7L 2s

From Xenharmonic Wiki
(Redirected from Superdiatonic)
Jump to navigation Jump to search
↖6L 1s↑7L 1s 8L 1s↗
←6L 2s7L 2s8L 2s→
↙6L 3s↓7L 3s 8L 3s↘
Brightest mode LLLLsLLLs
Period 2/1
Range for bright generator 5\9 (666.7¢) to 4\7 (685.7¢)
Range for dark generator 3\7 (514.3¢) to 4\9 (533.3¢)
TAMNAMS name armotonic
TAMNAMS prefix arm-
Parent MOS 2L 5s
Sister MOS 2L 7s
Daughter MOSes 9L 7s, 7L 9s
Equal tunings
Supersoft (L:s = 4:3) 19\34 (670.6¢)
Soft (L:s = 3:2) 14\25 (672¢)
Semisoft (L:s = 5:3) 23\41 (673.2¢)
Basic (L:s = 2:1) 9\16 (675¢)
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¢)
Brightest-mode tunings on xenpaper
Supersoft Soft Semisoft Basic Semihard Hard Superhard

7L 2s, named armotonic in TAMNAMS, is an octave-equivalent moment of symmetry scale containing 7 large steps and 2 small steps, repeating every octave. Generators that produce this scale range from 666.7¢ to 685.7¢, or from 514.3¢ to 533.3¢.

Scales of this form are strongly associated with Armodue theory, as applied to septimal mavila and Hornbostel temperaments.

Name

The TAMNAMS name for this pattern is armotonic, in reference to Armodue theory. Superdiatonic is also in use.

Intervals

This article assumes TAMNAMS for naming step ratios, mossteps, and mosdegrees.
Intervals of 7L 2s
Intervals (with relation to root) Size Abbrev.
Generic Specific L's and s's Range in cents
0-armstep (root) Perfect 0-armstep 0 0.0¢ P0ms
1-armstep Minor 1-armstep s 0.0¢ to 133.3¢ m1ms
Major 1-armstep L 133.3¢ to 171.4¢ M1ms
2-armstep Minor 2-armstep L + s 171.4¢ to 266.7¢ m2ms
Major 2-armstep 2L 266.7¢ to 342.9¢ M2ms
3-armstep Minor 3-armstep 2L + s 342.9¢ to 400.0¢ m3ms
Major 3-armstep 3L 400.0¢ to 514.3¢ M3ms
4-armstep Perfect 4-armstep 3L + s 514.3¢ to 533.3¢ P4ms
Augmented 4-armstep 4L 533.3¢ to 685.7¢ A4ms
5-armstep Diminished 5-armstep 3L + 2s 514.3¢ to 666.7¢ d5ms
Perfect 5-armstep 4L + s 666.7¢ to 685.7¢ P5ms
6-armstep Minor 6-armstep 4L + 2s 685.7¢ to 800.0¢ m6ms
Major 6-armstep 5L + s 800.0¢ to 857.1¢ M6ms
7-armstep Minor 7-armstep 5L + 2s 857.1¢ to 933.3¢ m7ms
Major 7-armstep 6L + s 933.3¢ to 1028.6¢ M7ms
8-armstep Minor 8-armstep 6L + 2s 1028.6¢ to 1066.7¢ m8ms
Major 8-armstep 7L + s 1066.7¢ to 1200.0¢ M8ms
9-armstep (octave) Perfect 9-armstep 7L + 2s 1200.0¢ P9ms

Note names

7L 2s, when viewed under Armodue theory, can be notated using Armodue notation.

Theory

Temperament interpretations

Mavila is an important harmonic entropy minimum here, insofar as 678¢ can be considered a fifth. Other temperaments include septimal mavila and Hornbostel.

Modes

Modes of 7L 2s
UDP Rotational order Step pattern Mode names
8|0 1 LLLLsLLLs Superlydian
7|1 6 LLLsLLLLs Superionian
6|2 2 LLLsLLLsL Supermixolidyan
5|3 7 LLsLLLLsL Supercorintihan
4|4 3 LLsLLLsLL Superolympian
3|5 8 LsLLLLsLL Superdorian
2|6 4 LsLLLsLLL Superaeolian
1|7 9 sLLLLsLLL Superphrygian
0|8 5 sLLLsLLLL Superlocrian

Scale tree

Todo: cleanup
Clean up scale tree

Optional types of 'JI Blown Fifth' Generators: 31/21, 34/23, 65/44, 71/48, 99/67, 105/71, 108/73, 133/90, 145/98, 176/119 & 250/169.


Generator ranges:

  • Chroma-positive generator: 666.6667 cents (5\9) to 685.7143 cents (4\7)
  • Chroma-negative generator: 514.2857 cents (3\7) to 533.3333 cents (4\9)
Generator Generator size (cents) Pentachord steps Comments
4\7 685.714 1 1 1 0
53\93 683.871 13 13 13 1
102\179 683.798 25 25 25 2 Approximately 0.03 cents away from 95/64
49\86 683.721 12 12 12 1
94\165 683.636 23 23 23 2
45\79 683.544 11 11 11 1
86\151 683.444 21 21 21 2
41\72 683.333 10 10 10 1
78\137 683.212 19 19 19 2
37\65 683.077 9 9 9 1
70\123 682.927 17 17 17 2
33\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.
62\109 682.569 15 15 15 2
29\51 682.353 7 7 7 1
54\95 682.105 13 13 13 2
25\44 681.818 6 6 6 1
46\81 681.4815 11 11 11 2
21\37 681.081 5 5 5 1
59\104 680.769 14 14 14 3
38\67 680.597 9 9 9 2
55\97 680.412 13 13 13 3
17\30 680 4 4 4 1 L/s = 4
47\83 679.518 11 11 11 3
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
678.569 π π π 1 L/s = π
108\191 678.534 25 25 25 8
121\214 678.505 28 28 28 9 28;9 Superdiatonic 1/28-tone (a slight exceeded representation of the ratio 262144/177147, the Pythagorean wolf Fifth)
134\237 678.481 31 31 31 10 HORNBOSTEL TEMPERAMENT (1/31-tone; Optimum high size of Hornbostel '6th')
13\23 678.261 3 3 3 1 HORNBOSTEL TEMPERAMENT (Armodue 1/3-tone)
126\223 678.027 29 29 29 10 HORNBOSTEL TEMPERAMENT

(Armodue 1/29-tone)

113\200 678 26 26 26 9 HORNBOSTEL (& OGOLEVETS) TEMPERAMENT (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' & '8')
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 (the Golden Tone System of Thorvald Kornerup and a temperament of the Alexei Ogolevets's list of temperaments)
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 (Optimum accuracy of Phi interval, the note '7')
677.562 e e e 1 L/s = e
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 φ+1 φ+1 φ+1 1 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\39 676.923 5 5 5 2 Armodue-Hornbostel 1/5-tone (Optimum low size of Hornbostel '6th')
75\133 676.692 17 17 17 7 17;7 Superdiatonic 1/17-tone (Note the very accuracy of the step 75 with the ratio 34/23 with an error of +0.011 Cents)
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\16 675 2 2 2 1 [BOUNDARY OF PROPRIETY: smaller generators are strictly proper]ARMODUE ESADECAFONIA (or Goldsmith Temperament)
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 (with an approximation of the Perfect Fifth + 1/5 Pyth.Comma [706.65 Cents]: 43\73 is 706.85 Cents)
32\57 673.684 7 7 7 4 Armodue-Mavila 1/7-tone (the 'Commatic' version of Armodue, because its high accuracy of the 7/4 interval, the note '8')
673.577 √3 √3 √3 1
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\41 673.171 5 5 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 φ φ φ 1 GOLDEN MAVILA (L/s = φ)
97\173 672.832 21 21 21 13 21;13 Golden Mavila 1/21-tone (Phi is the step 120\173)
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
672.523 π π π 2
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\25 672 3 3 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
80\143 671.329 17 17 17 12
33\59 671.186 7 7 7 5
52\93 670.968 11 11 11 8
19\34 670.588 4 4 4 3
43\77 670.13 9 9 9 7
24\43 669.767 5 5 5 4
53\95 669.474 11 11 11 9
29\52 669.231 6 6 6 5
63\113 669.0265 13 13 13 11
34\61 668.8525 7 7 7 6
73\131 668.702 15 15 15 13
39\70 668.571 8 8 8 7
83\149 668.456 17 17 17 15
44\79 668.354 9 9 9 8
93\167 668.2365 19 19 19 17
49\88 668.182 10 10 10 9
103\185 668.108 21 21 21 9
54\97 668.041 11 11 11 10
113\203 667.98 23 23 23 21
59\106 667.925 12 12 12 11
123\221 667.873 25 25 25 23
64\115 667.826 13 13 13 12
5\9 666.667 1 1 1 1