11-limit Lucy tuning

Charles never seems to use his tuning for anything but 5-limit
harmony. To boldly go where no Lucy tuning has gone before, I suggest
the following mapping:

<1200 1800+300/pi 2400+1200/pi 3400-100/pi 4400-800/pi|

This gives the following errors for odd primes to 11:

3: -6.462035
5: -4.341850
7: -0.656895
11: -5.965851

Here the minus sign indicates flatness.

If we apply the above tuning to a comma list, we find the following
are tempered out exactly: 81/80, 1029/1024, 1728/1715, 99/98,
385/384, 441/400, 3025/3024. This means we are tempering according to
11-limit mothra, the 26&31 temperament. The generator is 1/3 of a
fifth, equal to an 8/7, of (600+300/pi)/3 = 200+100/pi cents.

Trying to notate this in standard notation doesn't work since there
is no way to notate exactly 1/3 of a fifth. Adding a symbol pair for
the septimal comma (64/63) seems like a good start. C-D is +6
generators, so since it gets to the 8/7 generator, 64/63 is a -5
generator symbol. Conveniently, in mothra 33/32 and 64/63 are the
same, and so 11/8 is C-Ff. The same symbol serves for 36/35, 49/49,
55/54, and 56/55, and this may be all we really need. Any thoughts
from the notation experts on notating mothra?

Gene Ward Smith wrote:
> Charles never seems to use his tuning for anything but 5-limit > harmony. To boldly go where no Lucy tuning has gone before, I suggest > the following mapping:
> > <1200 1800+300/pi 2400+1200/pi 3400-100/pi 4400-800/pi|
> > This gives the following errors for odd primes to 11:
> > 3: -6.462035
> 5: -4.341850
> 7: -0.656895
> 11: -5.965851
> > Here the minus sign indicates flatness.
> > If we apply the above tuning to a comma list, we find the following > are tempered out exactly: 81/80, 1029/1024, 1728/1715, 99/98, > 385/384, 441/400, 3025/3024. This means we are tempering according to > 11-limit mothra, the 26&31 temperament. The generator is 1/3 of a > fifth, equal to an 8/7, of (600+300/pi)/3 = 200+100/pi cents.
> > Trying to notate this in standard notation doesn't work since there > is no way to notate exactly 1/3 of a fifth. Adding a symbol pair for > the septimal comma (64/63) seems like a good start. C-D is +6 > generators, so since it gets to the 8/7 generator, 64/63 is a -5 > generator symbol. Conveniently, in mothra 33/32 and 64/63 are the > same, and so 11/8 is C-Ff. The same symbol serves for 36/35, 49/49, > 55/54, and 56/55, and this may be all we really need. Any thoughts > from the notation experts on notating mothra?

Compound nominal notation runs into difficulties since the only thing that almost works is a 21-note MOS, but there are pairs of notes within the same 1/24 of the octave. So going with a chain-of-fifths Sagittal notation is probably the best choice here. Either |) !) 64/63, ~|) ~!) 49/48, or /|) \!) 36/35 could be used for the main accidental, covering everything from -6 to +6. For +7, +10, +13 I'll suggest (/| (\! 4096/3969 as a -10 accidental. Then +12, +15, etc. can be notated with ||\ !!/ 135/128 for a +21 accidental.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Compound nominal notation runs into difficulties since the only thing
> that almost works is a 21-note MOS, but there are pairs of notes
within
> the same 1/24 of the octave. So going with a chain-of-fifths Sagittal
> notation is probably the best choice here.

It's impossible to notate mothra in chain-of-fifths, because you only
get every third fifth. You can try to solve that by picking an edo, but
I've deliberately subverted that by proposing to notate Lucy tuning,
which happens to be a pretty decent mothra tuning.

Either |) !) 64/63, ~|) ~!)
> 49/48, or /|) \!) 36/35 could be used for the main accidental,
covering
> everything from -6 to +6. For +7, +10, +13 I'll suggest (/| (\!
> 4096/3969 as a -10 accidental. Then +12, +15, etc. can be notated
with
> ||\ !!/ 135/128 for a +21 accidental.

I'm not sure I'm getting the gist. Are you saying thst beyond 6 and 21.
we could pick a symbol pair for 10 and leave it at that?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> It's impossible to notate mothra in chain-of-fifths, because you only
> get every third fifth.

Every third note, sorry.

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >> Compound nominal notation runs into difficulties since the only thing >> that almost works is a 21-note MOS, but there are pairs of notes > within >> the same 1/24 of the octave. So going with a chain-of-fifths Sagittal >> notation is probably the best choice here. > > It's impossible to notate mothra in chain-of-fifths, because you only > get every third note.

That's the idea.

F 32/27 (+2, -9)
C 16/9 (+2, -6)
G 4/3 (+1, -3)
D 1/1 (+0, +0)
A 3/2 (+0, +3)
E 9/8 (-1, +6)
B 27/16 (-1, +9)

Then fill in the gaps, starting with an accidental for (+1, -5). Using ~|) 49/48 as an example:

F~|) 98/81 (+3, -14) F~!) 512/441 (+1, -4)
C~|) 49/27 (+3, -11) C~!) 256/147 (+1, -1)
G~|) 49/36 (+2, -8) G~!) 64/49 (+0, +2)
D~|) 49/48 (+1, -5) D~!) 96/49 (+0, +5)
A~|) 49/32 (+1, -2) A~!) 72/49 (-1, +8)
E~|) 147/128 (+0, +1) E~!) 54/49 (-2, +11)
B~|) 441/256 (+0, +4) B~!) 81/49 (-2, +14)

The next gap is at (-1, +7), (+2, -7). The simplest way to notate this appears to be G(\! 1323/1024, A(/| 2048/1323. Alternatively, G'!!) 80/63, A.||) 63/40 would work, if you don't mind accented symbols.

G'!!) 80/63 (-1, +7) F.||) 56/45 (+4, -19)
D'!!) 40/21 (-1, +10) C.||) 28/15 (+4, -16)
A'!!) 10/7 (-2, +13) G.||) 7/5 (+3, -13)
E'!!) 15/14 (-3, +16) D.||) 21/20 (+2, -10)
B'!!) 45/28 (-3, +19) A.||) 63/40 (+2, -7)

You can see where this is going; the next set of gaps is at (-2, +12), (+3, -12) which can conveniently be notated as F||\ 5/4 and B!!/ 8/5, although F)||( 100/81, B)!!( 81/50 is another possibility.

F||\ 5/4 (-2, +12)
C||\ 15/8 (-2, +15)
G||\ 45/32 (-3, +18)
D||\ 135/128 (-4, +21)
A||\ 405/256 (-4, +24)
E||\ 1215/1024 (-5, +27)
B||\ 3645/2048 (-5, +30)

The next set of gaps is at (-3, +17), (+4, -17), which can be notated as F)/|\ 60/49, B)\!/ 49/30. Then comes (-4, +22), (+5, -22), which can be notated F'|( 25/21, B.!( 42/25 (or if you really need unaccented symbols, F|\) 321489/262144 B!/) 524288/321489). Now when you get around to notating (-6, +33), (+7, -33) I really don't see any way to avoid accented symbols: G'\\! 125/96 and A.//| 192/125. The next set of gaps is at (-7, +38), (+8, -38), and we're getting to the point where you're going to have to use some of the more obscure accents that I'm not familiar with. But you can notate a chain of 75 notes of mothra temperament with the documented Sagittal symbols and the standard 7 nominals, which is a pretty good start.