205edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''205 equal divisions of the octave''' ('''205edo'''), or the '''205(-tone) equal temperament''' ('''205tet''', '''205et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 205 parts of 5.854 [[cent]]s each.
{{EDO intro|205}}


== Theory ==
== Theory ==
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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 [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx 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.
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 [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx 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.


=== Miscellaneous properties ===
=== Divisors ===
205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.
205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
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| 386.341<br>(5.85)
| 386.341<br>(5.85)
| 5/4<br>(32805/32768)
| 5/4<br>(32805/32768)
| [[Counterpyth]]
| [[Countercomp]]
|}
|}


== Scales ==
== Scales ==
=== Quanic (24\205) MOS ===
=== Quanic (24\205) mos ===
==== 17 note ====
; 17-note
11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13
: 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


==== 26 note ====
=== Amity (58\205) mos ===
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
; 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


=== Amity (58\205) MOS ===
=== 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


==== 11 note ====
=== Meantone (119\205) mos ===
27 27 4 27 27 4 27 27 4 27 4
; 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


==== 18 note ====
=== Myna (53\205) mos ===
23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4
; 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


==== 25 note ====
=== Porcupine (28\205) mos ===
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
; 15-note
 
: 19 9 19 9 19 9 19 9 19 9 19 9 19 9 9
=== Hemithirds (33\205) MOS ===
; 22-note
==== 13 note ====
: 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9
26 7 26 7 26 7 26 7 26 7 26 7 7
; 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
==== 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 notes ====
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 notes ====
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


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 13:57, 22 December 2022

← 204edo 205edo 206edo →
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

Template:EDO intro

Theory

205edo 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-limits, 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.

Divisors

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.

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
(Reduced) 		Cents
(Reduced) 		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) 		10125/8192
(16/15) 		Qintosec
41 66\205
(1\205) 		386.341
(5.85) 		5/4
(32805/32768) 		Countercomp

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
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