Re: 31-tET notation

on 2/14/01 5:39 AM, tuning@yahoogroups.com at tuning@yahoogroups.com wrote:

> C D E F G A B C
>
> Now we fill it up chromatically, first following 12-tet:
>
> C . D . E F . G . A . B C (the dot might be called # or b, just the same)
>
> Next following 19-tet:
>
> C . . D . . E . F . . G . . A . . B . C ( every first dot is called # and
> every secon dot is called b, except for the dot in between B and C, and E
> and F, which may be called either # or b). This doesn't really go against
> "basic training", for the ear still recognizes C D E F G A B C as an
> ordinary seven note scale. This works in the same way for 31-tet. You may
> take a 12-tet score for a traditional guitarpiece and sightread it
> instantaneaously in 19-tet. It will sound pretty close to meantone.
>
> However, it will go against "basic training" if we use this system for
> 21-tet:
>
> C . . D . . E . . F . . G . . A . . B . . C
>
> Now of course the seven note scale doesn't sound as the basic C D E F G A B
> C anymore. For example, the E# in this notation sounds virtually as an E in
> 12-tet! Dans la poubelle!

While on the topic of notation, I am curious if there is an already accepted
system of flat/sharp notation for 31-tET that keeps the standard C, D, E,
etc in meantone as you mentioned for 19-tET. My own shorthand has been
combination of fret numbers and/or ratio approximations, and it's a bit
cumbersome if I compose something for someone else to play besides myself.

Thanks in advance,

Seth

--
Seth Austen

http://www.sethausten.com
emails: seth@sethausten.com
klezmusic@earthlink.net

Seth Austen wrote,

>While on the topic of notation, I am curious if there is an already
accepted
>system of flat/sharp notation for 31-tET that keeps the standard C, D, E,
>etc in meantone as you mentioned for 19-tET.

There sure is, since 31-tET is _very_ meantone!

Using only standard notations symbols, it comes out as:

C-Dbb-C#-Db-Cx-D-Ebb-D#-Ev-Dx-E-Fb-E#-F-Gbb-F#-Gb-Fx-G-Abb-G#-Ab-Gx-A-Bbb-A#
-Bb-Ax-B-Cb-B#-C.

In other words, 7 naturals, 7 sharps, 7 flats, 5 double-sharps, and 5 double
flats, extending 15 fifths up and 15 fifths down from D.

Using a "half sharp" ^ and a "half flat" v, the following notation arises,
which may be more convenient if you're using 31-tET more "microtonally":

C-C^-C#-Db-Dv-D . . . etc.

You can see what this notation actually looks like (using sesqui-sharps and
sesqui-flats as well) here:

http://www.xs4all.nl/~huygensf/english/index.html

and click on "Theory" on the left.

Seth Austen wrote:

> While on the topic of notation, I am curious if there is an already
> accepted
> system of flat/sharp notation for 31-tET that keeps the standard C, D,
> E,
> etc in meantone as you mentioned for 19-tET. My own shorthand has been
> combination of fret numbers and/or ratio approximations, and it's a bit
> cumbersome if I compose something for someone else to play besides
> myself.

Indeed there is!

C C/ C# Db D\ D D/ D# Eb E\ E E/ F\ F F/ F# Gb ...
... F# Gb G\ G G/ G# Ab A\ A A/ A# Bb B\ B B/ C\ C

The / and \ are less accepted, but most people use some kind of
equivalent. You could use double sharps and flats if you really wanted
to. It's an obvious pattern. Anything there that seems to break it is
probably a typo.

Graham

Hi Seth!

Seth Austen wrote:
> While on the topic of notation, I am curious if there is an
> already accepted system of flat/sharp notation for 31-tET that
> keeps the standard C, D, E, etc in meantone as you mentioned for
> 19-tET. My own shorthand has been combination of fret numbers
> and/or ratio approximations, and it's a bit cumbersome if I
> compose something for someone else to play besides myself.

Thanks, Seth. I've been itching for someone to ask that.
By extending to 31, we get to use _all_ the notes that
common practice composers used.

Here we have our 31 standard note names, arranged in
the circle of fifths:
(I have kept Graham's intelligent idea to anchor the scale
at D to show the symmetry more easily.)

Gbb Dbb Abb Ebb Bbb Fb
Cb Gb Db Ab Eb Bb F C G
(D)
A E B F# C# G# D# A# E#
B# FX CX GX DX AX

But how do we sort them?

Note that the generating interval of 31tET -- as a
pseudo 1/4-comma-meantone temperament -- is it's
18th scale step of (2:1^(18/31)) = 658.058 cents.

348.387 cents = (1:2^(18/31))^15 * 512 = Gbb
1045.161 cents = (1:2^(18/31))^14 * 512 = Dbb
541.935 cents = (1:2^(18/31))^13 * 256 = Abb
38.710 cents = (1:2^(18/31))^12 * 128 = Ebb
735.484 cents = (1:2^(18/31))^11 * 128 = Bbb
232.258 cents = (1:2^(18/31))^10 * 64 = Fb
929.032 cents = (1:2^(18/31))^9 * 64 = Cb
425.806 cents = (1:2^(18/31))^8 * 32 = Gb
1122.581 cents = (1:2^(18/31))^7 * 32 = Db
619.355 cents = (1:2^(18/31))^6 * 16 = Ab
116.129 cents = (1:2^(18/31))^5 * 8 = Eb
812.903 cents = (1:2^(18/31))^4 * 8 = Bb
309.677 cents = (1:2^(18/31))^3 * 4 = F
1006.452 cents = (1:2^(18/31))^2 * 4 = C
503.226 cents = 1:2^(18/31) * 2 = G
0.000 cents = 1:1 = (D)
696.774 cents = 2:1^(18/31) = A
193.548 cents = (2:1^(18/31))^2 / 2 = E
890.323 cents = (2:1^(18/31))^3 / 2 = B
387.097 cents = (2:1^(18/31))^4 / 4 = F#
1083.871 cents = (2:1^(18/31))^5 / 4 = C#
580.645 cents = (2:1^(18/31))^6 / 8 = G#
77.419 cents = (2:1^(18/31))^7 / 16 = D#
774.194 cents = (2:1^(18/31))^8 / 16 = A#
270.968 cents = (2:1^(18/31))^9 / 32 = E#
967.742 cents = (2:1^(18/31))^10 / 32 = B#
464.516 cents = (2:1^(18/31))^11 / 64 = FX
1161.290 cents = (2:1^(18/31))^12 / 64 = CX
658.065 cents = (2:1^(18/31))^13 / 128 = GX
154.839 cents = (2:1^(18/31))^14 / 256 = DX
851.613 cents = (2:1^(18/31))^15 / 256 = AX

Now we just sort these to get a monotonically
increasing sequence:

0.000 cents = 1:1 = (D)
38.710 cents = (1:2^(18/31))^12 * 128 = Ebb
77.419 cents = (2:1^(18/31))^7 / 16 = D#
116.129 cents = (1:2^(18/31))^5 * 8 = Eb
154.839 cents = (2:1^(18/31))^14 / 256 = DX
193.548 cents = (2:1^(18/31))^2 / 2 = E
232.258 cents = (1:2^(18/31))^10 * 64 = Fb
270.968 cents = (2:1^(18/31))^9 / 32 = E#
309.677 cents = (1:2^(18/31))^3 * 4 = F
348.387 cents = (1:2^(18/31))^15 * 512 = Gbb
387.097 cents = (2:1^(18/31))^4 / 4 = F#
425.806 cents = (1:2^(18/31))^8 * 32 = Gb
464.516 cents = (2:1^(18/31))^11 / 64 = FX
503.226 cents = 1:2^(18/31) * 2 = G
541.935 cents = (1:2^(18/31))^13 * 256 = Abb
580.645 cents = (2:1^(18/31))^6 / 8 = G#
619.355 cents = (1:2^(18/31))^6 * 16 = Ab
658.065 cents = (2:1^(18/31))^13 / 128 = GX
696.774 cents = 2:1^(18/31) = A
735.484 cents = (1:2^(18/31))^11 * 128 = Bbb
774.194 cents = (2:1^(18/31))^8 / 16 = A#
812.903 cents = (1:2^(18/31))^4 * 8 = Bb
851.613 cents = (2:1^(18/31))^15 / 256 = AX
890.323 cents = (2:1^(18/31))^3 / 2 = B
929.032 cents = (1:2^(18/31))^9 * 64 = Cb
967.742 cents = (2:1^(18/31))^10 / 32 = B#
1006.452 cents = (1:2^(18/31))^2 * 4 = C
1045.161 cents = (1:2^(18/31))^14 * 512 = Dbb
1083.871 cents = (2:1^(18/31))^5 / 4 = C#
1122.581 cents = (1:2^(18/31))^7 * 32 = Db
1161.290 cents = (2:1^(18/31))^12 / 64 = CX

And there you have it.

- Jeff