Yahoo Tuning Groups Ultimate Backup tuning-math Notes on 7-limit temperament notation, part 1

Here are some possibilites for using Sagittal notation for the first few 7-limit temperaments in Paul's _Middle Path_ paper. I've tried to find sets of accidentals that add up correctly. In some cases I found more than one set. I also tried to stick with more familiar symbols if I could get them to work out. There may be other possibilities for some of these temperaments.

Blacksmith
5&10, 5&15, 10&15
[<5, 8, 12, 14], <0, 0, -1, 0]>
TOP P = 239.178693, G = 83.830599
TOP-RMS P = 239.445389, G = 87.030949

-2 -1 0 +1 +2
+0 D E!!/ F)!!(
+1 D)||( E\! E F/| G)!!(
+2 E)||( F||\ G A!!/ A)!!(
+3 G)||( G||\ A B!!/ C)!!(
+4 A)||( B\! C C/| D)!!(
+5 B)||( C||\ D

)||( [-3, -1, 2> (+1, -2)
/| [-4, 4, -1> (-0, +1)
||\ [-7, 3, 1> (+1, -1)

)||( + /| = ||\

F is equivalent to E, and B is equivalent to C. Depending on the key, it may make sense to substitute one for the other.

Note that /| 81/80 is not a good match for the size, but it makes more sense than the alternatives /|) and .||) -- the size mismatch between )||( and /| can be ignored if /| is used only with F and C, and )||( with D, E, G, A, B.

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Dimisept
4&12, 4&16, 12&16
[<4, 6, 9, 11], <0, 1, 1, 1]>
TOP P = 298.532115, G = 101.456140
TOP-RMS P = 299.054827, G = 99.209898

-3 -2 -1 0 +1 +2 +3
0 D E)!!( E
+1 E!!/ E\! F/| F||\ G/|
+2 F G)!!( G G)||( A A)||( B
+3 A\! B!!/ B\! C/| C||\
+4 C C)||( D

/| [-4, 4, -1> (-1, +3)
)||( [-3, -1, 2> (+0, +1)
||\ [-7, 3, 1> (-1, +4)

/| + )||( = ||\

Here's one of the disadvantages of using chain of fifth nominals: notes on the extreme ends of the chain (B, F) will not be likely to be needed at the same time. You could use F if your reference pitch is C, but in that case, B is so far out that it will never be used. So you end up needing 3 pairs of accidentals for a scale that typically would use no more than 12 notes.

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Dominant
5&12
[<1, 2, 4, 2], <0, -1, -4, 2]>
TOP P = 1195.228951, G = 495.881015
TOP-RMS P = 1195.412191, G = 496.521245

E)\! A)\! D)\! G)\!
B/||\ E/||\ A/||\ D/||\ G/||\ C/||\ F/||\
B E A D G C F
B\!!/ E\!!/ A\!!/ D\!!/ G\!!/ C\!!/ F\!!/
A)/| D)/| G)/| C)/|

)/| [-19, 12> (+5, -12)
/||\ [-11, 7> (+3, -7)

Other possibilities for -12 include '|) [-9, 6, 1, -1> and '(| [1, 0, 2, -2>.

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August
6&12, 6&18, 12&18
[<3, 5, 7, 9], <0, -1, 0, -2]>
TOP P = 399.992210, G = 107.311173
TOP-RMS P = 399.128482, G = 103.762811

-3 -2 -1 0 +1 +2 +3
0 D E!!/ E!) F
+1 D||\ E F|) F||\ G A!!/ A!)
+2 G|) G||\ A B!!/ B!) C D!!/
+3 B C|) C||\ D

|) [6, -2, 0, -1> (-1, +4)
||\ [-7, 3, 1> (+1, -3)

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Pajara
10&12, 10&22, 12&22
[<2, 3, 5, 6], <0, 1, -2, -2]>
TOP P = 598.446711, G = 106.566546
TOP-RMS P = 598.859347, G = 106.844183

12 A\! 01 D)||(
14 A)||( 03 E\!
16 B\! 05 F
18 C 07 F||\
20 C||\ 09 G
00 D 11 G||\
02 E!!/ 13 A
04 E 15 B!!/
06 F/| 17 B
08 G)!!( 19 C/|
10 G/| 21 D)!!(

/| [-4, 4, -1> (-1, +6)
)||( [-3, -1, 2> (+1, -5)
||\ [-7, 3, 1> (+0, +1)
/||\ [-11, 7> (-1, +7)

/| + )||( = ||\
/| + ||\ = /||\

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Semaphore
5&19
[<1, 2, 4, 3], <0, -2, -8, -1]>
TOP P = 1203.668841, G = 252.480358
TOP-RMS P = 1203.852847, G = 253.446134

02 E'!!) 06 F/||\ 10
10 A'!!) 14 B 18
18 D'!!) 03 E 07
07 G'!!) 11 A 15 B.||)
15 C'!!) 00 D 04 E.||)
04 F'!!) 08 G 12 A.||)
12 16 C 01 D.||)
01 05 F 09 G.||)
09 13 B\!!/ 17 C.||)

'|) [-9, 6, 1, -1> (+4, -19) or )|)
.||) [-1, 2, -1, 1> (-1, +5) or )/||
/||\ [-11, 7> (+3, -14)

'|) + .||) = /||\

I'm not entirely satisfied with this, but the alternatives don't look much better. Here are a couple of other sets where the arithmetic works out.

'|) [-9, 6, 1, -1> (+4, -19) or )|)
/|) [2, 2, -1, -1> (-1, +5)
~~|| [-7, 8, 0, -2> (+3, -14)

'|) + /|) = ~~||

'|) [-9, 6, 1, -1> (+4, -19) or )|)
.(|\ [2, -3, 0, 1> (-1, +5) or |\\
||\ [-7, 3, 1> (+3, -14)

'|) + .(|\ = ||\

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Meantone
12&19, 12&31, 12&43, 19&31, 19&50, 31&43, 31&50, 31&81, 50&81
[<1, 2, 4, 7], <0, -1, -4, -10]>
TOP P = 1201.698520, G = 504.134131
TOP-RMS P = 1201.242156, G = 504.026298

B(|\ E(|\ A(|\ D(|\ G(|\
C(|\ F(|\ B!) E!) A!) D!) G!)
C!) F!) A||\ D||\ G||\ C||\ F||\
B E A D G C F
B!!/ E!!/ A!!/ D!!/ G!!/ B|) E|)
A|) D|) G|) C|) F|) B(!/ E(!/
A(!/ D(!/ G(!/ C(!/ F(!/

(|\ [-13, 5, 1, 1> (+8, -19)
|) [6, -2, 0, -1> (-5, +12)
||\ [-7, 3, 1> (+3, -7)

(|\ + |) = ||\

Having |) represent something bigger than (|\ isn't very desirable, but here are a couple of the alternatives.

~|) [-4, -1, 0, 2> (+8, -19)
|) [6, -2, 0, -1> (-5, +12)
.(|\ [2, -3, 0, 1> (+3, -7) or |\\

~|) + |) = .(|\

~|) [-4, -1, 0, 2> (+8, -19)
'(| [1, 0, 2, -2> (-5, +12)
)||( [-3, -1, 2> (+3, -7)

~|) + '(| = )||(

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Injera
12&26
[<2, 3, 4, 5], <0, 1, 4, 4]>
TOP P = 600.888907, G = 93.609825
TOP-RMS P = 600.682602, G = 94.482670

14 A~!) 01 D|)
16 A|) 03 E~!)
18 B~!) 05 E|)
20 B|) 07 F
22 C 09 F.(|\
24 C.(|\ 11 G
00 D 13 G.(|\
02 E'(!/ 15 A
04 E 17 B'(!/
06 F!) 19 B
08 F~|) 21 C!)
10 G!) 23 C~|)
12 G~|) 25 D!)

~| [-10, 1, 0, 3> (-2, +13)
|) [6, -2, 0, -1> (+1, -6)
~|) [-4, -1, 0, 2> (-1, +7)
.(|\ [2, -3, 0, 1> (+0, +1) or |\\

~| + |) = ~|)
|) + ~|) = .(|\

One of these other two possibilities might be better if you don't need more than 26 notes.

/|) [2, 2, -1, -1> (+1, -6)
~|) [-4, -1, 0, 2> (-1, +7)
.||) [-2, 1, -1, 1> (+0, +1) or )/||

|) + ~|) = .||)

~|( [1, 2, -3, 1> (+1, -6)
)||( [-3, -1, 2> (-1, +7)
.||) [-2, 1, -1, 1> (+0, +1) or )/||

~|( + )||( = .||)

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Negrisept
9&10, 9&19, 10&19, 10&29, 19&29
[<1, 2, 2, 3], <0, -4, 3, -2]>
TOP P = 1203.187309, G = 124.841963
TOP-RMS P = 1203.503169, G = 125.974673

22 B 25 C/| 28 D)!!(
28 D)!!( 02 D||\ 05 E 08 F/| 11 G)!!(
11 G)!!( 14 G||\ 17 A 20 B!!/ 23 C)!!(
23 C)!!( 26 C||\ 00 D 03 E!!/ 06 E)||(
06 E)||( 09 F||\ 12 G 15 A!!/ 18 A)||(
18 A)||( 21 B\! 24 C 27 D!!/ 01 D)||(
01 D)||( 04 E\! 07 F

/| [-4, 4, -1> (+2, -19)
)||( [-3, -1, 2> (-1, +10)
||\ [-7, 3, 1> (+1, -9)

/| + )||( = ||\

This notation could also be used for negripent, since it only uses 5-limit accidentals, but there are a couple of other possibilities for the 7-limit version.

/| [-4, 4, -1> (+2, -19)
.(|\ [2, -3, 0, 1> (-1, +10) or |\\
.||) [-2, 1, -1, 1> (+1, -9) or )/||

/| + .(|\ = .||)

'|) [-9, 6, 1, -1> (+2, -19)
.(|\ [2, -3, 0, 1> (-1, +10)
||\ [-7, 3, 1> (+1, -9)

'|) + .(|\ = ||\

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Here are some possibilites for using Sagittal notation for the
first few
> 7-limit temperaments in Paul's _Middle Path_ paper. I've tried to
find
> sets of accidentals that add up correctly. In some cases I found
more
> than one set. I also tried to stick with more familiar symbols if I
> could get them to work out. There may be other possibilities for
some of
> these temperaments.

Hi Herman,

I looked at what you have for only a few of these (the ones I'm most
familiar with):

> ...
> Pajara

Looks good.

> ...
> Meantone
> 12&19, 12&31, 12&43, 19&31, 19&50, 31&43, 31&50, 31&81, 50&81
> [<1, 2, 4, 7], <0, -1, -4, -10]>
> TOP P = 1201.698520, G = 504.134131
> TOP-RMS P = 1201.242156, G = 504.026298
>
> B(|\ E(|\ A(|\ D(|\ G(|\
> C(|\ F(|\ B!) E!) A!) D!) G!)
> C!) F!) A||\ D||\ G||\ C||\ F||\
> B E A D G C F
> B!!/ E!!/ A!!/ D!!/ G!!/ B|) E|)
> A|) D|) G|) C|) F|) B(!/ E(!/
> A(!/ D(!/ G(!/ C(!/ F(!/
>
> (|\ [-13, 5, 1, 1> (+8, -19)
> |) [6, -2, 0, -1> (-5, +12)
> ||\ [-7, 3, 1> (+3, -7)
>
> (|\ + |) = ||\
>
> Having |) represent something bigger than (|\ isn't very desirable,
but
> here are a couple of the alternatives.
>
> ~|) [-4, -1, 0, 2> (+8, -19)
> |) [6, -2, 0, -1> (-5, +12)
> .(|\ [2, -3, 0, 1> (+3, -7) or |\\
>
> ~|) + |) = .(|\

Now |) represents something bigger than ~|), giving the flag ~: a
negative value, which may be even worse.

> ~|) [-4, -1, 0, 2> (+8, -19)
> '(| [1, 0, 2, -2> (-5, +12)
> )||( [-3, -1, 2> (+3, -7)
>
> ~|) + '(| = )||(

I prefer the first version, on account of its harmonic simplicity.

> ...
> Negrisept
> ...
> This notation could also be used for negripent, since it only uses
> 5-limit accidentals, but there are a couple of other possibilities
for
> the 7-limit version.

Looks good. I think it would be desirable to use the same notation
for both.

--George

🔗Herman Miller <hmiller@IO.COM>

Herman Miller wrote:

> Semaphore
> 5&19
> [<1, 2, 4, 3], <0, -2, -8, -1]>
> TOP P = 1203.668841, G = 252.480358
> TOP-RMS P = 1203.852847, G = 253.446134
> > 02 E'!!) 06 F/||\ 10
> 10 A'!!) 14 B 18
> 18 D'!!) 03 E 07
> 07 G'!!) 11 A 15 B.||)
> 15 C'!!) 00 D 04 E.||)
> 04 F'!!) 08 G 12 A.||)
> 12 16 C 01 D.||)
> 01 05 F 09 G.||)
> 09 13 B\!!/ 17 C.||)
> > '|) [-9, 6, 1, -1> (+4, -19) or )|)
> .||) [-1, 2, -1, 1> (-1, +5) or )/||
> /||\ [-11, 7> (+3, -14)
> > '|) + .||) = /||\

I've been playing with this on the virtual keyboard to get a feel for how it works, since I'm less familiar with this one. A -19 accidental is unlikely to be needed; the TOP-RMS tuning is practically 19-ET (with sharp octaves), and has only a very small (-4, +19) of about 0.065 cents. Since 19-ET is so close to an optimal tuning, there are likely to be usable tunings on either side of it; if a +19 or -19 accidental is needed, it can be a special case for each tuning that uses it.

On the other hand, a -9 accidental is needed for representing some of the thirds, such as D.||) G'!!) in the above notation. It would be nice to notate that as some kind of D 1/2 sharp F 3/2 sharp. The two best 7-limit symbols for (+2, -9) are |||( 15/14 and |||) 243/224. There are also a couple of single-shaft options for (-1, +5): /|) 36/35 and .(|\ 28/27. Of the possible combinations, I have a slight preference for the first one of these:

)||( [-3, -1, 2> (+3, -14)
/|) [2, 2, -1, -1> (-1, +5)
|||( [-1, 1, 1, -1> (+2, -9)

)/|\ [-3, 4, 1, -2> (+3, -14)
.(|\ [2, -3, 0, 1> (-1, +5) or |\\
|||( [-1, 1, 1, -1> (+2, -9)

||\ [-7, 3, 1> (+3, -14)
/|) [2, 2, -1, -1> (-1, +5)
|||) [-5, 5, 0 -1> (+2, -9)

~~|| [-7, 8, 0, -2> (+3, -14)
.(|\ [2, -3, 0, 1> (-1, +5) or |\\
|||) [-5, 5, 0 -1> (+2, -9)

The second one is also nice since both of the two smaller accidentals are single-shaft symbols. But it's got a 3-flag symbol and an accented one. The third one has ||\ which is misleadingly large, and the ~~|| 6561/6272 in the last one is not a very familiar symbol.