Yahoo Tuning Groups Ultimate Backup tuning Some more ideas and experiments note naming 16e

After responding to a post by Herman Miller (which dealt with a
notation for 16e) last night I tried to dig up an old piece of mine
that used a (>5/9 & <4/7) 7L&2s

(7) 5 3 1 8 6 4 2 9
16 (14) 12 10 17 15 13 11 18
25 23 (21) 19 26 24 22 20 27
34 32 30 (28)(35) 33 31 29 36
43 41 39 37 44 (42) 40 38 45
52 50 48 46 53 51 (49) 47 54
61 59 57 55 62 60 58 (56)(63)
70 68 66 64 71 69 67 65 72
...

scale (I think it was LLLsLLLLs):

*-------C-------*
/|\ /|\ / \
/ | \ / | \ / \
/ | \ / | \ / \
*---|---*---|---*-------*
\ | / \ | /
\ | / \ | /
\|/ \|/
*-------*

that I thought had a pretty consistent notation scheme for 16e... well
I still haven't been able to find it, but I did notice that if you
were to carry a circle of (log(5)-log(1))*(16/log(2^4)) or ~9.29/16
fifths out to what should be Bx & Dbbb, but were to then rearrange the
spellings to something on the order of:

0
F 7 9 G
Bb 13 3 D
Eb 4 12 A
Ab 11 5 E
Db 2 14 B
G\ 8 8 F/
C\ 15 1 C/
F\ 6 10 G/
Bbb 12 4 D#
Ebb 3 13 A#
A\ 10 6 E/
D\ 1 15 B/
Gbb 7 9 Fx
Cb 14 2 C#
Fb 5 11 G#
Bbbb 11 5 Dx
Ebbb 2 14 Ax
Abb 9 7 E#
Dbb 16 0 B#

or,

Dx------Ax------E#------B#
|\ /|\ /|\ /|
| \ / | \ / | \ / |
| \ / | \ / | \ / |
| Fx--|---C#--|---G# |
| /|\ | /|\ | /|\ |
| / | \ | / | \ | / | \ |
|/ | \|/ | \|/ | \|
D#--|---A#--|---E/--|---B/
|\ | /|\ | /|\ | /|
| \ | / | \ | / | \ | / |
| \|/ | \|/ | \|/ |
| F/--|---C/--|---G/ |
| /|\ | /|\ | /|\ |
| / | \ | / | \ | / | \ |
|/ | \|/ | \|/ | \|
D---|---A---|---E---|---B
|\ | /|\ | /|\ | /|
| \ | / | \ | / | \ | / |
| \|/ | \|/ | \|/ |
| F---|---C---|---G |
| /|\ | /|\ | /|\ |
| / | \ | / | \ | / | \ |
|/ | \|/ | \|/ | \|
Db--|---Ab--|---Eb--|---Bb
|\ | /|\ | /|\ | /|
| \ | / | \ | / | \ | / |
| \|/ | \|/ | \|/ |
| F\--|---C\--|---G\ |
| /|\ | /|\ | /|\ |
| / | \ | / | \ | / | \ |
|/ | \|/ | \|/ | \|
D\--|---A\--|---Ebb-|---Bbb
|\ | /|\ | /|\ | /|
| \ | / | \ | / | \ | / |
| \|/ | \|/ | \|/ |
| Fb--|---Cb--|---Gbb |
| / \ | / \ | / \ |
| / \ | / \ | / \ |
|/ \|/ \|/ \|
Dbb-----Abb-----Ebbb----Bbbb

you could then achieve some measure of a compromise between
consistency and traditional notation... though I suppose that the
emphasis here actually falls rather strongly on the side of enharmonic
consistency.

A 2L&5s mapping:

(2)(4)(6) 1 3 5 7
9 11 13 (8)(10)(12)(14)
16 18 20 15 17 19 21
...

(where the linear interval falls inside a 1/14 space that is >1/2 and
<4/7):

1/2 4/7
5/9
6/11 9/16
...

taken F to B (on the ~9.29/16 circle of fifths):

*---*---*---*
\ / \ / \ /
*---C---*

(as opposed to the ssLsssL Db to G):

*---C---*
/ \ / \ / \
*---*---*---*

could then be seen as something on the order of:

0 1 2 3 4 5 6 7 (8 8) 9 10 11 12 13 14 15 16
-----------------------------------------------------------
1,/1 b2,2 b3,3 \4,4,#4 b5,5,/5 b6,6 b7,7 \8,8

C,C/ Db,D Eb,E F\,F,F/ G\,G,G/ Ab,A Bb,B C\,C
C/,C# D,D# E,E/ F,F/,Fx G,,G/,G# A,A# B,B/ C,C/
Db,D Eb,E F\,F G\,G,G/ A\,Ab,A Bb,B C\,C D\,Db
D,D# E,E/ F,F/ G,G/,G# Ab,A,A# B,B/ C,C/ Db,D
Eb,E F\,F G\,G A\,Ab,A Bbb,Bb,B C\,C D\,Db Ebb,Eb
E,E/ F,F/ G,G/ Ab,A,A# Bb,B,B/ C,C/ Db,D Eb,E
F\,F G\,G A\,Ab Bbb,Bb,B Cb,C\,C D\,Db Ebb,Eb Fb,F\
F,F/ G,G/ Ab,A Bb,B,B/ C\,C,C/ Db,D Eb,E F\,F
F/,Fx G/,G# A,A# B,B/,B# C,C/,C# D,D# E,E/ F,F/

G\,G A\,Ab Bbb,Bb Cb,C\,C Dbb,D\,Db Ebb,Eb Fb,F\ F,G\
G,G/ Ab,A Bb,B C\,C,C/ D\,Db,D Eb,E F\,F G\,G
G/,G# A,A# B,B/ C,C/,C# Db,D,D# E,E/ F,F/ G,G/
Ab,A Bb,B C\,C D\,Db,D Ebb,Eb,E F\,F G\,G A\,Ab
A,A# B,B/ C,C/ Db,D,D# Eb,E,E/ F,F/ G,G/ Ab,A
Bb,B C\,C D\,Db Ebb,Eb,E Fb,F\,F G\,G A\,Ab Bbb,Bb
B,B/ C,C/ Db,D Eb,E,E/ F\,F,F/ G,G/ Ab,A Bb,B
C\,C D\,Db Ebb,Eb Fb,F\,F Gbb,G\,G A\,Ab Bbb,Bb Cb,C\
C,C/ Db,D Eb,E F\,F,F/ G\,G,G/ Ab,A Bb,B C\,C

Dan

Some more ideas and experiments note naming 16e

🔗D.Stearns <stearns@xxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗D.Stearns <stearns@xxxxxxx.xxxx>