9edf: Difference between revisions

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-irrelevant shit
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==Scale tree==
EDF scales can be approximated in [[EDO]]s by subdividing diatonic fifths. If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.
If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
Generator range: 76.1905 cents (4\7/9 = 4\63) to 80 cents (3\5/9 = 1\15)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||76.1905||
|-
| || || || || || ||27\47||76.596||
|-
| || || || || ||23\40|| ||76.{{Overline|6}}||
|-
| || || || || || ||42\73||76.712||
|-
| || || || ||19\33|| || ||76.{{Overline|76}}||
|-
| || || || || || ||53\92||76.812||
|-
| || || || || ||34\59|| ||76.836||
|-
| || || || || || ||49\85||76.893||
|-
| || || ||15\26|| || || ||76.932||
|-
| || || || || || ||56\97||76.976||
|-
| || || || || ||41\71|| ||76.995||
|-
| || || || || || ||67\116||77.0115||
|-
| || || || ||26\45|| || ||77.{{Overline|037}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||77.064||
|-
| || || || || ||37\64|| ||77.08{{Overline|3}}||
|-
| || || || || || ||48\83||77.108||
|-
| || ||11\19|| || || || ||77.193||
|-
| || || || || || ||51\88||77.{{Overline|27}}||
|-
| || || || || ||40\69|| ||77.295||
|-
| || || || || || ||69\119||77.311||
|-
| || || || ||29\50|| || ||77.{{Overline|3}}||
|-
| || || || || || ||76\131||77.354||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||77.366||
|-
| || || || || || ||65\112||77.381||
|-
| || || ||18\31|| || || ||77.419||[[Meantone]] is in this region
|-
| || || || || || ||61\105||77.460||
|-
| || || || || ||43\74|| ||77.4775||
|-
| || || || || || ||68\117||77.493||
|-
| || || || ||25\43|| || ||77.519||
|-
| || || || || || ||57\98||77.551||
|-
| || || || || ||32\55|| ||77.{{Overline|57}}||
|-
| || || || || || ||39\67||77.612||
|-
| ||7\12|| || || || || ||77.{{Overline|7}}||
|-
| || || || || || ||38\65||77.949||
|-
| || || || || ||31\53|| ||77.987||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||78.014||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||78.049||
|-
| || || || || || ||65\111||78.{{Overline|078}}||
|-
| || || || || ||41\70|| ||78.095||
|-
| || || || || || ||58\99||78.1145||
|-
| || || ||17\29|| || || ||78.161||
|-
| || || || || || ||61\104||78.205||
|-
| || || || || ||44\75|| ||78.{{Overline|2}}||
|-
| || || || || || ||71\121||78.237||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||78.261||[[Neogothic]] is in this region
The generator closest to a just [[11/7]] for EDOs less than 1800
|-
| || || || || || ||64\109||78.2875||
|-
| || || || || ||37\63|| ||78.307||
|-
| || || || || || ||47\80||78.{{Overline|3}}||
|-
| || ||10\17|| || || || ||78.431||
|-
| || || || || || ||43\73||78.539||
|-
| || || || || ||33\56|| ||78.571||
|-
| || || || || || ||56\95||78.5965||
|-
| || || || ||23\39|| || ||78.6325||
|-
| || || || || || ||59\100||78.{{Overline|6}}||
|-
| || || || || ||36\61|| ||78.6885||
|-
| || || || || || ||49\83||78.715||
|-
| || || ||13\22|| || || ||78.{{Overline|78}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||78.873||
|-
| || || || || ||29\49|| ||78.912||
|-
| || || || || || ||45\76||78.947||
|-
| || || || ||16\27|| || ||79.012||
|-
| || || || || || ||35\59||79.096||
|-
| || || || || ||19\32|| ||79.1{{Overline|6}}||
|-
| || || || || || ||22\37||79.{{Overline|279}}||
|-
|3\5|| || || || || || ||80.000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.
Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.


== Scales within 9edf ==
== Scales within 9edf ==

Revision as of 13:19, 7 May 2024

← 8edf 9edf 10edf →
Prime factorization 32
Step size 77.995 ¢ 
Octave 15\9edf (1169.93 ¢) (→ 5\3edf)
Twelfth 24\9edf (1871.88 ¢) (→ 8\3edf)
Consistency limit 3
Distinct consistency limit 3

9edf is the equal division of the just perfect fifth into 9 parts of 78 cents each, corresponding to 15.391524edo. It is nearly identical to Carlos Alpha.

Approximation of harmonics

Approximation of harmonics in 9edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Error Absolute (¢) -30.1 -30.1 +17.8 +21.5 +17.8 -15.0 -12.2 +17.8 -8.6 -17.6 -12.2 +5.2 +32.9 -8.6 +35.7 +8.7 -12.2 -27.8
Relative (%) -38.6 -38.6 +22.9 +27.6 +22.9 -19.3 -15.7 +22.9 -11.0 -22.5 -15.7 +6.7 +42.2 -11.0 +45.8 +11.2 -15.7 -35.7
Steps
(reduced) 		15
(6) 		24
(6) 		31
(4) 		36
(0) 		40
(4) 		43
(7) 		46
(1) 		49
(4) 		51
(6) 		53
(8) 		55
(1) 		57
(3) 		59
(5) 		60
(6) 		62
(8) 		63
(0) 		64
(1) 		65
(2)

Intervals

# Cents Approximate ratios Neptunian notation using 7\9edf
0 0.0 1/1 C
1 78.0 25/24, 21/20 C#
2 156.0 12/11, 11/10 Db
3 234.0 8/7 D
4 312.0 6/5 D#, Eb
5 390.0 5/4 E
6 468.0 21/16 E#, Fb
7 546.0 15/11, 11/8 F
8 624.0 10/7, 36/25 F#, Cb
9 702.0 3/2 C

Scales within 9edf

Livewire (this is the original/default tuning)

  • 77.995
  • 701.955
  • 779.950
  • 1403.905
  • 1481.905
  • 2105.865
  • 2183.860
  • 2807.820
  • 2885.815
  • 3509.775
  • 3587.770


Corrugated (this is the original/default tuning)

  • 155.990
  • 701.955
  • 857.945
  • 1403.910
  • 1559.900
  • 2105.865
  • 2261.855
  • 2807.820
  • 2963.810
  • 3509.775
  • 3665.765
  • 4211.730
  • 4367.720
  • 4913.685
  • 5069.675
  • 5615.640
  • 5771.630
  • 6317.595
  • 6473.585
  • 7019.550
  • 7175.540
  • 7721.505
  • 7877.495
  • 8423.460


Snowcone (this is the original/default tuning)

  • 233.985
  • 701.955
  • 935.940
  • 1403.910
  • 1637.895
  • 2105.865
  • 2339.850
  • 2807.820
  • 3041.805
  • 3509.775
  • 3743.760
  • 4211.730
  • 4445.715
  • 4913.685
  • 5147.670
  • 5615.640
  • 5849.625
  • 6317.595
  • 6551.580
  • 7019.550
  • 7253.535
  • 7721.505
  • 7955.490
  • 8423.460


Swan (this is the original/default tuning)

  • 311.980
  • 701.955
  • 1013.935
  • 1403.910
  • 1715.890
  • 2105.865
  • 2417.845


Cloudscape (this is the original/default tuning)

  • 389.975
  • 701.955
  • 1091.930
  • 1403.910
  • 1793.885
  • 2105.865
  • 2495.840
  • 2807.820
  • 3197.795
  • 3509.775
  • 3899.750
  • 4211.730
  • 4601.705
  • 4913.685
  • 5303.660
  • 5615.640
  • 6005.615


Pylon (this is the original/default tuning)

  • 467.970
  • 701.955
  • 1169.925
  • 1403.910
  • 1871.880
  • 2105.865
  • 2573.835
  • 2807.820
  • 3275.790
  • 3509.775
  • 3977.745
  • 4211.730
  • 4679.700
  • 4913.685
  • 5381.655
  • 5615.640
  • 6083.610
  • 6317.595
  • 6785.565
  • 7019.550
  • 7487.520
  • 7721.505
  • 8189.475
  • 8423.460


Quest (this is the original/default tuning)

  • 545.965
  • 701.955
  • 1247.920
  • 1403.910
  • 1949.875
  • 2105.865
  • 2651.830
  • 2807.820
  • 3353.785
  • 3509.775
  • 4055.740
  • 4211.730
  • 4757.695
  • 4913.685
  • 5459.650
  • 5615.640
  • 6161.605
  • 6317.595
  • 6863.560
  • 7019.550
  • 7565.515
  • 7721.505
  • 8267.470
  • 8423.460


Purgatory (this is the original/default tuning)

  • 623.960
  • 701.955
  • 1325.915
  • 1403.910
  • 2027.870
  • 2105.865
  • 2729.825
  • 2807.820
  • 3431.780
  • 3509.775
  • 4133.735
  • 4211.730
  • 4835.690


Molten Pelog (this is the original/default tuning)

  • 155.990
  • 311.980
  • 701.955
  • 857.945
  • 1247.920


Molten Slendro (exact tuning from 3edf)

  • 233.985
  • 467.970
  • 701.955
  • 935.940
  • 1169.925

Music

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