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'''330edo''' divides the octave into 330 equal parts of 3.6364 cents each.
{{Infobox ET}}
{{ED intro}}


330edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e [[val]] scores significantly better in [[TE error]] than its [[patent val]] and allows an extension to the 13-limit.  
== Theory ==
330edo has a flat tendency, with its [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.  


It tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 (landscape comma), 703125/702464 ([[meter comma|meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 (kalisma), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 (cuthbert) and 1001/1000 (sinbadma) in the [[13-limit]].  
The equal temperament tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].  


It provides a nice tuning for [[keenanismic family|keenanismic]], the rank-4 temperament that tempers out 385/384 (even better than its [[optimal patent val]] [[284edo]]), and actually a next-to-optimal tuning for 11-limit [[Canou family #semicanou|semicanou]], the rank-3 temperament that tempers out 9801/9800 and 14641/14580.  
It provides a nice tuning for [[keenanismic family|keenanismic]], the rank-4 temperament that tempers out 385/384 (even better than its [[optimal patent val]] [[284edo]]), and actually a close-to-optimal tuning for 11-limit [[Canou family #semicanou|semicanou]], the rank-3 temperament that tempers out 9801/9800 and 14641/14580.  
 
=== Prime harmonics ===
{{Harmonics in equal|330}}
 
=== Subsets and supersets ===
Since 330 factors into {{factorisation|330}}, 330edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.
 
== 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| -523 330 }}
| {{mapping| 330 523 }}
| +0.0432
| 0.0432
| 1.19
|-
| 2.3.5
| 32805/32768, {{monzo| -2 -50 35 }}
| {{mapping| 330 523 766 }}
| +0.1521
| 0.1581
| 4.35
|-
| 2.3.5.7
| 32805/32768, 250047/250000, 823543/819200
| {{mapping| 330 523 766 926 }}
| +0.2524
| 0.2212
| 6.08
|}
 
=== 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
| 137\330
| 498.182
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|-
| 3
| 137\330<br />(27\330)
| 498.182<br />(98.182)
| 4/3<br />(200/189)
| [[Term (temperament)|Term]]
|-
| 6
| 137\330<br />(27\330)
| 498.182<br />(98.182)
| 4/3<br />(200/189)
| [[Semiterm]] (330eff)
|-
| 22
| 137\330<br />(2\330)
| 498.182<br />(7.273)
| 4/3<br />({{monzo| 16 -13 2 }})
| [[Major arcana]] (5-limit)
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


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

Latest revision as of 13:32, 13 March 2026

← 329edo 330edo 331edo →
Prime factorization 2 × 3 × 5 × 11
Step size 3.63636 ¢ 
Fifth 193\330 (701.818 ¢)
Semitones (A1:m2) 31:25 (112.7 ¢ : 90.91 ¢)
Consistency limit 9
Distinct consistency limit 9

330 equal divisions of the octave (abbreviated 330edo or 330ed2), also called 330-tone equal temperament (330tet) or 330 equal temperament (330et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 330 equal parts of about 3.64 ¢ each. Each step represents a frequency ratio of 21/330, or the 330th root of 2.

Theory

330edo has a flat tendency, with its harmonics 3, 5, and 7 tuned progressively flatter. In the 11-limit, the 330e val ⟨330 523 766 926 1141] scores significantly better in TE error than its patent val ⟨330 523 766 926 1142] and allows an extension to the 13-limit.

The equal temperament tempers out 32805/32768 (schisma) in the 5-limit; 250047/250000 (landscape comma), 703125/702464 (meter) and 4802000/4782969 (canousma) in the 7-limit. Using the 330e val, it tempers out 385/384 (keenanisma), 9801/9800 (kalisma), and 14641/14580 (semicanousma) in the 11-limit; 847/845 (cuthbert) and 1001/1000 (sinbadma) in the 13-limit.

It provides a nice tuning for keenanismic, the rank-4 temperament that tempers out 385/384 (even better than its optimal patent val 284edo), and actually a close-to-optimal tuning for 11-limit semicanou, the rank-3 temperament that tempers out 9801/9800 and 14641/14580.

Prime harmonics

Approximation of prime harmonics in 330edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.14 -0.86 -1.55 +1.41 -0.53 +0.50 +0.67 +0.82 -0.49 +0.42
Relative (%) +0.0 -3.8 -23.6 -42.7 +38.8 -14.5 +13.7 +18.4 +22.5 -13.4 +11.5
Steps
(reduced) 		330
(0) 		523
(193) 		766
(106) 		926
(266) 		1142
(152) 		1221
(231) 		1349
(29) 		1402
(82) 		1493
(173) 		1603
(283) 		1635
(315)

Subsets and supersets

Since 330 factors into 2 × 3 × 5 × 11, 330edo has subset edos 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-523 330⟩ [⟨330 523]] +0.0432 0.0432 1.19
2.3.5 32805/32768, [-2 -50 35⟩ [⟨330 523 766]] +0.1521 0.1581 4.35
2.3.5.7 32805/32768, 250047/250000, 823543/819200 [⟨330 523 766 926]] +0.2524 0.2212 6.08

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 137\330 498.182 4/3 Helmholtz
3 137\330
(27\330) 		498.182
(98.182) 		4/3
(200/189) 		Term
6 137\330
(27\330) 		498.182
(98.182) 		4/3
(200/189) 		Semiterm (330eff)
22 137\330
(2\330) 		498.182
(7.273) 		4/3
([16 -13 2⟩) 		Major arcana (5-limit)

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

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