4349edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|4349}} == Theory == 4349edo is consistent to the 29-odd-limit, tempering out 12376/12375, 10241/10240, 13377/13376, 8937..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|4349}}
{{ED intro}}


== Theory ==
== Theory ==
4349edo is [[consistent]] to the [[29-odd-limit]], [[temper out|tempering out]] [[12376/12375]], 10241/10240, 13377/13376, 89376/89375, 25025/25024, [[123201/123200]], 76545/76544, 47125/47124 and 25840/25839. It [[support]]s [[counterquectismic]], [[chalmersic]], [[tritomere]] and [[girardic]]. [[Essentially tempered chords]] available in 4349et include [[vicetertismic chords]].
4349edo is a fairly strong [[29-limit]] system, [[consistent]] to the [[29-odd-limit]]. We may note it is a [[counterquectismic]], [[fortune]], and [[euzenius]] system. Some of the simpler commas in [[tempering out|tempers out]] in the higher limits include [[151263/151250]] and [[1771561/1771470]] in the 11-limit; [[123201/123200]] in the 13-limit; [[12376/12375]] in the 17-limit; 10241/10240, 13377/13376, 89376/89375 in the 19-limit; [[12168/12167]], 25025/25024, 76545/76544 in the 23-limit; 47125/47124 and 25840/25839. Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
4349edo is the 594th [[prime EDO]].
4349edo is the 594th [[prime edo]].


== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
|-
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve stretch (¢)
|-
! colspan="2" | Tuning error
![[TE error|Absolute]] (¢)
|-
![[TE simple badness|Relative]] (%)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo|-6893 4349}}
| {{monzo| -6893 4349 }}
| {{mapping|4349 6893}}
| {{mapping| 4349 6893 }}
| 0.0002
| +0.0002
| 0.0002
| 0.0002
| 0.07
| 0.07
|-
|-
| 2.3.5
| 2.3.5
| {{monzo|-107 47 14}}, {{monzo|-35 -79 69}}
| {{monzo| -107 47 14 }}, {{monzo| -35 -79 69 }}
| {{mapping|4349 6893 10098}}
| {{mapping| 4349 6893 10098 }}
| 0.0027
| +0.0027
| 0.0036
| 0.0036
| 1.30
| 1.30
|-
|-
| 2.3.5.7
| 2.3.5.7
| 78125000/78121827, {{monzo|-1 -18 -3 13}}, {{monzo|-52 17 12 -1}}
| 78125000/78121827, {{monzo| -1 -18 -3 13 }}, {{monzo| -52 17 12 -1 }}
| {{mapping|4349 6893 10098 12209}}
| {{mapping| 4349 6893 10098 12209 }}
| 0.0066
| +0.0066
| 0.0074
| 0.0074
| 2.68
| 2.68
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| 2.3.5.7.11
| 2.3.5.7.11
| 151263/151250, 21437500/21434787, 246071287/246037500, 369140625/369098752
| 151263/151250, 21437500/21434787, 246071287/246037500, 369140625/369098752
| {{mapping|4349 6893 10098 12209 15045}}
| {{mapping| 4349 6893 10098 12209 15045 }}
| 0.0064
| +0.0064
| 0.0067
| 0.0067
| 2.43
| 2.43
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 123201/123200, 196625/196608, 151263/151250, 492128/492075, 5175625/5174928
| 123201/123200, 151263/151250, 196625/196608, 492128/492075, 5175625/5174928
| {{mapping|4349 6893 10098 12209 15045 16093}}
| {{mapping| 4349 6893 10098 12209 15045 16093 }}
| 0.0079
| +0.0079
| 0.0070
| 0.0070
| 2.54
| 2.54
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 12376/12375, 123201/123200, 37180/37179, 194481/194480, 221221/221184, 1328125/1328096
| 12376/12375, 37180/37179, 123201/123200, 194481/194480, 221221/221184, 1328125/1328096
| {{mapping|4349 6893 10098 12209 15045 16093 17776}}
| {{mapping| 4349 6893 10098 12209 15045 16093 17776 }}
| 0.0104
| +0.0104
| 0.0089
| 0.0089
| 3.23
| 3.23
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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
|-
|-
|1
| 1
|803\4349
| 803\4349
|221.5682
| 221.5682
|8388608/7381125
| 8388608/7381125
|[[Fortune]]
| [[Fortune]]
|}
|}
<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


<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
== Music ==
; [[Francium]]
* "Naughty Girl Era" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/0ulmRt6G21G2X5RyxHLYHT Spotify] | [https://francium223.bandcamp.com/track/naughty-girl-era Bandcamp] | [https://www.youtube.com/watch?v=gJ9ZAKfgigA YouTube] – girardic in 4349edo

Latest revision as of 13:32, 13 March 2026

← 4348edo 4349edo 4350edo →
Prime factorization 4349 (prime)
Step size 0.275926 ¢ 
Fifth 2544\4349 (701.954 ¢)
Semitones (A1:m2) 412:327 (113.7 ¢ : 90.23 ¢)
Consistency limit 29
Distinct consistency limit 29

4349 equal divisions of the octave (abbreviated 4349edo or 4349ed2), also called 4349-tone equal temperament (4349tet) or 4349 equal temperament (4349et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4349 equal parts of about 0.276 ¢ each. Each step represents a frequency ratio of 21/4349, or the 4349th root of 2.

Theory

4349edo is a fairly strong 29-limit system, consistent to the 29-odd-limit. We may note it is a counterquectismic, fortune, and euzenius system. Some of the simpler commas in tempers out in the higher limits include 151263/151250 and 1771561/1771470 in the 11-limit; 123201/123200 in the 13-limit; 12376/12375 in the 17-limit; 10241/10240, 13377/13376, 89376/89375 in the 19-limit; 12168/12167, 25025/25024, 76545/76544 in the 23-limit; 47125/47124 and 25840/25839. Since it tempers out 12168/12167, it allows vicetertismic chords in the 23-odd-limit.

Prime harmonics

Approximation of prime harmonics in 4349edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.001 -0.018 -0.051 -0.019 -0.059 -0.104 -0.065 +0.008 -0.099 +0.055
Relative (%) +0.0 -0.2 -6.5 -18.7 -6.8 -21.2 -37.6 -23.7 +2.9 -35.9 +20.0
Steps
(reduced) 		4349
(0) 		6893
(2544) 		10098
(1400) 		12209
(3511) 		15045
(1998) 		16093
(3046) 		17776
(380) 		18474
(1078) 		19673
(2277) 		21127
(3731) 		21546
(4150)

Subsets and supersets

4349edo is the 594th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-6893 4349⟩ [⟨4349 6893]] +0.0002 0.0002 0.07
2.3.5 [-107 47 14⟩, [-35 -79 69⟩ [⟨4349 6893 10098]] +0.0027 0.0036 1.30
2.3.5.7 78125000/78121827, [-1 -18 -3 13⟩, [-52 17 12 -1⟩ [⟨4349 6893 10098 12209]] +0.0066 0.0074 2.68
2.3.5.7.11 151263/151250, 21437500/21434787, 246071287/246037500, 369140625/369098752 [⟨4349 6893 10098 12209 15045]] +0.0064 0.0067 2.43
2.3.5.7.11.13 123201/123200, 151263/151250, 196625/196608, 492128/492075, 5175625/5174928 [⟨4349 6893 10098 12209 15045 16093]] +0.0079 0.0070 2.54
2.3.5.7.11.13.17 12376/12375, 37180/37179, 123201/123200, 194481/194480, 221221/221184, 1328125/1328096 [⟨4349 6893 10098 12209 15045 16093 17776]] +0.0104 0.0089 3.23

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 803\4349 221.5682 8388608/7381125 Fortune

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

Music

Francium
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