7L 2s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Inthar (talk | contribs)
autopatrolled
15,004 edits
"Superdiatonic" is no longer the official TAMNAMS name for 7L2s.
Inthar (talk | contribs)
autopatrolled
15,004 edits
→Scale tree: Replaced scale tree with the standardized {{Scale tree}} template; Category:Superdiatonic is an otherwise empty category and an obsolete one now that the TAMNAMS name is officially "armotonic".
Line 22: Line 22:


==Scale tree==
==Scale tree==
{{Todo|cleanup|inline=1|comment=Clean up scale tree}}
{{Scale tree|Comments=
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.
1/1: near exact-7/6 [[Armodue]];
 
4/3: near exact-20/17 [[Pentagoth]];
 
7/5: near exact-5/4 [[Mavila]];
Generator ranges:
3/2: near exact-13/11 Pentagoth;
*Chroma-positive generator: 666.6667 cents (5\9) to 685.7143 cents (4\7)
7/4: near exact-7/4 [[Armodue]];
* Chroma-negative generator: 514.2857 cents (3\7) to 533.3333 cents (4\9)
10/3: near exact-6/5 [[Mavila]]; }}
 
{| class="wikitable"
|-
! colspan="3" | Generator
! |<span style="display: block; text-align: center;">'''Generator size (cents)'''</span>
! | Pentachord steps
! |Comments
|-
| |4\[[7edo|7]]
| |
| |
| |685.714
| |1 1 1 0
| |
|-
|53\93
|
|
| 683.871
|13 13 13 1
|
|-
| |
| |102\[[179edo|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\[[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.
|-
|
|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 <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
| |
| |
| | 678.261
| |3 3 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>
|-
| |
| |
| |
| | 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
| |<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
| |
| |
| |676.923
| | 5 5 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
| |
| |
| |675
| |2 2 2 1
| |<span style="display: block; text-align: left;">'''[BOUNDARY OF PROPRIETY: smaller generators are strictly proper]'''</span>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 <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]] interval, the note '8')</span>
|-
| |
| |
| |
| | 673.577
| |<span style="background-color: #ffffff;">√3 √3 √3 1</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
| |
| |
| |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
| |<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ φ φ 1</span>
| |GOLDEN MAVILA (L/s = <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ)</span>
|-
| |
| |
| |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
|-
| |
| |
| |
| |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\[[9edo|9]]
| |
| |
| | 666.667
| |1 1 1 1
| |
|}


[[Category:9-tone scales]]
[[Category:9-tone scales]]
[[Category:Mavila]]
[[Category:Mavila]]
[[Category:Superdiatonic]]

Revision as of 01:14, 28 February 2024

↖ 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, named armotonic in TAMNAMS, is a 2/1-equivalent (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 Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-armstep Perfect 0-armstep P0arms 0 0.0 ¢
1-armstep Minor 1-armstep m1arms s 0.0 ¢ to 133.3 ¢
Major 1-armstep M1arms L 133.3 ¢ to 171.4 ¢
2-armstep Minor 2-armstep m2arms L + s 171.4 ¢ to 266.7 ¢
Major 2-armstep M2arms 2L 266.7 ¢ to 342.9 ¢
3-armstep Minor 3-armstep m3arms 2L + s 342.9 ¢ to 400.0 ¢
Major 3-armstep M3arms 3L 400.0 ¢ to 514.3 ¢
4-armstep Perfect 4-armstep P4arms 3L + s 514.3 ¢ to 533.3 ¢
Augmented 4-armstep A4arms 4L 533.3 ¢ to 685.7 ¢
5-armstep Diminished 5-armstep d5arms 3L + 2s 514.3 ¢ to 666.7 ¢
Perfect 5-armstep P5arms 4L + s 666.7 ¢ to 685.7 ¢
6-armstep Minor 6-armstep m6arms 4L + 2s 685.7 ¢ to 800.0 ¢
Major 6-armstep M6arms 5L + s 800.0 ¢ to 857.1 ¢
7-armstep Minor 7-armstep m7arms 5L + 2s 857.1 ¢ to 933.3 ¢
Major 7-armstep M7arms 6L + s 933.3 ¢ to 1028.6 ¢
8-armstep Minor 8-armstep m8arms 6L + 2s 1028.6 ¢ to 1066.7 ¢
Major 8-armstep M8arms 7L + s 1066.7 ¢ to 1200.0 ¢
9-armstep Perfect 9-armstep P9arms 7L + 2s 1200.0 ¢

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 Cyclic
order		 Step
pattern
8|0 1 LLLLsLLLs
7|1 6 LLLsLLLLs
6|2 2 LLLsLLLsL
5|3 7 LLsLLLLsL
4|4 3 LLsLLLsLL
3|5 8 LsLLLLsLL
2|6 4 LsLLLsLLL
1|7 9 sLLLLsLLL
0|8 5 sLLLsLLLL

Scale tree

Template:Scale tree

Retrieved from "https://en.xen.wiki/index.php?title=7L_2s&oldid=137412"