A quasi-Pythagorean notation for 36 EDO

Below is a notation for 36-et using the nominals F-B, sharps, and
flats. Before people starting screaming in agony, let me point out its
remarkable property: you can play meantone music in it! The fifth has
been mapped to an approximate 7/4, which makes the major third an
approximate 7/6. The chord C-E-G is a consonance of the form
1-7/6-7/4; C-Eb-G becomes 1-3/2-7/4. Consonant 5-limit otonalities of
meantone are mapped to consonant 7-limit utonalities of a temperament
which tempers out 1029/1024, and utonalities are mapped to
otonalities. The mapping goes from meantone to a subset of miracle. If
you take a midi file in 12-equal, use scala to produce a seq file with
notenames, replace "E12" with "Q36" or whatever Manuel decides to call
this if he uses it, and add a line "0 equal 36", the meantone music
you started out with will be warped into an entirely different piece
in a {2,3,7} subset of miracle.

0: C
1: B
2: A#
3: Gx
4: Bbbb
5: Abb
6: Gb
7: F
8: E
9: D#
10: Cx
11: Ebbb
12: Dbb
13: Cb
14: Bb
15: A
16: G#
17: Fx
18: Ex
19: Gbb
20: Fb
21: Eb
22: D
23: C#
24: B#
25: Ax
26: Cbb
27: Bbb
28: Ab
29: G
30: F#
31: E#
32: Dx
33: Fbb
34: Ebb
35: Db
36: C

Gene, don't you mean C-Eb-Bb when you write 1-7/6-7/4, and C-G-Bb when you write 1-3/2-7/4?

I had myself prepared long ago a mapping of 35 Western tones, and it's in http://www.ozanyarman.com/mainpage/academic.html. But the more I see how jumbled the mapping you have chosen is, the more I am having goosebumps as we speak.

Cordially,
Ozan

----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 23 Nisan 2005 Cumartesi 23:26
Subject: [tuning] A quasi-Pythagorean notation for 36 EDO

Below is a notation for 36-et using the nominals F-B, sharps, and
flats. Before people starting screaming in agony, let me point out its
remarkable property: you can play meantone music in it! The fifth has
been mapped to an approximate 7/4, which makes the major third an
approximate 7/6. The chord C-E-G is a consonance of the form
1-7/6-7/4; C-Eb-G becomes 1-3/2-7/4. Consonant 5-limit otonalities of
meantone are mapped to consonant 7-limit utonalities of a temperament
which tempers out 1029/1024, and utonalities are mapped to
otonalities. The mapping goes from meantone to a subset of miracle. If
you take a midi file in 12-equal, use scala to produce a seq file with
notenames, replace "E12" with "Q36" or whatever Manuel decides to call
this if he uses it, and add a line "0 equal 36", the meantone music
you started out with will be warped into an entirely different piece
in a {2,3,7} subset of miracle.

0: C
1: B
2: A#
3: Gx
4: Bbbb
5: Abb
6: Gb
7: F
8: E
9: D#
10: Cx
11: Ebbb
12: Dbb
13: Cb
14: Bb
15: A
16: G#
17: Fx
18: Ex
19: Gbb
20: Fb
21: Eb
22: D
23: C#
24: B#
25: Ax
26: Cbb
27: Bbb
28: Ab
29: G
30: F#
31: E#
32: Dx
33: Fbb
34: Ebb
35: Db
36: C

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Gene, don't you mean C-Eb-Bb when you write 1-7/6-7/4, and C-G-Bb
when you write 1-3/2-7/4?

No, I'm afraid not. In the standard septimal meantone, the correct way
to write 1-7/6-7/4 would be C-D#-A#, and 1-3/2-7/4 would be C-G-A#. In
the meantone variant we've called "dominant", they would indeed be
C-Eb-Bb and C-G-Bb, but that comes with not very accurate tuning.

However, in this system I'm using C-G as the 36-equal approximation to
1-7/4, not 1-3/2. Hence four "fifths" of size about 7/4 up, and two
octaves down, is 2401/1024, which when you flatten it by 1024/1029,
becomes 7/3. Hence you can map 1-5/4-3/2 to 1-7/3-7/4, but this takes
us out of the range of an octave and won't work as a Scala-like
notation system. However, four 7/4s and *three* octaves down give us
an approximate 7/6 in 36-et, and so we can call this note "E".

The notation as given sends consonances to consonances. Sometimes you
might not like which octave you get, in which case you can adjust up
or down an octave. But as it stands it will take 5-limit meantone
music and make consonant but totally different music out of it. More
complex chords are dicier. The interval C-Bb is sent to a 21/16, so
C-E-G-Bb is approximately 1-7/6-21/16-7/4, which is pretty cool. C-A#
however becomes rather disasterously an approximate 28/27.

> I had myself prepared long ago a mapping of 35 Western tones, and
it's in http://www.ozanyarman.com/mainpage/academic.html. But the more
I see how jumbled the mapping you have chosen is, the more I am having
goosebumps as we speak.
>
> Cordially,
> Ozan
>
> ----- Original Message -----
> From: Gene Ward Smith
> To: tuning@yahoogroups.com
> Sent: 23 Nisan 2005 Cumartesi 23:26
> Subject: [tuning] A quasi-Pythagorean notation for 36 EDO
>
>
>
> Below is a notation for 36-et using the nominals F-B, sharps, and
> flats. Before people starting screaming in agony, let me point out its
> remarkable property: you can play meantone music in it! The fifth has
> been mapped to an approximate 7/4, which makes the major third an
> approximate 7/6. The chord C-E-G is a consonance of the form
> 1-7/6-7/4; C-Eb-G becomes 1-3/2-7/4. Consonant 5-limit otonalities of
> meantone are mapped to consonant 7-limit utonalities of a temperament
> which tempers out 1029/1024, and utonalities are mapped to
> otonalities. The mapping goes from meantone to a subset of miracle. If
> you take a midi file in 12-equal, use scala to produce a seq file with
> notenames, replace "E12" with "Q36" or whatever Manuel decides to call
> this if he uses it, and add a line "0 equal 36", the meantone music
> you started out with will be warped into an entirely different piece
> in a {2,3,7} subset of miracle.
>
>
> 0: C
> 1: B
> 2: A#
> 3: Gx
> 4: Bbbb
> 5: Abb
> 6: Gb
> 7: F
> 8: E
> 9: D#
> 10: Cx
> 11: Ebbb
> 12: Dbb
> 13: Cb
> 14: Bb
> 15: A
> 16: G#
> 17: Fx
> 18: Ex
> 19: Gbb
> 20: Fb
> 21: Eb
> 22: D
> 23: C#
> 24: B#
> 25: Ax
> 26: Cbb
> 27: Bbb
> 28: Ab
> 29: G
> 30: F#
> 31: E#
> 32: Dx
> 33: Fbb
> 34: Ebb
> 35: Db
> 36: C

Gene,

You answered Ozan as follows:
> > Gene, don't you mean C-Eb-Bb when you write 1-7/6-7/4, and C-G-Bb
> > when you write 1-3/2-7/4?

> No, I'm afraid not.
...
> However, in this system I'm using C-G as the 36-equal approximation to
> 1-7/4, not 1-3/2. Hence four "fifths" of size about 7/4 up, and two
> octaves down, is 2401/1024, which when you flatten it by 1024/1029,
> becomes 7/3. Hence you can map 1-5/4-3/2 to 1-7/3-7/4, but this takes
> us out of the range of an octave and won't work as a Scala-like
> notation system. However, four 7/4s and *three* octaves down give us
> an approximate 7/6 in 36-et, and so we can call this note "E".

[Yahya]
Gene, don't you think that this kind of naming is very confusing?

When I see C-E-G, I expect a C major triad.
When I see C-Eb-G, I expect a C minor triad.
Call me naïve! :-)

What's more, I expect Eb to E, flattened, and therefore lower than E.
But your E and Eb don't conform to this: your E is 7/6 and your Eb is 3/2,
which is 9/6, giving Eb:E = 9:7.

To my mind, this kind of naming simply makes little musical sense.

You wrote: "The chord C-E-G is a consonance of the form 1-7/6-7/4;
C-Eb-G becomes 1-3/2-7/4." If we multiply through by the LCM of 6 and
4, namely 12, we get C:E:G = 12:14:21, while C:Eb:G = 12:18:21.

Why would you want your Eb to be several steps higher in the scale than
your E?

-----

Nomenclature aside, I do find this kind of mathematical warping very
interesting. You wrote, in part:
"Consonant 5-limit otonalities of meantone are mapped to consonant
7-limit utonalities of a temperament which tempers out 1029/1024,
and utonalities are mapped to otonalities. The mapping goes from
meantone to a subset of miracle. If you ..., the meantone music you
started out with will be warped into an entirely different piece in a
{2,3,7} subset of miracle. "

Which leads me to wonder ...:

What other mappings between temperaments give us this inversion
of u- and o-tonalities? And are there any such inverting mappings
from one temperament to itself?

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Gene, don't you think that this kind of naming is very confusing?

Whether or not it is confusing is pretty much irrelevant to the point
of it.

> To my mind, this kind of naming simply makes little musical sense.

How much musical sense it makes depends on using it as I suggested it
be used, as a way of retuning a piece in 12 equal or meantone.

> Why would you want your Eb to be several steps higher in the scale than
> your E?

It's just the way it works out.

> What other mappings between temperaments give us this inversion
> of u- and o-tonalities? And are there any such inverting mappings
> from one temperament to itself?

Sounds like a tuning math question. I've looked at these kind of
mappings, but without paying attention to otonalities vs utonalities.