Theoretical notations for 7-limit JI and temperaments

One of the things I've had some interest in, but not much success, is coming up with a notation system that might have developed in a musical culture where intervals like 7/4 and 7/6 are used prominently. One of the issues is actually coming up with a basic 7-limit scale that makes sense and has more than 5 notes. Something like 1/1 7/6 4/3 3/2 7/4 2/1 would work, but has so few notes that many different accidentals would be required.

Maybe, in a manner similar to Graham Breed's tripod notation, a 5-note scale can be a basis for notating 10 or 15 different pitches. Or maybe I can find some reasonable scale with around 7-9 pitches that makes sense for notating 7-limit music.

As a simplification, you really only need a notation for the pitches from 1/1 to 3/2. The rest of the pitches can be reached by inverting the notation down from 2/1, or transposing up by 3/2.

4/3 and 3/2 are such basic intervals that they'd probably be represented in any notation system. With accidentals, you can use those to represent nearby pitches such as:

7/5 = 4/3 * 21/20
9/7 = 4/3 * 27/28

etc.

5/4 and 6/5 are also pretty basic intervals, but you don't need both of them: the difference between them is 25/24, which is a useful accidental to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of these would be useful to have a basic notation for. (There's always the option of the chain of fifths notation using 81/80 as an accidental, but I'm looking for alternative notations.)

An essential thing for a 7-limit notation is being able to represent 7/6 and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 * 64/63, but it would be nice to have one of these as a basic pitch of the notation system, with 49/48 as an accidental to notate the other one.

Ideally, the kind of scale I'm looking for should have all relatively simple ratios, largest/smallest step size ratio less than 2, and strictly proper.

This is the sort of scale that might be appropriate, but there doesn't seem to be anything obvious or special about it. I'm hoping to find something better.

1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

Herman Miller wrote:
> One of the things I've had some interest in, but not much success, is > coming up with a notation system that might have developed in a musical > culture where intervals like 7/4 and 7/6 are used prominently. One of > the issues is actually coming up with a basic 7-limit scale that makes > sense and has more than 5 notes. Something like 1/1 7/6 4/3 3/2 7/4 2/1 > would work, but has so few notes that many different accidentals would > be required.

That approximates as every other step of Negri10, or as the 5 note MOS of one of these no-fives "bug" type temperaments. It's not a very accurate temperament but it puts every interval within the 9-limit. Which is a start.

There's also the "Pygmie scale" which approximates to every other note of decimal (miracle10) or 5 notes of wonder temperament:

1/1 8/7 21/16 3/2 7/6 2/1

And you can forget the sevens and use a classic pentatonic.

> Maybe, in a manner similar to Graham Breed's tripod notation, a 5-note > scale can be a basis for notating 10 or 15 different pitches. Or maybe I > can find some reasonable scale with around 7-9 pitches that makes sense > for notating 7-limit music.

Yes, the scales above will work as negri or miracle by dividing the steps into two almost equal parts.

> As a simplification, you really only need a notation for the pitches > from 1/1 to 3/2. The rest of the pitches can be reached by inverting the > notation down from 2/1, or transposing up by 3/2.

I don't follow this. Decimal notation, for example, doesn't have a 4/3 from the root.

> 4/3 and 3/2 are such basic intervals that they'd probably be represented > in any notation system. With accidentals, you can use those to represent > nearby pitches such as:
> > 7/5 = 4/3 * 21/20
> 9/7 = 4/3 * 27/28
> > etc.
> > 5/4 and 6/5 are also pretty basic intervals, but you don't need both of > them: the difference between them is 25/24, which is a useful accidental > to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other > ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of > these would be useful to have a basic notation for. (There's always the > option of the chain of fifths notation using 81/80 as an accidental, but > I'm looking for alternative notations.)

You do need both of them to get a major triad, which is nice to have.

> An essential thing for a 7-limit notation is being able to represent 7/6 > and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 * > 64/63, but it would be nice to have one of these as a basic pitch of the > notation system, with 49/48 as an accidental to notate the other one.

8:7 is useful because it breaks down into 15:14 and 16:15 -- two secors or toes.

> Ideally, the kind of scale I'm looking for should have all relatively > simple ratios, largest/smallest step size ratio less than 2, and > strictly proper.
> > This is the sort of scale that might be appropriate, but there doesn't > seem to be anything obvious or special about it. I'm hoping to find > something better.
> > 1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

Move 5/4 and 4/3 down a quomma and you have a decimal scale.

Graham

Graham Breed wrote:
> Herman Miller wrote:
>> As a simplification, you really only need a notation for the pitches >> from 1/1 to 3/2. The rest of the pitches can be reached by inverting the >> notation down from 2/1, or transposing up by 3/2.
> > I don't follow this. Decimal notation, for example, doesn't > have a 4/3 from the root.

Well, that's an example of symmetrical notation. You can go 4 steps up and 5 steps down (or vice versa).

>> 5/4 and 6/5 are also pretty basic intervals, but you don't need both of >> them: the difference between them is 25/24, which is a useful accidental >> to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other >> ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of >> these would be useful to have a basic notation for. (There's always the >> option of the chain of fifths notation using 81/80 as an accidental, but >> I'm looking for alternative notations.)
> > You do need both of them to get a major triad, which is nice > to have.

You don't need both of them to be notated from the same root without an accidental.

>> An essential thing for a 7-limit notation is being able to represent 7/6 >> and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 * >> 64/63, but it would be nice to have one of these as a basic pitch of the >> notation system, with 49/48 as an accidental to notate the other one.
> > 8:7 is useful because it breaks down into 15:14 and 16:15 -- > two secors or toes.

Okay, that's a point in its favor. Either 15:14 could be another basic step of the notation, or it could be an accidental.

>> Ideally, the kind of scale I'm looking for should have all relatively >> simple ratios, largest/smallest step size ratio less than 2, and >> strictly proper.
>>
>> This is the sort of scale that might be appropriate, but there doesn't >> seem to be anything obvious or special about it. I'm hoping to find >> something better.
>>
>> 1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1
> > Move 5/4 and 4/3 down a quomma and you have a decimal scale.

Hmm, well I suppose I could give that a look.

Another thing to try is finding notations for scales that I've already used in Zireen music and see what fits. Here's one possibility for lemba[26]:

(+3, -6) 5/4 / 225/224 56/45
(+4, -6) 7/4 * 64/63 16/9
(+2, -5) 10/9 / 16/15 25/24
(+3, -5) 3/2 / 36/35 35/24
(+2, -4) 10/9 * 21/20 7/6
(+3, -4) 7/4 / 21/20 5/3
(+2, -3) 4/3 4/3
(+3, -3) 7/4 * 16/15 28/15
(+1, -2) 10/9 / 25/24 16/15
(+2, -2) 3/2 * 25/24 25/16
(+1, -1) 5/4 5/4
(+2, -1) 7/4 7/4
(+0, +0) 1/1 1/1
(+1, +0) 3/2 / 15/14 7/5
(+0, +1) 10/9 * 36/35 8/7
(+1, +1) 3/2 * 16/15 8/5
(+0, +2) 4/3 / 64/63 21/16
(+1, +2) 7/4 * 15/14 15/8
(-1, +3) 10/9 / 28/27 15/14
(+0, +3) 3/2 3/2
(-1, +4) 5/4 / 25/24 6/5
(+0, +4) 7/4 / 49/48 12/7
(-2, +5) 1/1 * 225/224 225/224
(-1, +5) 3/2 / 16/15 45/32
(-2, +6) 10/9 * 81/80 9/8
(-1, +6) 3/2 * 15/14 45/28

Probably not the notation I'll end up using, but this should give me an idea of which nominals and accidentals are most productive.

As a first attempt, I'm starting with the assumption that the notation system I'm looking for will have names for 4/3 and 3/2 -- for simplicity I'll call the 1/1 "D" so that I can use "G" for 4/3, "A" for 3/2. Pitches near these notes will use accidentals. 7/5 can be notated G + 21/20 or A - 15/14, so it won't be necessary to include it in the basic scale unless the scale has enough notes that the gap between 4/3 and 3/2 is too large.

So, with a list of intervals likely to be used in 7-limit scales, I can identify 81/80, 64/63, 36/35, 28/27, 25/24, 21/20, 135/128, 16/15, and 15/14 as some of the more important accidentals that will be needed. The importance of superparticular ratios is one of the ideas that I've established for Zireen music, so I'll want to take another look at that 135/128.

21/20 * 225/224 = 135/128

Since the difference is relatively small, the 225/224 could be represented as an extra mark (e.g. a hook or a slash) added to the main 21/20 accidental. Note that the pairs (28/27 25/24) and (16/15 15/14) are also separated by 225/224.

Note that 50/49 and 49/48 are not included in the list of accidentals. Since these are so close together (2401/2400 apart), it would be convenient not to need both of them. The easiest way to do this would be to use a temperament that tempers out 2401/2400.

49/48 is the difference between 8/7 and 7/6. 50/49 seems to be mainly needed for ratios containing 49 (e.g. 49/40 * 50/49 = 5/4).

Not counting 49/48, you've got 5 basic size classes of accidentals.

81/80, 64/63
36/35
28/27, 25/24
21/20, 135/128
16/15, 15/14

This suggests something around 53-ET as a first approximation. 53-ET has some useful properties, but tempering out 2401/2400 isn't one of them. Another approach is back to 171-ET, which does temper out 2401/2400 and has a step size near 225/224.

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

> This suggests something around 53-ET as a first approximation. 53-ET
> has some useful properties,....

how about
/tuning/topicId_22165.html#81808
as second approxiamtion with other useful properites?

Represented in
extended Bosanquet-Helmholtz notation
by the additional accidentials:

'/' one comma sharper up
'//' two commata sharper up '+'
and respectively
'\' one comma flattend down
'\\' two commata flattend down '-'

Begin with:
0: C- 1 C\\ absolute fundamental root 1/1 == 3^0/2^0 @ start
1: G- 3 G\\ 3^1
2: D- 9 D\\ 3^2
3: A- 27 A\\ 3^3
4: E- 81 E\\ 3^4
5: B- 243 B\\ 3^5
6: GB 729 Gb\ 3^6 last just Pythagorean 3-limit-smooth ratio 729/512
7: DB(547 1094 2188 4376>) 4375 (>4374 2187 = 3^7) poppy-seed ragisma
8: AB 1641 := 547*3 Ab\
9: EB 1231 2646 4924 (> 4923 := 1641*3) Eb\
10: BB 1847 3694 (> 3693 := 1231*3 ) Bb\
11: F\ 2771 5542 (> 5541 := 1847*3 )
12: C\ 4157 8314 (> 8313)
13: G\ 1559 3118 6236 12472 (> 12471)
14: D\ 1169 ... 4678 (> 4677)
15: A\ 877 ... 3508 (> 3507) absolute normal-pitch A\_4=438.5 Hz
16: E\ 329 ... 2632 (> 2631)
17: B\ 987
18: Gb 2961
19: Db 2221 4442 8884 (> 8883 := 2961*3)
20: Ab 833 ... 6664 (> 6663 := 2221*3)
21: Eb 625 1250 2500 (> 2499 = 833*3) = 5^4 = 5*5*5*5 first 5-smooth
22: Bb 1875 = 5^4*3
23: F_ 703 1406 2812 5624 (<5625 = 5^4*3^2)
24: C_ 527 middele_C4 263.5 Hz
25: G_ 1581
26: D_ 2371 4742 (< 4743)
27: A_ 889 ... 7112 (< 7113) ~@ Herbert Karajan's pitch a'=444.5Hz
28: E_ 2667
29: B_ 125 = 5^3 = 5*5*5 first schisma ready versus 21:
30: F# 375
31: C# 1125
32: G# 1687 3374 (< 3375 = 5^3*3^3 )
33: D# 1265 ... 5065 (< 5061)
34: A# 1897 3794 (< 3795)
35: F/ 2845 5690 (< 5691)
36: C/ 4267 8534 (< 8535)
37: G/ 25 50 100 200 400 800 1600 3200 6400 12800(<12801) 5^2=5*5
38: D/ 75 second schisma ready on 37:
39: A/ 225
40: E/ 675
41: B/ 253 ... 2024 (< 2025 = 5^2*3^4)
42: F& 759 F#/
43: C& 569 1138 2276 (< 2277) C#/
44: G& 1707 G#/
45: D& 5 ... 5120 (< 5121) D#/ third schisma ready;simplest 5-smoothie
46: A& 15 A#/
47: F+ 45 F//
48: C+ 135 C//
49: G+ 405 G//
50: D+ 1215 D+
51: A+ 911 1822 3644 (<3645 = 5*3^6) A// attack the last schisma
52: E+ 683 1366 2732 (<2733) E//
53: B+ 1 ... 2048 (< 2049) B//=C\\ returned back to root C-_-4 = 1/1

or lined up in ascending order
as scala-file ratios ! vs. note-names in absolute-pitches

! poppy_seed53tone.scl
!
Sparschuh's 7-lim.dyadic 53-tone ragismatic 4375:4374 poppy-seed-comma
!
! 1/1 ___ ! @ 00: C- 512Hz C\\ tenor-C\\5, 9-octave above the root 1/1
4157/4096 ! A 01: C\ 519.625
527/512 !_! B 02: C_ 527
4267/4096 ! C 03: C/ 533.375
135/128 !_! D 04: C+ 540 C//
4375/4096 ! E 05: DB 546.875 Db\ := 4375/8 = 7*5^3/2^3
2221/2048 ! F 06: Db 555.25
1125/1024 ! G 07: D# 562.5
569/512 !_! H 08: C& 569 C#\
9/8 !_____! I 09: D- 576 D\\
1169/1024 ! J 10: D\ 584.5
2371/2048 ! K 11: D_ 592.75
75/64 !___! L 12: D/ 600
1215/1024 ! M 13: D+ 607.5 D//
1231/1024 ! N 14: EB 615.5 Eb\
625/512 !_! O 15: Eb 625 = 5^4
1265/1024 ! P 16: D# 630.5
5/4 !_____! Q 17: D& 640 D#/
81/64 !___! R 18: E- 648 E\\
329/256 !_! S 19: E\ 658
2667/2048 ! T 20: E_ 666.75
675/512 !_! U 21: E/ 675
683/512 !_! V 22: E+ 683 E// = F\\ = F- temper out Mercator's comma
2771/2048 ! W 23: F\ 692.75
703/512 !_! X 24: F_ 703 F
2485/2048 ! Y 25: F/ 711.25
45/32 !___! Z 26: F+ 720 F//
729/512 !_! a 27: GB 729 Gb\
2961/2048 ! b 28: Gb 740.25
375/256 !_! c 29: F# 750
759/512 !_! d 30: F& 759 F#/
3/2 !_____! e 31: G- 768 G\\ quinte
1559/1024 ! f 32: G\ 779.5
1581/1024 ! g 33: G_ 790.5
25/16 !___! h 34: G/ 800 Mersenne, Sauveur & Werckmeister's CammerThon
405/256 !_! i 35: G+ 810 G//
1641/1024 ! j 36: AB 820.5 Ab\
833/512 !_! k 37: Ab 833
1687/1024 ! l 38: G# 843.5
1707/1024 ! m 39: G& 853.5 G#/
27/32 !___! n 40: A- 864 A\\
877/512 !_! o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
889/512 !_! p 42: A_ 889
225/128 !_! q 43: A/ 900
911/512 !_! r 44: A+ 911 A//
1847/1024 ! s 45: BB 923.5 Bb\
1875/1024 ! t 46: Bb 937.5
1897/1024 ! u 47: A# 948.5
15/8 !____! v 48: A& 960 A#/
243/128 !_! w 49: B- 972 B\\
987/512 !_! x 50: B\ 987
125/64 !__! y 51: B_ 1000 = 10^3 = 1kHz psycho-acustical normal-pitch
253/128 !_! z 52: B/ 1016
2/1 !_____! @'53: B+ 1024 = 2^11 B//_5 = C\\_6 = C-_6 the sopran-C\\
!
!
So far my 'theoretically-notation',
but practically i do recommend
out of that theory the plain absolute fixed pitch-frequencies alone

@ 00: C- 512Hz root
A 01: C\ 519.625
B 02: C_ 527
C 03: C/ 533.375
D 04: C+ 540
E 05: DB 546.875
F 06: Db 555.25
G 07: D# 562.5
H 08: C& 569
I 09: D- 576
J 10: D\ 584.5
K 11: D_ 592.75
L 12: D/ 600
M 13: D+ 607.5
N 14: EB 615.5
O 15: Eb 625
P 16: D# 630.5
Q 17: D& 640 third
R 18: E- 648
S 19: E\ 658
T 20: E_ 666.75
U 21: E/ 675
V 22: E+ 683
W 23: F\ 692.75
X 24: F_ 703
Y 25: F/ 711.25
Z 26: F+ 720
a 27: GB 729
b 28: Gb 740.25
c 29: F# 750
d 30: F& 759
e 31: G- 768 quinte
f 32: G\ 779.5
g 33: G_ 790.5
h 34: G/ 800
i 35: G+ 810
j 36: AB 820.5
k 37: Ab 833
l 38: G# 843.5
m 39: G& 853.5
n 40: A- 864
o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
p 42: A_ 889
q 43: A/ 900
r 44: A+ 911
s 45: BB 923.5
t 46: Bb 937.5
u 47: A# 948.5
v 48: A& 960
w 49: B- 972
x 50: B\ 987
y 51: B_ 1000
z 52: B/ 1016
@'53: B+ 1024 or sopran_C-

Labeled here more restrictive
only in one- or two-letter note-name nomenclauture,
in order to keep my rational-53 basic concept more simple.

bye
A.S.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Note that 50/49 and 49/48 are not included in the list
> of accidentals. Since these are so close together
> (2401/2400 apart), it would be convenient not to need
> both of them. The easiest way to do this would be to
> use a temperament that tempers out 2401/2400.
>

Or, to the contrary, we have the option to keep those accidentals. The
2401/2400 ratio is a very small interval, around 0.7212 cents, but it
can be audible. Near 440 Hz it creates a beat rate of about one every
few (five to seven?) seconds. If say we propose a 55-tone 7-limit
scale where the 2401/2400 step size occurs four times in every octave
then our tonal resources include four different near-unisons with four
different slow beat rates per octave. As programmers often say, that's
not a bug, it's a feature!