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==Scale tree==
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: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" | Fifth
! Cents
! Comments
|-
| 4\7 || || || ||  || || || 62.3377 ||
|-
| ||  || || || || || 27\47 || 62.66925 ||
|-
| || || ||  || || 23\4 0 || || 62.{{Overline|72}} ||
|-
| || ||  || || || || 42\73 || 62.7646 ||
|-
| || || || || 19\33 || || || 62.8099 ||
|-
| || || || || || || 53\92 || 62.84585 ||
|-
| || ||  || || || 34\59 || || 62.86595 ||
|-
| || || || ||  || || 49\85 || 62.8877 ||
|-
| || ||  || 15\26 || ||  || || 62.9371 ||
|-
| || || || || ||  || 56\97 || 62.9803 || The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  || 41\71 || || 62.9962 ||
|-
| || || || || || || 67\116 || 63.0094 ||
|-
| || || || || 26\45 || || || 63.{{Overline|03}} || [[Flattone]] is in this region
|-
| || || || || || || 63\109 || 63.0525 ||
|-
| || || || || || 37\64 || || 63.06{{Overline|81}} ||
|-
| || || || || || || 48\83 || 63.0887 ||
|-
| || || 11\19 || || || || || 63.1579 ||
|-
| || || || || || || 51\88 || 63.2231 ||
|-
| || || || || || 40\69 || || 63.2411 ||
|-
| || || || || || || 69\119 || 63.2544 ||
|-
| || || || || 29\50 || || || 63.{{Overline|27}} ||
|-
| || || || || || || 76\131 || 63.2893 || [[Golden meantone]] (696.2145¢)
|-
| || || || || || 47\81 || || 63.2997 ||
|-
| || || || || || || 65\112 || 63.3117 ||
|-
| || || || 18\31 || || || || 63.3431 || [[Meantone]] is in this region
|-
| || || || || || || 61\105 || 63.3766 ||
|-
| || || || || || 43\74 || || 63.3907 ||
|-
| || || || || || || 68\117 || 63.4033 ||
|-
| || || || || 25\43 || || || 63.42495 ||
|-
| || || || || || || 57\98 || 63.4508 ||
|-
| || || || || || 32\55 || || 63.{{Overline|45}} ||
|-
| || || || || || || 39\67 || 63.5007 ||
|-
| || 7\12 || || || || || || 63.{{Overline|63}} ||
|-
| || || || || || || 38\65 || 63.7762 ||
|-
| || || || || || 31\53 || || 63.8079 || The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || || 55\94 || 63.8298 || [[Garibaldi]] / [[Cassandra]]
|-
| || || || || 24\41 || || || 63.8581 ||
|-
| || || || || || || 65\111 || 63.8821 ||
|-
| || || || || || 41\70 || || 63.8951 ||
|-
| || || || || || || 58\99 || 63.91185 ||
|-
| || || || 17\29 || || || || 63.9499 ||
|-
| || || || || || || 61\104 || 63.8960 ||
|-
| || || || || || 44\75 || || 64.000 ||
|-
| || || || || || || 71\121 || 64.0120 || Golden neogothic (704.0956¢)
|-
| || || || || 27\46 || || || 64.0316 || [[Neogothic]] is in this region
|-
| || || || || || || 64\109 || 64.0534 ||
|-
| || || || || || 37\63 || || 64.0693 ||
|-
| || || || || || || 47\80 || 64.{{Overline|09}} ||
|-
| || || 10\17 || || || || || 64.1711 ||
|-
| || || || || || || 43\73 || 64.2590 ||
|-
| || || || || || 33\56 || || 64.2857 ||
|-
| || || || || || || 56\95 || 64.3062 ||
|-
| || || || || 23\39 || || || 64.3357 ||
|-
| || || || || || || 59\100 || 64.{{Overline|36}} ||
|-
| || || || || || 36\61 || || 64.3815 ||
|-
| || || || || || || 49\83 || 64.4031 ||
|-
| || || || 13\22 || || || || 64.4628 || [[Archy]] is in this region
|-
| || || || || || || 42\71 || 64.53265 ||
|-
| || || || || || 29\49 || || 64.5640 ||
|-
| || || || || || || 45\76 || 64.5933 ||
|-
| || || || || 16\27 || || || 64.{{Overline|64}} ||
|-
| || || || || || || 35\59 || 64.71495 ||
|-
| || || || || || 19\32 || || 64.7{{Overline|72}} ||
|-
| || || || || || || 22\37 || 64.{{Overline|864}} ||
|-
| 3\5 || || || || || || || 65.{{Overline|45}} ||
|}
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:Todo:expand]]

Revision as of 13:10, 7 May 2024

← 10edf 11edf 12edf →
Prime factorization 11 (prime)
Step size 63.8141 ¢ 
Octave 19\11edf (1212.47 ¢)
Twelfth 30\11edf (1914.42 ¢)
Consistency limit 7
Distinct consistency limit 7

11EDF is the equal division of the just perfect fifth into 11 parts of 63.8141 cents each, corresponding to 18.8046 edo (similar to every fifth step of 94edo). It is similar to 19edo and nearly identical to Carlos Beta.

Harmonics

Approximation of harmonics in 11edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +12.47 +12.47 +24.94 +21.51 +24.94 +13.32 -26.41 +24.94 -29.84 -3.40 -26.41
Relative (%) +19.5 +19.5 +39.1 +33.7 +39.1 +20.9 -41.4 +39.1 -46.8 -5.3 -41.4
Steps
(reduced) 		19
(8) 		30
(8) 		38
(5) 		44
(0) 		49
(5) 		53
(9) 		56
(1) 		60
(5) 		62
(7) 		65
(10) 		67
(1)

Intervals

Degree Cent value Corresponding
JI intervals 		Comments
0 exact 1/1
1 63.8141 (28/27), (27/26)
2 127.6282 14/13
3 191.4423
4 255.2564
5 319.07045 6/5
6 382.8845 5/4
7 446.6986
8 510.5127
9 574.3268 39/28
10 638.1409 (13/9)
11 701.955 exact 3/2 just perfect fifth
12 765.7691 14/9, 81/52
13 828.5732 21/13
14 893.3973
15 956.2114
16 1020.0255 9/5
17 1084.8395 15/8
18 1148.6536
19 1211.4677
20 1276.2816 117/56
21 1340.0959 13/6
22 1403.91 exact 9/4
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