5L 7s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
+modes
→Scale tree: improvement
Line 34: Line 34:


== Scale tree ==
== Scale tree ==
{| class="wikitable"
{| class="wikitable center-all"
! colspan="6" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
|-
! colspan="8" | Generator
| 7\12 || || || || || || 700.000 || 1 || 1 || 2.000 ||  
! | in cents
! | Comments
|-
|-
| style="text-align:center;" | 5\12
| || || || || || 38\65 || 701.539 || 6 || 5 || 1.200 || Photia
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 500
| |
|-
|-
| |
| || || || || 31\53 || || 701.887 || 5 || 4 || 1.250 || Helmholtz
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 37\89
| style="text-align:center;" | 498.876
| |
|-
|-
| |
| || || || || || 55\94 || 702.128 || 9 || 7 || 1.286 || Garibaldi
| |
| |
| |
| |
| |
| style="text-align:center;" | 32\77
| |
| style="text-align:center;" | 498.702
| |
|-
|-
| |
| || || || 24\41 || || || 702.409 || 4 || 3 || 1.333 || Andromeda
| |
| |
| |
| |
| style="text-align:center;" | 27\65
| |
| |
| style="text-align:center;" | 498.462
| style="text-align:center;" | Photia
|-
|-
| |
| || || || || || 65\111 || 702.703 || 11 || 8 || 1.375 ||
| |
| |
| |
| |
| |
| style="text-align:center;" | 49\118
| |
| style="text-align:center;" | 498.305
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
|-
|-
| |
| || || || || 41\70 || || 702.857 || 7 || 5 || 1.400 ||
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 71\171
| style="text-align:center;" | 498.246
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
|-
|-
| |
| || || || || || 58\99 || 703.030 || 10 || 7 || 1.428 || Undecental
| |
| |
| |
| style="text-align:center;" | 22\53
| |
| |
| |
| style="text-align:center;" | 498.113
| style="text-align:center;" | Helenus
|-
|-
| |
| || || 17\29 || || || || 703.448 || 3 || 2 || 1.500 || L/s = 3/2 Edson
| |
| |
| |
| |
| style="text-align:center;" | 39\94
| |
| |
| style="text-align:center;" | 497.872
| style="text-align:center;" | Garibaldi
|-
|-
| |
| || || || || || 61\104 || 703.846 || 11 || 7 || 1.571 ||
| |
| |
| style="text-align:center;" | 17\41
| |
| |
| |
| |
| style="text-align:center;" | 497.591
| style="text-align:center;" | Cassandra
|-
|-
| |
| || || || || 44\75 || || 704.000 || 8 || 5 || 1.600 ||
| |
| |
| |
| |
| style="text-align:center;" | 46\111
| |
| |
| style="text-align:center;" | 497.297
| |
|-
|-
| |
| || || || || || 71\121 || 704.132 || 13 || 8 || 1.625 || Golden parapyth
| |
| |
| |
| style="text-align:center;" | 29\70
| |
| |
| |
| style="text-align:center;" | 497.143
| style="text-align:center;" | Undecental
|-
|-
| |
| || || || 27\46 || || || 704.348 || 5 || 3 || 1.667 || Leapday
| |
| |
| |
| |
| style="text-align:center;" | 41\99
| |
| |
| style="text-align:center;" | 496.97
| style="text-align:center;" | Undecental
|-
|-
| |
| || || || || || 64\109 || 704.587 || 12 || 7 || 1.714 ||
| |
| style="text-align:center;" | 12\29
| |
| |
| |
| |
| |
| style="text-align:center;" | 496.552
| style="text-align:center;" | Optimum rank range (L/s=3/2)
 
Edson
|-
|-
| |
| || || || || 37\63 || || 704.762 || 7 || 4 || 1.750 ||
| |
| |
| |
| |
| style="text-align:center;" | 43\104
| |
| |
| style="text-align:center;" | 496.154
| |
|-
|-
| |  
| || || || || || 47\80 || 705.000 || 9 || 5 || 1.800 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 496.157
| style="text-align:center;" | L/s = pi/2
|-
|-
| |
| || 10\17 || || || || || 705.882 || 2 || 1 || 2.000 || Basic p-chromatic<br>(Generators smaller than this are proper)
| |
| |
| |
| style="text-align:center;" | 31\75
| |
| |
| |
| style="text-align:center;" | 496
| |
|-
|-
| |  
| || || || || || 43\73 || 706.849 || 9 || 4 || 2.250 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 495.904
| style="text-align:center;" | L/s = phi
|-
|-
| |
| || || || || 33\56 || || 707.143 || 7 || 3 || 2.333 || Supra
| |
| |
| |
| |
| style="text-align:center;" | 50\121
| |
| |
| style="text-align:center;" | 495.868
| style="text-align:center;" | Leapday/[[Peppermint-24|Peppermint]]/Pepperoni
|-
|-
| |
| || || || || || 56\95 || 707.368 || 12 || 5 || 2.400 ||
| |
| |
| style="text-align:center;" | 19\46
| |
| |
| |
| |
| style="text-align:center;" | 495.652
| style="text-align:center;" | Leapday
|-
|-
| |
| || || || 23\39 || || || 707.692 || 5 || 2 || 2.500 || Quasisupra
| |
| |
| |
| |
| style="text-align:center;" | 45\109
| |
| |
| style="text-align:center;" | 495.413
| |
|-
|-
| |  
| || || || || || 59\100 || 708.000 || 13 || 5 || 2.600 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 495.325
| style="text-align:center;" | L/s = sqrt(3)
|-
|-
| |
| || || || || 36\61 || || 708.197 || 8 || 3 || 2.667 ||
| |
| |
| |
| style="text-align:center;" | 26\63
| |
| |
| |
| style="text-align:center;" | 495.238
| |
|-
|-
| |
| || || || || || 49\83 || 708.434 || 11 || 4 || 2.750 ||
| |
| |
| |
| |
| style="text-align:center;" | 33\80
| |
| |
| style="text-align:center;" | 495
| |
|-
|-
| |
| || || 13\22 || || || || 709.091 || 3 || 1 || 3.000 || L/s = 3/1 Suprapyth
| style="text-align:center;" | 7\17
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 494.118
| style="text-align:center;" | Boundary of propriety (generators larger than this are proper)
 
Supraphon
|-
|-
| |
| || || || || || 42\71 || 709.859 || 10 || 3 || 3.333 ||
| |
| |
| |
| |
| style="text-align:center;" | 30\73
| |
| |
| style="text-align:center;" | 493.151
| |
|-
|-
| |
| || || || || 29\49 || || 710.204 || 7 || 2 || 3.500 || Superpyth
| |
| |
| |
| style="text-align:center;" | 23\56
| |
| |
| |
| style="text-align:center;" | 492.857
| |
|-
|-
| |
| || || || || || 45\76 || 710.526 || 11 || 3 || 3.667 ||
| |
| |
| |
| |
| style="text-align:center;" | 39\95
| |
| |
| style="text-align:center;" | 492.632
| |
|-
|-
| |
| || || || 16\27 || || || 711.111 || 4 || 1 || 4.000 ||
| |
| |
| style="text-align:center;" | 16\39
| |
| |
| |
| |
| style="text-align:center;" | 492.308
| |
|-
|-
| |
| || || || || || 35\59 || 711.864 || 9 || 2 || 4.500 ||
| |
| |
| |
| |
| style="text-align:center;" | 41\100
| |
| |
| style="text-align:center;" | 492
| |
|-
|-
| |  
| || || || || 19\32 || || 712.500 || 5 || 1 || 5.000 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 491.946
| style="text-align:center;" | L/s = phi+1
|-
|-
| |
| || || || || || 22\37 || 713.514 || 6 || 1 || 6.000 ||
| |
| |
| |
| style="text-align:center;" | 25\61
| |
| |
| |
| style="text-align:center;" | 491.803
| |
|-
|-
| |
| 3\5 || || || || || || 720.000 || 1 || 0 || → inf ||
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 491.655
| style="text-align:center;" | L/s = e
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 34\83
| |  
| |  
| style="text-align:center;" | 491.566
| |
|-
| |
| |
| style="text-align:center;" | 9\22
| |
| |
| |
| |
| |
| style="text-align:center;" | 490.909
| style="text-align:center;" | Suprapyth/Supra
|-
| |
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 490.569
| style="text-align:center;" | L/s = pi
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 29\71
| |
| |
| style="text-align:center;" | 490.141
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 20\49
| |
| |
| |
| style="text-align:center;" | 489.796
| style="text-align:center;" | Superpyth
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 31\76
| |
| |
| style="text-align:center;" | 489.474
| |
|-
| |
| |
| |
| style="text-align:center;" | 11\27
| |
| |
| |
| |
| style="text-align:center;" | 488.889
| style="text-align:center;" | Archy
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 24\59
| |
| |
| style="text-align:center;" | 488.136
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 13\32
| |
| |
| |
| style="text-align:center;" | 487.500
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 15\37
| |
| |
| style="text-align:center;" | 486.486
| |
|-
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 17\42
| |
| style="text-align:center;" | 485.714
| |
|-
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 19\47
| style="text-align:center;" | 485.106
| |
|-
| style="text-align:center;" | 2\5
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 480.000
| |
|}
|}


Revision as of 17:35, 7 April 2021

↖ 4L 6s ↑ 5L 6s 6L 6s ↗
← 4L 7s 5L 7s 6L 7s →
↙ 4L 8s ↓ 5L 8s 6L 8s ↘ 
┌╥┬╥┬╥┬┬╥┬╥┬┬┐
│║│║│║││║│║│││
││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLssLsLss
ssLsLssLsLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\12 to 3\5 (700.0 ¢ to 720.0 ¢)
Dark 2\5 to 5\12 (480.0 ¢ to 500.0 ¢)
TAMNAMS information
Related to 5L 2s (diatonic)
With tunings 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 5L 2s
Sister 7L 5s
Daughters 12L 5s, 5L 12s
Neutralized 10L 2s
2-Flought 17L 7s, 5L 19s
Equal tunings
Equalized (L:s = 1:1) 7\12 (700.0 ¢)
Supersoft (L:s = 4:3) 24\41 (702.4 ¢)
Soft (L:s = 3:2) 17\29 (703.4 ¢)
Semisoft (L:s = 5:3) 27\46 (704.3 ¢)
Basic (L:s = 2:1) 10\17 (705.9 ¢)
Semihard (L:s = 5:2) 23\39 (707.7 ¢)
Hard (L:s = 3:1) 13\22 (709.1 ¢)
Superhard (L:s = 4:1) 16\27 (711.1 ¢)
Collapsed (L:s = 1:0) 3\5 (720.0 ¢)

5L 7s is the MOS pattern of the Pythagorean/Helmholtz/Garibaldi chromatic scale, and also the superpyth chromatic scale. In contrast to the meantone chromatic scale, in which diatonic semitones are larger than chromatic semitones, here the reverse is true: diatonic semitones are smaller than chromatic semitones, so the diatonic scale subset is actually improper.

The two distinct harmonic entropy minima with this MOS pattern are, on the one hand, scales very close to Pythagorean such that 64/63 is not tempered out, such as the schismatic temperaments known as "Helmholtz" and "Garibaldi", and on the other hand, the much simpler and less accurate scale known as "superpyth" in which 64/63 is tempered out.

The Pythagorean/schismatic version is proper, but the superpyth version is improper (it doesn't become proper until you add 5 more notes to form the superpyth "enharmonic" scale, superpyth[17]).

Modes

  • 11|0 LsLsLssLsLss
  • 10|1 LsLssLsLsLss
  • 9|2 LsLssLsLssLs
  • 8|3 LssLsLsLssLs
  • 7|4 LssLsLssLsLs
  • 6|5 sLsLsLssLsLs
  • 5|6 sLsLssLsLsLs
  • 4|7 sLsLssLsLssL
  • 3|8 sLssLsLsLssL
  • 2|9 sLssLsLssLsL
  • 1|10 ssLsLsLssLsL
  • 0|11 ssLsLssLsLsL

Scales

Scale tree

Generator Cents L s L/s Comments
7\12 700.000 1 1 2.000
38\65 701.539 6 5 1.200 Photia
31\53 701.887 5 4 1.250 Helmholtz
55\94 702.128 9 7 1.286 Garibaldi
24\41 702.409 4 3 1.333 Andromeda
65\111 702.703 11 8 1.375
41\70 702.857 7 5 1.400
58\99 703.030 10 7 1.428 Undecental
17\29 703.448 3 2 1.500 L/s = 3/2 Edson
61\104 703.846 11 7 1.571
44\75 704.000 8 5 1.600
71\121 704.132 13 8 1.625 Golden parapyth
27\46 704.348 5 3 1.667 Leapday
64\109 704.587 12 7 1.714
37\63 704.762 7 4 1.750
47\80 705.000 9 5 1.800
10\17 705.882 2 1 2.000 Basic p-chromatic
(Generators smaller than this are proper)
43\73 706.849 9 4 2.250
33\56 707.143 7 3 2.333 Supra
56\95 707.368 12 5 2.400
23\39 707.692 5 2 2.500 Quasisupra
59\100 708.000 13 5 2.600
36\61 708.197 8 3 2.667
49\83 708.434 11 4 2.750
13\22 709.091 3 1 3.000 L/s = 3/1 Suprapyth
42\71 709.859 10 3 3.333
29\49 710.204 7 2 3.500 Superpyth
45\76 710.526 11 3 3.667
16\27 711.111 4 1 4.000
35\59 711.864 9 2 4.500
19\32 712.500 5 1 5.000
22\37 713.514 6 1 6.000
3\5 720.000 1 0 → inf
Retrieved from "https://en.xen.wiki/index.php?title=5L_7s&oldid=65923"