439edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|439}}
{{ED intro}}


== Theory ==
== Theory ==
Line 17: Line 17:
== 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|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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| {{monzo| 696 -439 }}
| {{monzo| 696 -439 }}
| {{mapping| 439 696 }}
| {{mapping| 439 696 }}
| -0.1737
| −0.1737
| 0.1737
| 0.1737
| 6.36
| 6.36
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5
| 2.3.5
| style="border-top: double;" | 15625/15552, {{monzo| 115 -74 1 }}
| 15625/15552, {{monzo| 115 -74 1 }}
| style="border-top: double;" | {{mapping| 439 696 1019 }} (439)
| {{mapping| 439 696 1019 }} (439)
| style="border-top: double;" | +0.0122
| +0.0122
| style="border-top: double;" | 0.2988
| 0.2988
| style="border-top: double;" | 10.9
| 10.9
|-
|-
| 2.3.5.7
| 2.3.5.7
Line 46: Line 47:
| 0.3127
| 0.3127
| 11.4
| 11.4
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5
| 2.3.5
| style="border-top: double;" | {{monzo| 24 -21 4 }}, {{monzo| 36 11 -23 }}
| {{monzo| 24 -21 4 }}, {{monzo| 36 11 -23 }}
| style="border-top: double;" | {{mapping| 439 696 1020 }} (439c)
| {{mapping| 439 696 1020 }} (439c)
| style="border-top: double;" | -0.3801
| −0.3801
| style="border-top: double;" | 0.3244
| 0.3244
| style="border-top: double;" | 11.9
| 11.9
|-
|-
| 2.3.5.7
| 2.3.5.7
| 3136/3125, 2097152/2083725, 43046721/42875000
| 3136/3125, 2097152/2083725, 43046721/42875000
| {{mapping| 439 696 1020 1233 }} (439cd)
| {{mapping| 439 696 1020 1233 }} (439cd)
| -0.4241
| −0.4241
| 0.2911
| 0.2911
| 10.7
| 10.7

Latest revision as of 12:41, 21 February 2025

← 438edo 439edo 440edo →
Prime factorization 439 (prime)
Step size 2.73349 ¢ 
Fifth 257\439 (702.506 ¢)
Semitones (A1:m2) 43:32 (117.5 ¢ : 87.47 ¢)
Consistency limit 3
Distinct consistency limit 3

439 equal divisions of the octave (abbreviated 439edo or 439ed2), also called 439-tone equal temperament (439tet) or 439 equal temperament (439et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 439 equal parts of about 2.73 ¢ each. Each step represents a frequency ratio of 21/439, or the 439th root of 2.

Theory

439edo is inconsistent in the 5-odd-limit. In the 7-limit, the patent val ⟨439 696 1019 1232], and the 439cd val ⟨439 696 1020 1233] are about as viable.

The patent val tempers out 10976/10935, 15625/15552 and 703125/702464 in the 7-limit; 3025/3024, 4375/4356, 14700/14641, 59290/59049, 131072/130977, and 184877/184320 in the 11-limit.

The 439cd val tempers out 3136/3125, 2097152/2083725, and 4096000/4084101 in the 7-limit; 3136/3125, 15488/15435, 16384/16335, 19712/19683, 24057/24010, 43923/43750, 43923/43904, and 46656/46585 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 439edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.55 -0.89 -1.17 +0.85 -1.35 -1.08 +0.44 +0.43 +0.95 +0.29
Relative (%) +0.0 +20.1 -32.6 -42.9 +31.0 -49.3 -39.6 +16.0 +15.6 +34.6 +10.8
Steps
(reduced) 		439
(0) 		696
(257) 		1019
(141) 		1232
(354) 		1519
(202) 		1624
(307) 		1794
(38) 		1865
(109) 		1986
(230) 		2133
(377) 		2175
(419)

Subsets and supersets

439edo is the 85th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [696 -439⟩ [⟨439 696]] −0.1737 0.1737 6.36
2.3.5 15625/15552, [115 -74 1⟩ [⟨439 696 1019]] (439) +0.0122 0.2988 10.9
2.3.5.7 10976/10935, 15625/15552, [-25 6 -3 8⟩ [⟨439 696 1019 1232]] (439) +0.1135 0.3127 11.4
2.3.5 [24 -21 4⟩, [36 11 -23⟩ [⟨439 696 1020]] (439c) −0.3801 0.3244 11.9
2.3.5.7 3136/3125, 2097152/2083725, 43046721/42875000 [⟨439 696 1020 1233]] (439cd) −0.4241 0.2911 10.7

Scales

Music

Francium
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