7L 3s: Difference between revisions

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{{Infobox MOS
| Name = Neapolitan-dicotonic
| Periods = 1
| nLargeSteps = 7
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLLsLLsLLs
| Equave = 15/7}}
{{Infobox MOS
{{Infobox MOS
| Name = dicotonic
| Name = dicotonic
Line 18: Line 9:
}}
}}


'''7L 3s(<15/7>)''' refers to the structure of [[MOSScales|moment of symmetry scales]] built from a 10-tone chain of (neutral) thirds (assuming a period of an octave or a minor ninth):
'''7L 3s''' refers to the structure of [[MOSScales|moment of symmetry scales]] built from a 10-tone chain of neutral thirds (assuming a period of an octave):


L s L L L s L L s L
L s L L L s L L s L
Line 25: Line 16:


t q t t t q t t q t
t q t t t q t t q t
==Names==
== Names==
This MOS is called '''dicotonic''' (named after the abstract temperaments [[dicot]] and more specifically 11-limit [[Dicot_family#Dichotic|dichotic]]) in [[TAMNAMS]].
This MOS is called '''dicotonic''' (named after the abstract temperaments [[dicot]] and more specifically 11-limit [[Dicot_family#Dichotic|dichotic]]) in [[TAMNAMS]].


==Intervals==
==Intervals==
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo]] and 360 cents (3\10 - three degrees of [[10edo]]), hence a neutral third.
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo|7edo]] and 360 cents (3\10 - three degrees of [[10edo|10edo]]), hence a neutral third.


2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
Line 43: Line 34:
!# generators up
!# generators up
!Notation (1/1 = 0)
!Notation (1/1 = 0)
!name
! name
!In L's and s's
!In L's and s's
!# generators up
!# generators up
!Notation of 2/1 inverse
! Notation of 2/1 inverse
!name
! name
!In L's and s's
! In L's and s's
|-
|-
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
|-
|-
|0
| 0
|0
|0
|perfect unison
|perfect unison
Line 61: Line 52:
|7L+3s
|7L+3s
|-
|-
|1
| 1
|7
| 7
|perfect 7-step
|perfect 7-step
|5L+2s
|5L+2s
Line 82: Line 73:
|1
|1
|major (1-)step
|major (1-)step
|1L
| 1L
| -3
| -3
|9v
|9v
|minor 9-step
| minor 9-step
|6L+3s
|6L+3s
|-
|-
Line 91: Line 82:
|8
|8
|major 8-step
|major 8-step
|6L+2s
| 6L+2s
| -4
| -4
|2v
|2v
Line 101: Line 92:
|major 5-step
|major 5-step
|4L+1s
|4L+1s
| -5
| -5
|5v
|5v
|minor 5-step
| minor 5-step
|3L+2s
| 3L+2s
|-
|-
|6
|6
|2
|2
|major 2-step
| major 2-step
|2L
|2L
| -6
| -6
Line 118: Line 109:
|9
|9
|major 9-step
|major 9-step
|7L+2s
| 7L+2s
| -7
| -7
|1v
|1v
|minor (1-)step
| minor (1-)step
|1s
|1s
|-
|-
Line 127: Line 118:
|6^
|6^
|major 6-step
|major 6-step
|5L+1s
| 5L+1s
| -8
| -8
|4v
|4v
|minor 4-step
|minor 4-step
|2L+2s
| 2L+2s
|-
|-
|9
|9
|3^
|3^
|augmented 3-step
| augmented 3-step
|3L
|3L
| -9
| -9
|7v
|7v
|diminished 7-step
|diminished 7-step
| 4L+3s
|4L+3s
|-
|-
|10
|10
|0^
|0^
|augmented unison
| augmented unison
|1L-1s
|1L-1s
| -10
| -10
Line 153: Line 144:
|11
|11
|7^
|7^
|augmented 7-step
| augmented 7-step
|6L+1s
|6L+1s
| -11
| -11
|3v
|3v
|diminished 3-step
|diminished 3-step
| 1L+2s
|1L+2s
|-
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s]], [[10L 7s]], or [[17edo]]) also has the following intervals (from some root):
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s]], [[10L 7s]], or [[17edo]]) also has the following intervals (from some root):
Line 164: Line 155:
|12
|12
|4^
|4^
|augmented 4-step
| augmented 4-step
|4L
|4L
| -12
| -12
|6v
| 6v
|diminished 6-step
|diminished 6-step
|3L+3s
|3L+3s
Line 176: Line 167:
|2L-1s
|2L-1s
| -13
| -13
|9w
| 9w
|diminished 9-step
|diminished 9-step
|5L+4s
|5L+4s
Line 184: Line 175:
|augmented 8-step
|augmented 8-step
|8L+1s
|8L+1s
| -14
| -14
|2w
|2w
|diminished 2-step
|diminished 2-step
Line 191: Line 182:
|15
|15
|5^
|5^
| augmented 5-step
|augmented 5-step
|5L
| 5L
| -15
| -15
|5w
|5w
|diminished 5-step
| diminished 5-step
|2L+3s
|2L+3s
|-
|-
|16
|16
|2^
|2^
|augmented 2-step
| augmented 2-step
| 3L-1s
|3L-1s
| -16
| -16
|8w
|8w
Line 207: Line 198:
|4L+4s
|4L+4s
|}
|}
==Scale tree ==
== Scale tree ==
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:


{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="4" rowspan="2" |Generator
! colspan="6" rowspan="2" | Generator
! colspan="2" | Cents
! colspan="2" | Cents
! rowspan="2" |''ed17\16''
! rowspan="2" | L
! rowspan="2" |L
! rowspan="2" | s
! rowspan="2" |s
! rowspan="2" | L/s
! rowspan="2" |L/s
! rowspan="2" | Comments
! rowspan="2" |Comments
|-
|-
!edo
! Chroma-positive
!Neapolitan (normalized)
! Chroma-negative
|-
|-
|7\10|| || || ||840.000||933.333 (1171.242 śata)
| 7\10 || || || || || || 840.000 || 360.000 || 1 || 1 || 1.000 ||  
|''1190.000''||1||1||1.000||
|-
|-
| 40\57|| || || ||842.105||923.077 (1158.371 śata)
| || || || || || 40\57 || 842.105 || 357.895 || 6 || 5 || 1.200 || Restles↑
|''1192.9825''|| 6||5||1.200||(Neapolitan-)Restles↑
|-
|-
|
| || || || || 33\47 || || 842.553 || 357.447 || 5 || 4 || 1.250 ||  
|73\104
|
|
|842.308
|922.105 (1157.152 śata)
|''1193.269''
|11
|9
|1.222
|
|-
|-
|
| || || || || || 59\84 || 842.857 || 357.143 || 9 || 7 || 1.286 ||  
|
|106\151
|
|842.384
|921.739 (1156.692 śata)
|''1193.3775''
|16
|13
|1.231
|
|-
|-
| 33\47|| || || || 842.553||920.930 (1155.677 śata)
| || || || 26\37 || || || 843.243 || 356.757 || 4 || 3 || 1.333 ||  
|''1193.617''||5|| 4||1.250 ||  
|-
|-
|
| || || || || || 71\101 || 843.564 || 356.436 || 11 || 8 || 1.375 ||  
|92\131
|
|
|842.748
|920.000 (1154.510 śata)
|''1193.893''
|14
|11
|1.273
|
|-
|-
| 59\84|| || || ||842.857 ||919.4805 (1153.858 śata)
| || || || || 45\64 || || 843.750 || 356.250 || 7 || 5 || 1.400 || Beatles
|''1194.048''||9||7||1.286||
|-
|-
|
| || || || || || 64\91 || 843.956 || 356.044 || 10 || 7 || 1.428 ||  
|85\121
|
|
|842.975
|918.919 (1153.153 śata)
|''1194.215''
|13
|10
|1.300
|
|-
|-
|
| || || 19\27 || || || || 844.444 || 355.556 || 3 || 2 || 1.500 || L/s = 3/2, suhajira/ringo
|
|111\158
|
|843.038
|918.621 (1152.779 śata)
|''1194.304''
|17
|13
|1.308
|
|-
|-
| 26\37|| || || ||843.243||917.647 (1151.557 śata)
| || || || || || 69\98 || 844.698 || 355.102 || 11 || 7 || 1.571 ||  
|''1194.595''||4 || 3||1.333 ||
|-
|-
|
| || || || || 50\71 || || 845.070 || 354.930 || 8 || 5 || 1.600 ||  
|97\138
|
|
|843.478
|916.535 (1150.162 śata)
|''1194.9275''
|15
|11
|1.364
|
|-
|-
|71\101|| || || ||843.564||916.129 (1149.652 śata)
| || || || || || 81\115 || 845.217 || 354.783 || 13 || 8 || 1.625 || Golden suhajira
|''1195.0495''||11||8||1.375||
|-
|-
|45\64|| || || || 843.750||915.254 (1148.554 śata)
| || || || 31\44 || || || 845.455 || 354.545 || 5 || 3 || 1.667 ||  
|''1195.3125''||7|| 5||1.400||(Neapolitan-)Beatles
|-
|-
|64\91|| || || ||843.956||914.286 (1147.339 śata)
| || || || || || 74\105 || 845.714 || 354.286 || 12 || 7 || 1.714 ||  
|''1195.604''||10||7 || 1.428 ||  
|-
|-
|
| || || || || 43\61 || || 845.902 || 354.098 || 7 || 4 || 1.750 ||
|83\118
|
|
|844.068
|913.7615 (1146.681 śata)
|''1195.763''
|13
|9
|1.444
|
|-
|-
|
| || || || || || 55\78 || 846.154 || 353.846 || 9 || 5 || 1.800 ||
|
|102\145
|
|844.138
|913.433 (1146.269 śata)
|''1195.862''
|16
|11
|1.4545
|
|-
|
|
|
|121\172
|844.186
|913.2075 (1145.986 śata)
|''1195.930''
|19
|13
|1.4615
|
|-
|19\27|| || || || 844.444|| 912.000 (1144.471 śata)
|''1196.296''||3||2 || 1.500||L/s = 3/2, (Neapolitan-)suhajira/ringo
|-
|
|
|
|107\152
|844.736
|910.638 (1142.762 śata)
|''1196.7105''
|17
|11
|1.5455
|
|-
|
|
|88\125
|
|844.8
|910.345 (1142.3935 śata)
|''1196.800''
|14
|9
|1.556
|
|-
| || 69\98||  || ||844.898||909.890 (1141.823 śata)
|''1196.939''||11||7 || 1.571||
|-
|50\71|| || || ||845.070|| 909.091 (1140.820 śata)
|''1197.183''||8||5||1.600||
|-
|  ||81\115|| || ||845.217||908.411 (1139.967 śata)
|''1197.391''||13||8 || 1.625||Golden (Neapolitan-)suhajira
|-
| 31\44|| || || ||845.455||907.317 (1138.594 śata)
|''1197.727''||5|| 3||1.667 ||
|-
| ||74\105|| || ||845.714 || 906.122 (1137.095 śata)
|''1198.095''||12 || 7||1.714 ||
|-
|43\61|| || || ||845.902||905.263 (1136.0165 śata)
|''1198.361''||7||4||1.750||
|-
|
|
|98\139
|
|846.043
|904.615 (1135.204 śata)
|''1198.561''
|16
|9
|1.778
|
|-
| ||55\78||  ||  ||846.154||904.110 (1134.569 śata)
|''1198.718''||9||5||1.800 ||
|-
|
|
|67\95
|
|846.316
|903.371 (1133.642 śata)
|''1198.947''
|11
|6
|1.833
|
|-
|
|
|
|79\112
|846.429
|902.857 (1132.997 śata)
|''1199.107''
|13
|7
|1.857
|
|-
|-
|12\17|| || || || 847.059||900.000 (1129.412 śata)
| || 12\17 || || || || || 847.059 || 352.941 || 2 || 1 || 2.000 || Basic dicotonic<br>(Generators smaller than this are proper)
|''1200.000''||2||1||2.000 ||Basic (Neapolitan-)dicotonic<br>(Generators smaller than this are proper)
|-
|-
| || ||53\75||  ||848.000||352.000
| || || || || || 53\75 || 848.000 || 352.000 || 9 || 4 || 2.250 ||
| ||9||4||2.250||
|-
|-
| ||41\58|| || ||848.273||351.724
| || || || || 41\58 || || 848.273 || 351.724 || 7 || 3 || 2.333 ||  
| ||7||3||2.333||
|-
|-
| || ||70\99|| ||848.485||351.515
| || || || || || 70\99 || 848.485 || 351.515 || 12 || 5 || 2.400 || Hemif/hemififths
| ||12||5||2.400||(Neapolitan-)Hemif/hemififths
|-
|-
|29\41|| || || || 848.780 || 351.220
| || || || 29\41 || || || 848.780 || 351.220 || 5 || 2 || 2.500 || Mohaha/neutrominant
| ||5|| 2||2.500||(Neapolitan-)Mohaha/neutrominant
|-
|-
| || ||75\106|| ||849.057 || 350.943
| || || || || || 75\106 || 849.057 || 350.943 || 13 || 5 || 2.600 || Hemif/salsa/karadeniz
| || 13||5 || 2.600|| (Neapolitan-)Hemif/salsa/karadeniz
|-
|-
| ||46\65|| ||  ||849.231 ||350.769
| || || || || 46\65 || || 849.231 || 350.769 || 8 || 3 || 2.667 || Mohaha/mohamaq
| ||8||3 || 2.667 || (Neapolitan-)Mohaha/mohamaq
|-
|-
| || ||63\89|| ||849.438||350.562
| || || || || || 63\89 || 849.438 || 350.562 || 11 || 4 || 2.750 ||
| ||11||4||2.750||
|-
|-
|17\24|| || || ||850.000||350.000
| || || 17\24 || || || || 850.000 || 350.000 || 3 || 1 || 3.000 || L/s = 3/1
| ||3 || 1||3.000 ||L/s = 3/1
|-
|-
| || ||56\79|| ||850.633||349.367
| || || || || || 56\79 || 850.633 || 349.367 || 10 || 3 || 3.333 ||
| ||10||3 ||3.333||
|-
|-
| ||39\55|| || ||850.909||349.091
| || || || || 39\55 || || 850.909 || 349.091 || 7 || 2 || 3.500 ||  
| ||7||2 || 3.500||  
|-
|-
| || ||61\86|| || 851.613|| 358.837
| || || || || || 61\86 || 851.613 || 358.837 || 11 || 3 || 3.667 ||  
| ||11||3||3.667||
|-
|-
|22\31|| ||  || || 851.613||348.387
| || || || 22\31 || || || 851.613 || 348.387 || 4 || 1 || 4.000 || Mohaha/migration/mohajira
| ||4 || 1||4.000||(Neapolitan-)Mohaha/migration/mohajira
|-
|-
| || ||49\69|| ||852.174|| 347.826
| || || || || || 49\69 || 852.174 || 347.826 || 9 || 2 || 4.500 ||  
| ||9||2 ||4.500||
|-
|-
| ||27\38|| ||  ||852.632||347.368
| || || || || 27\38 || || 852.632|| 347.368 || 5 || 1 || 5.000 ||  
| ||5||1||5.000 ||
|-
|-
| || ||32\45|| ||853.333||346.667
| || || || || || 32\45 || 853.333 || 346.667 || 6 || 1 || 6.000 || Mohaha/ptolemy
| || 6||1 ||6.000||(Neapolitan-)Mohaha/ptolemy
|-
|-
|5\7|| || || ||857.143|| 342.867
| 5\7 || || || || || || 857.143 || 342.867 || 1 || 0 || → inf ||
| || 1||0 ||→ inf||
|}
|}


Line 496: Line 285:
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).


==Rank-2 temperaments==
== Rank-2 temperaments ==
==7-note subsets ==
==7-note subsets==
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s]]:
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L_4s|3L 4s]]:


L s s L s L s
L s s L s L s
Line 512: Line 301:
which is not a complete moment of symmetry scale in itself, but a subset of one.
which is not a complete moment of symmetry scale in itself, but a subset of one.


== Tetrachordal structure==
==Tetrachordal structure==
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.


Revision as of 09:04, 2 April 2022

↖ 6L 2s ↑ 7L 2s 8L 2s ↗
← 6L 3s 7L 3s 8L 3s →
↙ 6L 4s ↓ 7L 4s 8L 4s ↘ 
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\10 to 5\7 (840.0 ¢ to 857.1 ¢)
Dark 2\7 to 3\10 (342.9 ¢ to 360.0 ¢)
TAMNAMS information
Name dicoid
Prefix dico-
Abbrev. di
Related MOS scales
Parent 3L 4s
Sister 3L 7s
Daughters 10L 7s, 7L 10s
Neutralized 4L 6s
2-Flought 17L 3s, 7L 13s
Equal tunings
Equalized (L:s = 1:1) 7\10 (840.0 ¢)
Supersoft (L:s = 4:3) 26\37 (843.2 ¢)
Soft (L:s = 3:2) 19\27 (844.4 ¢)
Semisoft (L:s = 5:3) 31\44 (845.5 ¢)
Basic (L:s = 2:1) 12\17 (847.1 ¢)
Semihard (L:s = 5:2) 29\41 (848.8 ¢)
Hard (L:s = 3:1) 17\24 (850.0 ¢)
Superhard (L:s = 4:1) 22\31 (851.6 ¢)
Collapsed (L:s = 1:0) 5\7 (857.1 ¢)

7L 3s refers to the structure of moment of symmetry scales built from a 10-tone chain of neutral thirds (assuming a period of an octave):

L s L L L s L L s L

Graham Breed has a page on his website dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:

t q t t t q t t q t

Names

This MOS is called dicotonic (named after the abstract temperaments dicot and more specifically 11-limit dichotic) in TAMNAMS.

Intervals

The generator (g) will fall between 343 cents (2\7 - two degrees of 7edo and 360 cents (3\10 - three degrees of 10edo), hence a neutral third.

2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of diatonic fifths.

The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a sinaic.

The "small step" will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.

The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.

Note: In TAMNAMS, a k-step interval class in dicotonic may be called a "k-step", "k-mosstep", or "k-dicostep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

# generators up Notation (1/1 = 0) name In L's and s's # generators up Notation of 2/1 inverse name In L's and s's
The 10-note MOS has the following intervals (from some root):
0 0 perfect unison 0 0 0 perfect 10-step 7L+3s
1 7 perfect 7-step 5L+2s -1 3 perfect 3-step 2L+1s
2 4 major 4-step 3L+1s -2 6 minor 6-step 4L+2s
3 1 major (1-)step 1L -3 9v minor 9-step 6L+3s
4 8 major 8-step 6L+2s -4 2v minor 2-step 1L+1s
5 5 major 5-step 4L+1s -5 5v minor 5-step 3L+2s
6 2 major 2-step 2L -6 8v minor 8-step 5L+3s
7 9 major 9-step 7L+2s -7 1v minor (1-)step 1s
8 6^ major 6-step 5L+1s -8 4v minor 4-step 2L+2s
9 3^ augmented 3-step 3L -9 7v diminished 7-step 4L+3s
10 0^ augmented unison 1L-1s -10 0v diminished 10-step 6L+4s
11 7^ augmented 7-step 6L+1s -11 3v diminished 3-step 1L+2s
The chromatic 17-note MOS (either 7L 10s, 10L 7s, or 17edo) also has the following intervals (from some root):
12 4^ augmented 4-step 4L -12 6v diminished 6-step 3L+3s
13 1^ augmented (1-)step 2L-1s -13 9w diminished 9-step 5L+4s
14 8^ augmented 8-step 8L+1s -14 2w diminished 2-step 2s
15 5^ augmented 5-step 5L -15 5w diminished 5-step 2L+3s
16 2^ augmented 2-step 3L-1s -16 8w diminished 8-step 4L+4s

Scale tree

The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of 17edo:

Generator Cents L s L/s Comments
Chroma-positive Chroma-negative
7\10 840.000 360.000 1 1 1.000
40\57 842.105 357.895 6 5 1.200 Restles↑
33\47 842.553 357.447 5 4 1.250
59\84 842.857 357.143 9 7 1.286
26\37 843.243 356.757 4 3 1.333
71\101 843.564 356.436 11 8 1.375
45\64 843.750 356.250 7 5 1.400 Beatles
64\91 843.956 356.044 10 7 1.428
19\27 844.444 355.556 3 2 1.500 L/s = 3/2, suhajira/ringo
69\98 844.698 355.102 11 7 1.571
50\71 845.070 354.930 8 5 1.600
81\115 845.217 354.783 13 8 1.625 Golden suhajira
31\44 845.455 354.545 5 3 1.667
74\105 845.714 354.286 12 7 1.714
43\61 845.902 354.098 7 4 1.750
55\78 846.154 353.846 9 5 1.800
12\17 847.059 352.941 2 1 2.000 Basic dicotonic
(Generators smaller than this are proper)
53\75 848.000 352.000 9 4 2.250
41\58 848.273 351.724 7 3 2.333
70\99 848.485 351.515 12 5 2.400 Hemif/hemififths
29\41 848.780 351.220 5 2 2.500 Mohaha/neutrominant
75\106 849.057 350.943 13 5 2.600 Hemif/salsa/karadeniz
46\65 849.231 350.769 8 3 2.667 Mohaha/mohamaq
63\89 849.438 350.562 11 4 2.750
17\24 850.000 350.000 3 1 3.000 L/s = 3/1
56\79 850.633 349.367 10 3 3.333
39\55 850.909 349.091 7 2 3.500
61\86 851.613 358.837 11 3 3.667
22\31 851.613 348.387 4 1 4.000 Mohaha/migration/mohajira
49\69 852.174 347.826 9 2 4.500
27\38 852.632 347.368 5 1 5.000
32\45 853.333 346.667 6 1 6.000 Mohaha/ptolemy
5\7 857.143 342.867 1 0 → inf

The scale produced by stacks of 5\17 is the 17edo neutral scale. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a chromatic pair.

Other compatible edos include: 37edo, 27edo, 44edo, 41edo, 24edo, 31edo.

You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).

Rank-2 temperaments

7-note subsets

If you stop the chain at 7 tones, you have a heptatonic scale of the form 3L 4s:

L s s L s L s

The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for "tone," uppercase because it is a wider tone than t. Thus, we have:

T t t T t T t

This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:

T t t T t t T

which is not a complete moment of symmetry scale in itself, but a subset of one.

Tetrachordal structure

Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a tetrachordal scale. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.

I ( - Andrew Heathwaite) offer "a" to refer to a step of 2t (for "augmented second")

Thus, the possible tetrachords are:

T t t

t T t

t t T

a q t

a t q

t a q

t q a

q a t

q t a

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