a consonant interval. Because the introduction of the dreaded a-word
is *required*, I can't see any way your idea can make genuine musical
sense. It's true that after eight fourths along the chain of fourths,
you get to an interval you can test with your hearing which is not a
fifth or a fourth. It's not a just major third, either, since it's
1.95 cents flat. 53 differs in that it has this interval 1.41 cents
flat. This makes a triad with 53 sound slightly different, but only
very slightly, and then because it is *more* in tune.

> The intermediate results become more and more of of tune
> by each step arising detuned
> versus the correct chain of just (3/2)^n pure 5ths.

This is a purely theoretical claim, not one based on hearing. Yes, 53
gradually moves away from the Pythagorean chain, but why is the
Pythagorean chain right and 53 wrong? Your ears won't tell you that.

> Retrace that by experiment in order to comprehend that acoustical too.
> Then you have a clue of my own acustics experiments.

I don't think it is possible you came to this conclusion based on
acoustic experiments with consonant intervals; instead, you seem to be
concerned with listening to commas. But whether the Pythagorean comma
and the 41-circle comma are precisely the same or not is irrelevant so
far as music goes, so your acoustic experiment is meaningless. Why is
the JI tuning musically better? Better in what way?

> Learn to distingusih the different commas one from the other:
> 3^12/2^19 ~23.4 cents 13 steps in 665 (~~12 in 612)
> 2^65/3^41 ~19.8 cents 11 steps in 665 (~~10 in 612)
> instead of assuming them fuzzily equal,

I'm not assuming them to be equal. They in fact *are* equal if you
tune the fifth 0.068 cents flat, and close the circle of fifths, but
this fact has to do with the structural organization of music in a
tuning system (the 53 circle closes) not with any "acoustic
phenonmenon". They are not intervals having their own, independent
musical life, whose exact tuning matters.

> i'm fully aware in respecting Jing-Fangs comma discrimination
> 3^53/2^84 ~3.6 cents ~2 steps in 665 (or 612 too)
> as mental in mind as acustical by ears.

It's acoustic in the sense you can hear it. It's not acoustic in the
sense your ears will tell you what size it ought to be.

> Already Moritz Wilhelm Drobisch gained
> the same accuracy of discrimination in the 19. century
> http://de.wikipedia.org/wiki/Moritz_Wilhelm_Drobisch
> theoretical and practical.

Looks like an English page on Drobisch should be added to Wikipedia,
but the German page doesn't say anything relevant.

> If you alreay have grasped 612,
> why not the better choice?
> Superior 665!

Inferior 665 if we go beyond the 3-limit.

> > Why would anyone care to make the Pythagorean ratios exact and
>ignore
> > all others?
> 5 arises from 3^-8, 7 from 3^-14, 11 from 3^23 ...........
> already in 53P.

Terrific. 5 arises from 3^-314, 7 from 3^-67, 11 from 3^129 in 665.
These are not easy to get to, and when you do get to them they are not
as accurately tuned as you seem to want. 5 is 1/7 cent flat, 7 is 1/5
cent sharp, and 11 is 6/7 cents sharp. That's good enough for many
people, but not for JI advocates. Moreover, pure fifths have the *same
problem*.

> Are you doubting about my championship of 665?

I'm talking specifically about fifths and listening tests.

> Counterexample:
> Just try to tune my Werckmeister concept in an real acustic organ:
> @ A4=456 Hz as exact by ear as you are able, as your ears can:

I'd like to investigate this, but you give more than one number per
note. They could also be reduced to the same octave. If I ignore the
stuff in parenthesis, I get

! spars.scl
Sparschuh circulating scale
12
!
2304/2173
2448/2173
2592/2173
2736/2173
2916/2173
3072/2173
80/53
3456/2173
3648/2173
3888/2173
4104/2173
2

Is this what you wanted? 665-et does not stand out as a particularly
great way to approximate this scale, which is 53-limit. I also don't
see why it is a "Werckmeister concept", with one flat and one sharp
fifth, so that I suspect I've gotten it a little wrong.

Gene Ward Smith wrote:
[snip]
> probably best left to something like Csound. A single beat of a C-G
> interval, with the C at 256 Hz, would take 11 hours to complete. By
> comparison, 306 would take 13 minutes, and 53 66 seconds; the last is
> still slow enough that it already would give problems. The MIDI tuning
> standard, by the way, can in theory make the beat 40 minutes long, but
> no longer.
[snip]

That's kind of an interesting idea in itself. Sounds like something John Cage or La Monte would do. Just have a bunch of drones that slowly change phase, to the movements of the planets or something. =P

Keenan