349edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|349}}
{{ED intro}}


== Theory ==
== Theory ==
349edo is only [[consistent]] to the [[5-odd-limit]]. Omitting the [[harmonic]] [[7/1|7]], it is consistent to the [[13-odd-limit]] with a flat tendency. In the 2.3.5.11.13 [[subgroup]], the equal temperament [[tempering out|tempers out]] [[625/624]], 17303/17280, 28561/28512, 41067/40960, 43940/43923, 85293/85184, 131625/131072 and 166375/165888.
349edo is only [[consistent]] to the [[5-odd-limit]]. Omitting the [[harmonic]] [[7/1|7]], it is consistent to the [[13-odd-limit]] with a flat tendency. In the 2.3.5.11.13 [[subgroup]], the equal temperament [[tempering out|tempers out]] [[625/624]], 17303/17280, 28561/28512, 41067/40960, 43940/43923, 85293/85184, 131625/131072, and 166375/165888.


=== Odd harmonics ===
=== Odd harmonics ===
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== 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]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
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| {{monzo| -553 349 }}
| {{monzo| -553 349 }}
| {{mapping| 349 553 }}
| {{mapping| 349 553 }}
| 0.1648
| +0.1648
| 0.1648
| 0.1648
| 4.79
| 4.79
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| 2109375/2097152, {{monzo| -31 43 -16 }}
| 2109375/2097152, {{monzo| -31 43 -16 }}
| {{mapping| 349 553 810 }}
| {{mapping| 349 553 810 }}
| 0.2841
| +0.2841
| 0.2158
| 0.2158
| 6.28
| 6.28
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| 166375/165888, 1366875/1362944, 1953125/1948617
| 166375/165888, 1366875/1362944, 1953125/1948617
| {{mapping| 349 553 810 1207 }}
| {{mapping| 349 553 810 1207 }}
| 0.2980
| +0.2980
| 0.1884
| 0.1884
| 5.48
| 5.48
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| 625/624, 17303/17280, 41067/40960, 216513/216320
| 625/624, 17303/17280, 41067/40960, 216513/216320
| {{mapping| 349 553 810 1207 1291 }}
| {{mapping| 349 553 810 1207 1291 }}
| 0.3227
| +0.3227
| 0.1756
| 0.1756
| 5.11
| 5.11
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Orson]]
| [[Orson]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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
 
== Music ==
; [[Francium]]
* "chill futuristic lounge" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/3p2RNlmQieZA2fh3ZwQzVw Spotify] | [https://francium223.bandcamp.com/track/chill-futuristic-lounge Bandcamp] | [https://www.youtube.com/watch?v=B_P7zp5Zmio YouTube]
 
[[Category:Listen]]

Latest revision as of 13:32, 13 March 2026

← 348edo 349edo 350edo →
Prime factorization 349 (prime)
Step size 3.4384 ¢ 
Fifth 204\349 (701.433 ¢)
Semitones (A1:m2) 32:27 (110 ¢ : 92.84 ¢)
Consistency limit 5
Distinct consistency limit 5

349 equal divisions of the octave (abbreviated 349edo or 349ed2), also called 349-tone equal temperament (349tet) or 349 equal temperament (349et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 349 equal parts of about 3.44 ¢ each. Each step represents a frequency ratio of 21/349, or the 349th root of 2.

Theory

349edo is only consistent to the 5-odd-limit. Omitting the harmonic 7, it is consistent to the 13-odd-limit with a flat tendency. In the 2.3.5.11.13 subgroup, the equal temperament tempers out 625/624, 17303/17280, 28561/28512, 41067/40960, 43940/43923, 85293/85184, 131625/131072, and 166375/165888.

Odd harmonics

Approximation of odd harmonics in 349edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.52 -1.21 +0.80 -1.04 -1.17 -1.56 +1.70 +1.63 +1.63 +0.28 +0.95
Relative (%) -15.2 -35.3 +23.3 -30.4 -34.2 -45.3 +49.5 +47.5 +47.3 +8.1 +27.7
Steps
(reduced) 		553
(204) 		810
(112) 		980
(282) 		1106
(59) 		1207
(160) 		1291
(244) 		1364
(317) 		1427
(31) 		1483
(87) 		1533
(137) 		1579
(183)

Subsets and supersets

349edo is the 70th prime edo. 1047edo, which triples it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-553 349⟩ [⟨349 553]] +0.1648 0.1648 4.79
2.3.5 2109375/2097152, [-31 43 -16⟩ [⟨349 553 810]] +0.2841 0.2158 6.28
2.3.5.11 166375/165888, 1366875/1362944, 1953125/1948617 [⟨349 553 810 1207]] +0.2980 0.1884 5.48
2.3.5.11.13 625/624, 17303/17280, 41067/40960, 216513/216320 [⟨349 553 810 1207 1291]] +0.3227 0.1756 5.11

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 79\349 271.63 75/64 Orson

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

Music

Francium
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