205edo

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← 204edo205edo206edo →
Prime factorization 5 × 41
Step size 5.85366¢ 
Fifth 120\205 (702.439¢) (→24\41)
Semitones (A1:m2) 20:15 (117.1¢ : 87.8¢)
Consistency limit 9
Distinct consistency limit 9

205 equal divisions of the octave (abbreviated 205edo or 205ed2), also called 205-tone equal temperament (205tet) or 205 equal temperament (205et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 205 equal parts of about 5.85 ¢ each. Each step represents a frequency ratio of 21/205, or the 205th root of 2.

Theory

205 = 5 × 41, and 205edo shares its fifth with 41edo. It can serve as a tuning for various temperaments, such as amity or laka, and supplies the optimal patent val for quanic in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments tempering out the huntma, 640/637, the rank-5 temperament for which it also supplies the optimal patent val.

In the 5-limit it tempers out 1600000/1594323, the amity comma, and [38 -2 -15, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val 205 325 476 576 709 759] tempers out 4375/4374, 5120/5103, 6144/6125 in the 7-limit; 540/539, 1331/1323, and 2420/2401 in the 11-limit; 352/351, 640/637, 729/728, 847/845, and 1188/1183 in the 13-limit.

Using its alternative mapping 205 325 476 575] (205d) it can also be used for hemithirds temperament. This extension tempers out 385/384, 441/440, and 3388/3375 in the 11-limit. The 13-limit version of this, 205 325 476 575 709 759] (205d), is especially noteworthy, where it tempers out 196/195 and 1001/1000. Another 13-limit extension is 205 325 476 575 709 758] (205df), where it adds 325/324 and 364/363 to the comma list.

Anyway, assume the patent val, 205et tempers out 540/539, so that it allows swetismic chords; 729/728, so that it allows squbemic chords; 640/637, so that it allows huntmic chords; 352/351, so that it allows minthmic chords; 1188/1183, so that it allows kestrel chords; and 847/845, so that it allows the cuthbert triad. In the alternative 205df val, it allows marveltwin chords, keenanismic chords, gentle chords, and werckismic chords. This makes it a tuning of exceptional fludity for its degree of accuracy.

Odd harmonics

Approximation of odd harmonics in 205edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.48 +0.03 +2.88 +0.97 -1.07 +2.40 +0.51 +0.41 +1.02 -2.49 -1.93
Relative (%) +8.3 +0.5 +49.2 +16.5 -18.3 +41.0 +8.7 +7.0 +17.5 -42.5 -33.0
Steps
(reduced)
325
(120)
476
(66)
576
(166)
650
(35)
709
(94)
759
(144)
801
(186)
838
(18)
871
(51)
900
(80)
927
(107)

Temperament generators and Tonal Plexus

205edo is the default tuning for the Tonal Plexus midi controller. See the theory part on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent meantone fifth in 119\205, an excellent myna generator in 53\205, and a very good porcupine generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.

The 119\205 meantone fifth is extremely close to the 1/4-comma fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing Nicola Vicentino's adaptive-JI scheme of 1555. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

Subsets and supersets

205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 minas.

Notation

Ups and downs

205edo can be notated with ups and downs representing 5\205 = 1\41, and lifts and drops (written as / and \) representing 1\205. This has the advantage of building on a familiarity with 41edo, and is especially useful for Kite guitarists who want to notate microbends more precisely.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
P1 /1 //1 ^\\1 ^\1 ^1 ^/1 ^//1 v\\m2 v\m2 vm2 v/m2 v//m2 \\m2 \m2 m2 /m2

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 1600000/1594323, [38 -2 -15 [205 325 476]] -0.106 0.141 2.41
2.3.5.11 5632/5625, 14641/14580, 1600000/1594323 [181 287 420 508]] -0.002 0.218 3.72

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 6\205 35.122 45/44 Gammic (205e)
1 24\205 140.488 13/12 Quanic (205)
1 33\205 193.171 28/25 Luna / lunatic (205) / hemithirds (205d)
1 58\205 339.512 128/105 Amity (205)
5 63\205
(19\205)
368.780
(111.220)
1024/891
(16/15)
Quintosec
41 66\205
(1\205)
386.341
(5.85)
5/4
(32805/32768)
Countercomp

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

Quanic (24\205) mos

17-note
11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13
26-note
11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2

Amity (58\205) mos

11-note
27 27 4 27 27 4 27 27 4 27 4
18-note
23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4
25-note
19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4

Hemithirds (33\205) mos

13-note
26 7 26 7 26 7 26 7 26 7 26 7 7
19-note
19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7
25-note
12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7
31-notes
5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7

Meantone (119\205) mos

12-note
13 20 13 20 13 20 20 13 20 13 20 20
19-note
13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7
31-note
6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7

Myna (53\205) mos

11-note
7 7 39 7 7 39 7 7 39 7 39
15-note
7 7 7 32 7 7 7 32 7 7 7 32 7 7 32
19-note
7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25
23-note
7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18
27-note
7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11
31-note
7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4

Porcupine (28\205) mos

15-note
19 9 19 9 19 9 19 9 19 9 19 9 19 9 9
22-note
10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9
29-note
1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 9