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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: 97.9592 cents (4\7/7 = 4\49) to 102.8571 cents (3\5/7 = 3\35)
{| class="wikitable center-all"
! colspan="7" |Fifth
! Cents
! Comments
|-
| 4\7 || || ||  || || || || 97.959 ||
|-
| || || ||  || || || 27\47 || 98.480 ||
|-
| || || || || || 23\40 || || 98.571 ||
|-
| || || || ||  || || 42\73 || 98.630 ||
|-
| || || || || 19\33 || || || 98.701 ||
|-
| || || || || || || 53\92 || 98.758 ||
|-
| || || || || || 34\59 || || 98.789 ||
|-
| || || || || || || 49\85 || 98.8235 ||
|-
| || || || 15\26 || || || || 98.901 ||
|-
| || || || || || || 56\97 || 98.969 || The generator closest to a just [[18/17]] for EDOs less than 1400
|-
| || || || || || 41\71 || || 98.994 ||
|-
| || || || || || || 67\116 || 99.015 ||
|-
| || || || || 26\45 || || || 99.048 || [[Flattone]] is in this region
|-
| || || || || || || 63\109 || 99.083 ||
|-
| || || || || || 37\64 || || 99.107 ||
|-
| || || || || || || 48\83 || 99.139 ||
|-
| || || 11\19 || || || || || 99.248 ||
|-
| || || || || || || 51\88 || 99.351 ||
|-
| || || || || || 40\69 || || 99.379 ||
|-
| || || || || || || 69\119 || 99.400 ||
|-
| || || || || 29\50 || || || 99.429 ||
|-
| || || || || || || 76\131 || 99.455 || [[Golden meantone]] (696.2145¢)
|-
| || || || || || 47\81 || || 99.471 ||
|-
| || || || || || || 65\112 || 99.490 ||
|-
| || || || 18\31 || || || || 99.539 || [[Meantone]] is in this region
|-
| || || || || || || 61\105 || 99.592 ||
|-
| || || || || || 43\74 || || 99.613 || The generator closest to a just [[16/9]] for EDOs less than 1400
|-
| || || || || || || 68\117 || 99.634 ||
|-
| || || || || 25\43 || || || 99.668 ||
|-
| || || || || || || 57\98 || 99.7085 ||
|-
| || || || || || 32\55 || || 99.740 ||
|-
| || || || || || || 39\67 || 99.787 ||
|-
| || 7\12 || || || || || || 100.000 ||
|-
| || || || || || || 38\65 || 100.220 ||
|-
| || || || || || 31\53 || || 100.2695 || The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || || 55\94 || 100.304 || [[Garibaldi]] / [[Cassandra]]
|-
| || || || || 24\41 || || || 100.348 ||
|-
| || || || || || || 65\111 || 100.361 ||
|-
| || || || || || 41\70 || || 100.408 ||
|-
| || || || || || || 58\99 || 100.433 ||
|-
| || || || 17\29 || || || || 100.493 ||
|-
| || || || || || || 61\104 || 100.5495 ||
|-
| || || || || || 44\75 || || 100.571 ||
|-
| || || || || || || 71\121 || 100.590 || Golden neogothic (704.0956¢)
|-
| || || || || 27\46 || || || 100.621 || [[Neogothic]] is in this region
|-
| || || || || || || 64\109 || 100.655 ||
|-
| || || || || || 37\63 || || 100.680 ||
|-
| || || || || || || 47\80 || 100.714 ||
|-
| || || 10\17 || || || || || 100.840 ||
|-
| || || || || || || 43\73 || 100.9785 ||
|-
| || || || || || 33\56 || || 101.020 ||
|-
| || || || || || || 56\95 || 101.053 ||
|-
| || || || || 23\39 || || || 101.099 ||
|-
| || || || || || || 59\100 || 101.143 ||
|-
| || || || || || 36\61 || || 101.171 ||
|-
| || || || || || || 49\83 || 101.205 ||
|-
| || || || 13\22 || || || || 101.299 || [[Archy]] is in this region
|-
| || || || || || || 42\71 || 101.4085 ||
|-
| || || || || || 29\49 || || 101.458 ||
|-
| || || || || || || 45\76 || 101.504 ||
|-
| || || || || 16\27 || || || 101.587 ||
|-
| || || || || || || 35\59 || 101.695 ||
|-
| || || || || || 19\32 || || 101.786 || The generator closest to a just [[9/5]] for EDOs less than 1400
|-
| || || || || || || 22\37 || 101.9305 ||
|-
| 3\5 || || || || || || || 102.857 ||
|}
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.
[[Category:Edf]]
[[Category:Edonoi]]
[[Category:todo:improve synopsis]]

Revision as of 13:11, 7 May 2024

← 6edf 7edf 8edf →
Prime factorization 7 (prime)
Step size 100.279 ¢ 
Octave 12\7edf (1203.35 ¢)
(convergent)
Twelfth 19\7edf (1905.31 ¢)
(convergent)
Consistency limit 10
Distinct consistency limit 6

Division of the just perfect fifth into 7 equal parts (7EDF) is related to 12 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 100.2793 cents. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13.

Lookalikes: 12edo, 19ed3, 31ed6

Harmonics

Approximation of harmonics in 7edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.35 +3.35 +6.70 +21.51 +6.70 +40.67 +10.05 +6.70 +24.86 -39.87 +10.05
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1)

Intervals

# Cents 12edo notation
1 100.2793 C#, Db
2 200.5586 D
3 300.8379 D#, Eb
4 401.1171 E
5 501.3964 F
6 601.6757 F#, Gb
7 701.955 G
8 802.2343 G#, Ab
9 902.5136 A
10 1002.7929 A#, Bb
11 1103.0721 B
12 1203.3514 C
13 1303.6307 C#, Db
14 1403.91 D
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