Moving from 53-EDO to 72-EDO

53 Equal Divisions of the Octave (EDO) is great for modulating. A circle of fifths ends up one step above where you start after 12 modulations. You have to go around the circle 12 * 53 times to get back where you started. Not that you'd want to. I once tried it in a piece and found it not so useful...

The problem with 53-EDO is the ratio of 11:8. I think it is too much of a compromise to be so far off just. If I use the 24th step in 53-EDO, it's 11 cents off. If I use the 25th step, it's 20 cents off. And the 14:11 and 9:7 are the same in 53. I've used both, and have been uphappy with each.

In 72 that problem goes away, with all the 15 limit ratios represented very well, within 7 cents, and a distinct 14:11 and 9:7. I use those ratios a lot in my music. Anyone else using 72-EDO?

Prent Rodgers

Hi Prent,

You are right with the issues of the 53 division, let me put the matter in
more "classical" terms. 53 is the oldest multiple division, known in China
BC, and the lowest division to approximate very well all the main intervals
of classical harmony: fifths, fourths, both major thirds and both major
sixths.

The Fifth has 31 steps, deviation from purity only -0.07 Cents.

The Major Third has 17 steps, deviation from purity only -1.41 Cents

The Minor third has 14 steps, deviation from purity only +1.34 Cents.

Thus the Major Triad is consistent: 17+14=31.

However, a nasty Syntonic comma issue subsists.
If you start from C say and play the usual 4 consecutive fifths, you get an
e' 31x4=124 steps up.
Then you get down 2 octaves (2x53=106) and you expect to land in E, but you
actually get in 124-106=18 steps, i.e. you are ONE STEP HIGHER than the
major third.
So you do not need to go around the circle, you are one step above VERY SOON
with a few modulations.
This seems IMHO to be the reason why classical musicians did not adopt the
53 division.

Regards

Claudio

http://temper.braybaroque.ie/

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Prent Rodgers
Sent: 26 April 2009 21:56
To: tuning@yahoogroups.com
Subject: [tuning] Moving from 53-EDO to 72-EDO

53 Equal Divisions of the Octave (EDO) is great for modulating. A circle of
fifths ends up one step above where you start after 12 modulations. You have
to go around the circle 12 * 53 times to get back where you started. Not
that you'd want to. I once tried it in a piece and found it not so useful...

The problem with 53-EDO is the ratio of 11:8. I think it is too much of a
compromise to be so far off just. If I use the 24th step in 53-EDO, it's 11
cents off. If I use the 25th step, it's 20 cents off. And the 14:11 and 9:7
are the same in 53. I've used both, and have been uphappy with each.

In 72 that problem goes away, with all the 15 limit ratios represented very
well, within 7 cents, and a distinct 14:11 and 9:7. I use those ratios a lot
in my music. Anyone else using 72-EDO?

Prent Rodgers

You're right; while 5-limit ratios are more closely approximated by 53-edo, 72-edo excels at 7-limit and higher, while the 5-limit approximations are still very good.

72-edo is particularly attractive for me given that I'm pretty well committed to using a duodecimally-divided octave, with each division conveniently being '02 octave.

--- In tuning@yahoogroups.com, "Prent Rodgers" <prentrodgers@...> wrote:
>
> 53 Equal Divisions of the Octave (EDO) is great for modulating. A circle of fifths ends up one step above where you start after 12 modulations. You have to go around the circle 12 * 53 times to get back where you started. Not that you'd want to. I once tried it in a piece and found it not so useful...
>
> The problem with 53-EDO is the ratio of 11:8. I think it is too much of a compromise to be so far off just. If I use the 24th step in 53-EDO, it's 11 cents off. If I use the 25th step, it's 20 cents off. And the 14:11 and 9:7 are the same in 53. I've used both, and have been uphappy with each.
>
> In 72 that problem goes away, with all the 15 limit ratios represented very well, within 7 cents, and a distinct 14:11 and 9:7. I use those ratios a lot in my music. Anyone else using 72-EDO?
>
> Prent Rodgers
>

Claudio is absolutely right about the comma shifts ... Try to carefully compare the first and the last chord in this example ... You would suppose they should be the same, when they actually aren't:
www.sendspace.com/file/rxcm79

Petr

From: "Prent Rodgers" <prentrodgers@...>

> 53 Equal Divisions of the Octave (EDO) is great for modulating. A > circle of fifths ends up one step above where you start after 12 > modulations. You have to go around the circle 12 * 53 times to get > back where you started. Not that you'd want to. I once tried it in a > piece and found it not so useful...
>
> The problem with 53-EDO is the ratio of 11:8. I think it is too much > of a compromise to be so far off just. If I use the 24th step in > 53-EDO, it's 11 cents off. If I use the 25th step, it's 20 cents off. > And the 14:11 and 9:7 are the same in 53. I've used both, and have > been uphappy with each.
>
> In 72 that problem goes away, with all the 15 limit ratios represented > very well, within 7 cents, and a distinct 14:11 and 9:7. I use those > ratios a lot in my music. Anyone else using 72-EDO?

I've been talking about 72-et so much; I shouldn't answer this... (just kidding)

Yes, I've been working on a very large symphonic work, parts of it being in 72-et. Like Tony, I settled on 72 partly for convenience, it being a multiple of 12. It does indeed work better than 53 for the primes 7 and 11, and though 53 approximates 13 more precisely, 72 does fine with it, and I sometimes use 13/10 (c. 454.21 cents) as a diminished fourth, for example.

Both 53 and 72 are kleismic temperaments, and I'm using a modified octatonic of six pairs of alternating steps of 12 and 7 in 72-et and 9 and 5 in 53. The period is the perfect twelfth, the "tritave" of Bohlen-Pierce, but with twelve non-equal steps instead of thirteen equal. What should I call it anyway?

~D.

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:

> However, a nasty Syntonic comma issue subsists.
> If you start from C say and play the usual 4 consecutive fifths,
> you get ane' 31x4=124 steps up.
> Then you get down 2 octaves (2x53=106) and you expect to land in E,
But i expect to land after 4 times of 5ths within the
http://en.wikipedia.org/wiki/Pythagorean_interval
s at the ditone:
http://de.wikipedia.org/wiki/Ditonus
(9/8)^2 = 81/64 ~408Cents

> but you
> actually get in 124-106=18 steps,
Taht's concrete in 4 steps of 5ths upwards:
1/1=C-G-D-A=81/64 an ditonic melodic-3rd of ~408Cents.

> i.e. you are ONE STEP HIGHER than the
5/4 ~384Cents harmonic
> major third.

http://en.wikipedia.org/wiki/53_equal_temperament
"Instead, the major triads are chords like C-Fb-G, where the major third is a diminished fourth; this is the defining characteristic of schismatic temperament."
as obtained from a chain of 8 ascending consecutive fourths lined up:

1/1=C-F-Bb-Eb-Ab-Db-Gb-Cb=f-Fb=e=8192/6516 ~384Cents = (5/4)/schisma

or ((4/3)^8)/2 = 4.99436062... versus 81/16 = 5,0625 an SC=81/80 higher above the 'diminished-fourth'.

when compareing that both 3-limit approximations
against the intended 5th-partial 5/1 out of the overtone-series.

Conclusion:
The the 3-limit
http://en.wikipedia.org/wiki/Pythagorean_interval
diminished fourth 8192/6561 ~384.36
approximates the harmonic-3rd: 5/4 ~386Cents much better
than its counterpart,
the melodic-3rd 81/64 ~408Cents, the ditone.

Historically
the accurate discrimination inbetween
the 81/64_melodic-ditone-3rd ~408Cents
and the 5/4_harmoiic-5th-partial-3rd ~384Cents
or its schismatic dim-4th approximation 2^13/3^8 ~386Cents
was probably lost during the transition from
Renaissance JI to Baroque meantonics,
but was still used in Descartes's & Newton's drawings.

bye
A.S.