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These scales are strongly associated with [[mavila]] system, which can be divided into two systems:
These scales are strongly associated with [[mavila]] system, which can be divided into two systems:
* the [[Armodue|Armodue]] project/system and its associated [[armodue]] temperament.
* the [[Armodue|Armodue]] project/system and its associated [[armodue]] temperament, with fifths sharper than 5\9 (666.7¢) and flatter than 9\16 (675¢).
* Hornbostel temperament.
* Hornbostel temperament, with fifths sharper than 9\16 (675¢) and flatter than 4\7 (685.71¢).


Some high JI approximations of the generator: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119, 250/169. These could be used to guide the construction of neji versions of superdiatonic scales or edos.
Some high JI approximations of the generator: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119, 250/169. These could be used to guide the construction of neji versions of superdiatonic scales or edos.
 
== Scale tree ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! colspan="3" | Generator
! colspan="5" | Generator
! | <span style="display: block; text-align: center;">'''Generator size (cents)'''</span>
! | Generator size (cents)
! | Pentachord steps
! | L/s
! | Comments
! | Comments
|-
|-
Line 29: Line 29:
| |  
| |  
| |  
| |  
| |
| |
| | 685.714
| | 685.714
| | 1 1 1 0
| | 1/0
| |
|-
| |  
| |  
| |
| | 102\[[179edo|179]]
| | 683.798
| | 25 25 25 2
| | Approximately 0.03 cents away from [[95/64]]
|-
| | 33\[[58edo|58]]
| |
| |
| | 682.758
| | 8 8 8 1
| | 2 generators equal 11/10, 6 equal 4/3, creating a hybrid Mavila/Porcupine scale with three perfect 5ths as well as the flat ones.
|-
|-
| |
| |
| |
| | 21\37
| | 21\37
| |
| |
| |  
| | 681.08
| | 681.081
| | 5/1
| | 5 5 5 1
| |
| |  
|-
|-
| |
| |
| |
| | 17\30
| | 17\30
| |
| |
| |  
| | 680
| | 680
| | 4 4 4 1
| | 4/1
| | L/s = 4
| |
|-
| |
| | 30\53
| |
| | 679.245
| | 7 7 7 2
| |
|-
| |
| | 43\76
| |
| | 678.947
| | 10 10 10 3
| |
|-
| |
| | 56\99
| |
| | 678.788
| | 13 13 13 4
| |
|-
| |
| | 69\122
| |
| | 678.6885
| | 16 16 16 5
| |
|-
| |
| | 82\145
| |
| | 678.621
| | 19 19 19 6
| |
|-
| |
| | 95\168
| |
| | 678.571
| | 22 22 22 7
| |
|-
| |
| | 108\191
| |
| | 678.534
| | 25 25 25 8
| |
|-
| |
| | 121\214
| |
| | 678.505
| | 28 28 28 9
| | 28;9 Superdiatonic 1/28-tone <span style="font-size: 12.8000001907349px;">(a slight exceeded representation of the ratio 262144/177147, the Pythagorean wolf Fifth)</span>
|-
| |
| | 134\237
| |  
| | 678.481
| | 31 31 31 10
| | HORNBOSTEL TEMPERAMENT <span style="font-size: 12.8000001907349px;">(1/31-tone; Optimum high size of Hornbostel '6th')</span>
|-
|-
| |
| |
| | 13\23
| | 13\23
| |  
| |
| |  
| |
| | 678.261
| | 678.261
| | 3 3 3 1
| | 3/1
| | HORNBOSTEL TEMPERAMENT <span style="font-size: 12.8000001907349px;">(Armodue 1/3-tone)</span>
| |
|-
|-
| |
| |  
| |  
| | 126\223
| |
| |
| | 678.027
| | 29 29 29 10
| | HORNBOSTEL TEMPERAMENT
 
<span style="font-size: 12.8000001907349px;">(Armodue 1/29-tone)</span>
|-
| |
| | 113\200
| |
| | 678
| | 26 26 26 9
| | HORNBOSTEL (&amp; [[Alexei_Stepanovich_Ogolevets|OGOLEVETS]]) TEMPERAMENT <span style="font-size: 12.8000001907349px;">(Armodue 1/26-tone; Best equillibrium between 6/5, Phi (833.1 Cent) and Square root of Pi (990.9 Cent), the notes '3', '7' &amp; '8')</span>
|-
| |
| | 100\177
| |
| | 677.966
| | 23 23 23 8
| |
|-
| |
| | 87\154
| |
| | 677.922
| | 20 20 20 7
| |
|-
| |
| | 74\131
| |
| | 677.863
| | 17 17 17 6
| | Armodue-Hornbostel 1/17-tone <span style="font-size: 12.8000001907349px;">(the Golden Tone System of Thorvald Kornerup and a temperament of the Alexei Ogolevets's list of temperaments)</span>
|-
| |
| | 61\108
| |
| | 677.778
| | 14 14 14 5
| | Armodue-Hornbostel 1/14-tone
|-
| |
| | 109\193
| |
| | 677.720
| | 25 25 25 9
| | Armodue-Hornbostel 1/25-tone
|-
| |
| | 48\85
| |
| | 677.647
| | 11 11 11 4
| | Armodue-Hornbostel 1/11-tone <span style="font-size: 12.8000001907349px;">(Optimum accuracy of Phi interval, the note '7')</span>
|-
| |
| | 35\62
| |
| | 677.419
| | 8 8 8 3
| | Armodue-Hornbostel 1/8-tone
|-
| |
| | 92\163
| |
| | 677.301
| | 21 21 21 8
| | 21;8 Superdiatonic 1/21-tone
|-
| |
| |
| |
| | 677.28
| | <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ+1 φ+1 φ+1 1</span>
| | Split φ superdiatonic relation (34;13 - 55;21 - 89;34 - 144;55 - 233;89 - 377;144 - 610;233..)
|-
| |
| | 57\101
| |
| | 677.228
| | 13 13 13 5
| | 13;5 Superdiatonic 1/13-tone
|-
| | 22\39
| | 22\39
| |
| |
| |  
| | 676.923
| | 676.923
| | 5 5 5 2
| | 5/2
| | Armodue-Hornbostel 1/5-tone <span style="font-size: 12.8000001907349px;">(Optimum low size of Hornbostel '6th')</span>
|-
| |
| | 75\133
| |
| | 676.692
| | 17 17 17 7
| | 17;7 Superdiatonic 1/17-tone <span style="font-size: 12.8000001907349px;">(Note the very accuracy of the step 75 with the ratio 34/23 with an error of +0.011 Cents)</span>
|-
| |
| | 53\94
| |
| | 676.596
| | 12 12 12 5
| |
|-
| |
| | 31\55
| |
| | 676.364
| | 7 7 7 3
| | 7;3 Superdiatonic 1/7-tone
|-
| |
| | 40\71
| |
| | 676.056
| | 9 9 9 4
| | 9;4 Superdiatonic 1/9-tone
|-
| |
| | 49\87
| |
| | 675.862
| | 11 11 11 5
| | 11;5 Superdiatonic 1/11-tone
|-
| |
| | 58\103
| |
| | 675.728
| | 13 13 13 6
| | 13;6 Superdiatonic 1/13-tone
|-
|-
| |
| | 9\16
| | 9\16
| |  
| |  
| |  
| |  
| |
| | 675
| | 675
| | 2 2 2 1
| | 2/1
| | <span style="display: block; text-align: left;">'''[BOUNDARY OF PROPRIETY: smaller generators are strictly proper]'''</span>ARMODUE ESADECAFONIA (or Goldsmith Temperament)
| | Boundary of propriety; smaller generators are strictly proper
|-
| |
| | 59\105
| |
| | 674.286
| | 13 13 13 7
| | Armodue-Mavila 1/13-tone
|-
| |
| | 50\89
| |
| | 674.157
| | 11 11 11 6
| | Armodue-Mavila 1/11-tone
|-
| |
| | 41\73
| |
| | 673.973
| | 9 9 9 5
| | Armodue-Mavila 1/9-tone <span style="font-size: 12.8000001907349px;">(with an approximation of the Perfect Fifth + 1/5 Pyth.Comma [706.65 Cents]: 43\73 is 706.85 Cents)</span>
|-
| |
| | 32\57
| |
| | 673.684
| | 7 7 7 4
| | Armodue-Mavila 1/7-tone <span style="font-size: 12.8000001907349px;">(the 'Commatic' version of Armodue, because its high accuracy of the [[7/4|7/4]] interval, the note '8')</span>
|-
| |
| | 55\98
| |
| | 673.469
| | 12 12 12 7
| |
|-
| |
| | 78\139
| |
| | 673.381
| | 17 17 17 10
| | Armodue-Mavila 1/17-tone
|-
| |
| | 101\180
| |
| | 673.333
| | 22 22 22 13
| |
|-
|-
| |
| |
| | 23\41
| | 23\41
| |  
| |  
| |  
| |  
| | 673.171
| | 673.171
| | 5 5 5 3
| | 5/3
| | 5;3 Golden Armodue-Mavila 1/5-tone
|-
| |  
| |  
| | 60\107
| |
| | 672.897
| | 13 13 13 8
| | 13;8 Golden Mavila 1/13-tone
|-
|-
| |  
| |  
| |  
| |  
| |  
| |  
| |
| |
| | 672.85
| | 672.85
| | <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ φ φ 1</span>
| | φ/1
| | GOLDEN Mavila (L/s = <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ)</span>
| | Golden mavila
|-
| |
| |
| | 97\173
| | 672.832
| | 21 21 21 13
| | 21;13 Golden Mavila 1/21-tone <span style="font-size: 12.8000001907349px;">(Phi is the step 120\173)</span>
|-
| |
| | 37\66
| |
| | 672.727
| | 8 8 8 5
| | 8;5 Golden Mavila 1/8-tone
|-
| |
| | 51\91
| |
| | 672.527
| | 11 11 11 7
| | 11;7 Superdiatonic 1/11-tone
|-
| |
| |
| | 116\207
| | 672.464
| | 25 25 25 16
| | 25;16 Superdiatonic 1/25-tone
|-
| |
| | 65\116
| |
| | 672.414
| | 14 14 14 9
| | 14;9 Superdiatonic 1/14-tone
|-
| |
| | 79\141
| |
| | 672.340
| | 17 17 17 11
| | 17;11 Superdiatonic 1/17-tone
|-
| |
| | 93\166
| |
| | 672.289
| | 20 20 20 13
| |
|-
| |
| | 107\191
| |
| | 672.251
| | 23 23 23 15
| |
|-
| |
| | 121\216
| |
| | 672.222
| | 26 26 26 17
| | 26;17 Superdiatonic 1/26-tone
|-
| |
| | 135\241
| |
| | 672.199
| | 29 29 29 19
| | 29;19 Superdiatonic 1/29-tone
|-
|-
| |
| |
| | 14\25
| | 14\25
| |  
| |  
| |  
| |
| | 672
| | 672
| | 3 3 3 2
| | 3/2
| | 3;2 Golden Armodue-Mavila 1/3-tone
|-
| |
| | 145\259
| |
| | 671.815
| | 31 31 31 21
| | 31;21 Superdiatonic 1/31-tone
|-
| |
| | 131\234
| |
| | 671.795
| | 28 28 28 19
| | 28;19 Superdiatonic 1/28-tone
|-
| |
| | 117\209
| |
| | 671.770
| | 25 25 25 17
| |
|-
| |
| | 103\184
| |
| | 671.739
| | 22 22 22 15
| |
|-
| |
| | 89\159
| |
| | 671.698
| | 19 19 19 13
| |
|-
| |
| | 75\134
| |
| | 671.642
| | 16 16 16 11
| |
|-
| |
| | 61\109
| |
| | 671.560
| | 13 13 13 9
| |
|-
| | 47\84
| |
| |
| | 671.429
| | 10 10 10 7
| |
|-
| | 33\59
| |
| |
| | 671.186
| | 7 7 7 5
| |  
| |  
|-
|-
| |
| |
| |
| | 19\34
| | 19\34
| |
| |  
| |  
| | 670.588
| | 670.588
| | 4 4 4 3
| | 4/3
| |  
| |  
|-
|-
| |
| |
| |
| |
| | 24\43
| | 24\43
| |
| |
| | 669.767
| | 669.767
| | 5 5 5 4
| | 5/4
| |  
| |  
|-
|-
Line 500: Line 127:
| |  
| |  
| |  
| |  
| |
| |
| | 666.667
| | 666.667
| | 1 1 1 1
| | 1/1
| |  
| |  
|}
|}

Revision as of 22:04, 26 March 2021

↖ 6L 1s ↑ 7L 1s 8L 1s ↗
← 6L 2s 7L 2s 8L 2s →
↙ 6L 3s ↓ 7L 3s 8L 3s ↘ 
┌╥╥╥╥┬╥╥╥┬┐
│║║║║│║║║││
│││││││││││
└┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLLsLLLs
sLLLsLLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\9 to 4\7 (666.7 ¢ to 685.7 ¢)
Dark 3\7 to 4\9 (514.3 ¢ to 533.3 ¢)
TAMNAMS information
Name armotonic
Prefix arm-
Abbrev. arm
Related MOS scales
Parent 2L 5s
Sister 2L 7s
Daughters 9L 7s, 7L 9s
Neutralized 5L 4s
2-Flought 16L 2s, 7L 11s
Equal tunings
Equalized (L:s = 1:1) 5\9 (666.7 ¢)
Supersoft (L:s = 4:3) 19\34 (670.6 ¢)
Soft (L:s = 3:2) 14\25 (672.0 ¢)
Semisoft (L:s = 5:3) 23\41 (673.2 ¢)
Basic (L:s = 2:1) 9\16 (675.0 ¢)
Semihard (L:s = 5:2) 22\39 (676.9 ¢)
Hard (L:s = 3:1) 13\23 (678.3 ¢)
Superhard (L:s = 4:1) 17\30 (680.0 ¢)
Collapsed (L:s = 1:0) 4\7 (685.7 ¢)

7L 2s, mavila superdiatonic or superdiatonic refers to the structure of octave-equivalent MOS scales with generators ranging from 4\7 (four degrees of 7edo = 685.71¢) to 5\9 (five degrees of 9edo = 666.67¢). In the case of 9edo, L and s are the same size; in the case of 7edo, s becomes so small it disappears (and all that remains are the seven equal L's).

From a regular temperament perspective (i.e. approximating low JI intervals), this MOS pattern is essentially synonymous to mavila. If you're looking for highly accurate scales (that is, ones that approximate low JI closely), there are much better scale patterns to look at. However, if 678 cents is an acceptable 3/2 to you, then mavila is an important harmonic entropy minimum here. So a general name for this MOS pattern could be "mavila superdiatonic" or simply 'Superdiatonic'.

These scales are strongly associated with mavila system, which can be divided into two systems:

  • the Armodue project/system and its associated armodue temperament, with fifths sharper than 5\9 (666.7¢) and flatter than 9\16 (675¢).
  • Hornbostel temperament, with fifths sharper than 9\16 (675¢) and flatter than 4\7 (685.71¢).

Some high JI approximations of the generator: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119, 250/169. These could be used to guide the construction of neji versions of superdiatonic scales or edos.

Scale tree

Generator Generator size (cents) L/s Comments
4\7 685.714 1/0
21\37 681.08 5/1
17\30 680 4/1
13\23 678.261 3/1
22\39 676.923 5/2
9\16 675 2/1 Boundary of propriety; smaller generators are strictly proper
23\41 673.171 5/3
672.85 φ/1 Golden mavila
14\25 672 3/2
19\34 670.588 4/3
24\43 669.767 5/4
5\9 666.667 1/1

Primodal theory

Neji versions of superdiatonic modes

  • 40:48:52:54:59:64:70:77:80 Pental Superionian

16nejis

23nejis

25nejis

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