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The '''359 equal divisions of the octave''' ('''359edo''') is the [[EDO|equal division of the octave]] into 359 parts of 3.34262 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
359edo contains a very close approximation of the pure [[3/2]] fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. It provides the [[optimal patent val]] for the 11-limit [[hera]] temperament.  
359edo contains a very close approximation of the pure [[3/2]] fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. In the 5-limit it tempers out the [[würschmidt comma]] and the [[counterschisma]]; in the 7-limit [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]]; in the 11-limit, [[8019/8000]], providing the [[optimal patent val]] for 11-limit [[hera]]. Due to the fifth being reached at the extremely divisible number of 210 steps, 359edo turns out to be important as an accurate supporting edo of various temperaments that divide the fifth into multiple parts.


359edo [[support]]s a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America; the Pythagorean fifth (701.955¢) minus the Pythagorean comma (23.46¢) = 678.495¢; in 359edo this is the step 203\359 of 678.55153¢.
359edo [[support]]s a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America{{citation needed}}; the 678.495{{c}} [[262144/177147|Pythagorean diminished sixth]]; in 359edo this is reached using 203 steps, or 678.55153{{c}}.


Pythagorean diatonic scale: 61 61 27 61 61 61 27
Pythagorean diatonic scale: 61 61 27 61 61 61 27
Line 10: Line 11:
Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one]{{clarify}}).
Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one]{{clarify}}).


359edo is the 72nd [[prime EDO]].
=== Prime harmonics ===
{{Harmonics in equal|359|columns=11}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|359}}
359edo is the 72nd [[prime edo]]. [[718edo]], which doubles it, provides a good correction to the harmonics 5, 13, 17, and 31.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -569 359 }}
| {{mapping| 359 569 }}
| +0.0016
| 0.0016
| 0.05
|-
| 2.3.5
| 393216/390625, {{monzo| -69 45 -1 }}
| {{mapping| 359 569 834 }}
| −0.2042
| 0.2910
| 8.71
|-
| 2.3.5.7
| 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }}
| {{mapping| 359 569 834 1008 }}
| −0.2007
| 0.2521
| 7.54
|-
| 2.3.5.7.11
| 2401/2400, 3136/3125, 8019/8000, 42592/42525
| {{mapping| 359 569 834 1008 1242 }}
| −0.1729
| 0.2322
| 6.95
|-
| 2.3.5.7.11.13
| 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125
| {{mapping| 359 569 834 1008 1242 1328 }} (359f)
| −0.2257
| 0.2426
| 7.26
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 58\359
| 193.87
| 28/25
| [[Hemiwürschmidt]]
|-
| 1
| 116\359
| 387.74
| 5/4
| [[Würschmidt]] (5-limit)
|-
| 1
| 149\359
| 498.05
| 4/3
| [[Counterschismic]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Francium]]
* "This Madness Won't Stop!" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/50O9nTxeMafR8AyBtsPSKa Spotify] | [https://francium223.bandcamp.com/track/this-madness-wont-stop Bandcamp] | [https://www.youtube.com/watch?v=UJyIKzgLVQU YouTube]


[[Category:Equal divisions of the octave]]
[[Category:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Hera]]
[[Category:Hera]]
[[Category:Listen]]

Latest revision as of 12:47, 12 July 2025

← 358edo 359edo 360edo →
Prime factorization 359 (prime)
Step size 3.34262 ¢ 
Fifth 210\359 (701.95 ¢)
(semiconvergent)
Semitones (A1:m2) 34:27 (113.6 ¢ : 90.25 ¢)
Consistency limit 11
Distinct consistency limit 11

359 equal divisions of the octave (abbreviated 359edo or 359ed2), also called 359-tone equal temperament (359tet) or 359 equal temperament (359et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 359 equal parts of about 3.34 ¢ each. Each step represents a frequency ratio of 21/359, or the 359th root of 2.

Theory

359edo contains a very close approximation of the pure 3/2 fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. In the 5-limit it tempers out the würschmidt comma and the counterschisma; in the 7-limit 2401/2400 and 3136/3125, supporting hemiwürschmidt; in the 11-limit, 8019/8000, providing the optimal patent val for 11-limit hera. Due to the fifth being reached at the extremely divisible number of 210 steps, 359edo turns out to be important as an accurate supporting edo of various temperaments that divide the fifth into multiple parts.

359edo supports a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America[citation needed]; the 678.495 ¢ Pythagorean diminished sixth; in 359edo this is reached using 203 steps, or 678.55153 ¢.

Pythagorean diatonic scale: 61 61 27 61 61 61 27

Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one][clarification needed]).

Prime harmonics

Approximation of prime harmonics in 359edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.01 +1.43 +0.53 +0.21 -1.53 -1.33 -0.02 +0.14 -0.05 +1.48
Relative (%) +0.0 -0.2 +42.8 +16.0 +6.4 -45.8 -39.9 -0.6 +4.1 -1.5 +44.4
Steps
(reduced) 		359
(0) 		569
(210) 		834
(116) 		1008
(290) 		1242
(165) 		1328
(251) 		1467
(31) 		1525
(89) 		1624
(188) 		1744
(308) 		1779
(343)

Subsets and supersets

359edo is the 72nd prime edo. 718edo, which doubles it, provides a good correction to the harmonics 5, 13, 17, and 31.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-569 359⟩ [⟨359 569]] +0.0016 0.0016 0.05
2.3.5 393216/390625, [-69 45 -1⟩ [⟨359 569 834]] −0.2042 0.2910 8.71
2.3.5.7 2401/2400, 3136/3125, [-18 24 -5 -3⟩ [⟨359 569 834 1008]] −0.2007 0.2521 7.54
2.3.5.7.11 2401/2400, 3136/3125, 8019/8000, 42592/42525 [⟨359 569 834 1008 1242]] −0.1729 0.2322 6.95
2.3.5.7.11.13 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125 [⟨359 569 834 1008 1242 1328]] (359f) −0.2257 0.2426 7.26

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 58\359 193.87 28/25 Hemiwürschmidt
1 116\359 387.74 5/4 Würschmidt (5-limit)
1 149\359 498.05 4/3 Counterschismic

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

Music

Francium
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