743edo: Difference between revisions

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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" |[[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]] (%)
|-
|-
| 2.9
| 2.9
| {{monzo|-2355 743}}
| {{monzo|-2355 743}}
| {{mapping|743 2355}}
| {{mapping|743 2355}}
| 0.0648
| +0.0648
| 0.0648
| 0.0648
| 4.01
| 4.01
Line 32: Line 33:
| 32805/32768, {{monzo|-120 -83 165}}
| 32805/32768, {{monzo|-120 -83 165}}
| {{mapping|743 2355 1725}}
| {{mapping|743 2355 1725}}
| 0.0878
| +0.0878
| 0.0621
| 0.0621
| 3.85
| 3.85
Line 39: Line 40:
| 32805/32768, 30517578125/30474952704, 247669456896/247165842875
| 32805/32768, 30517578125/30474952704, 247669456896/247165842875
| {{mapping|743 2355 1725 2086}}
| {{mapping|743 2355 1725 2086}}
| 0.0464
| +0.0464
| 0.0897
| 0.0897
| 5.55
| 5.55
Line 46: Line 47:
| 32805/32768, 1890625/1889568, 539055/537824, 102487/102400
| 32805/32768, 1890625/1889568, 539055/537824, 102487/102400
| {{mapping|743 2355 1725 2086 2570}}
| {{mapping|743 2355 1725 2086 2570}}
| 0.0705
| +0.0705
| 0.0936
| 0.0936
| 5.80
| 5.80
Line 53: Line 54:
| 67392/67375, 35750/35721, 32805/32768, 91125/91091, 557568/557375
| 67392/67375, 35750/35721, 32805/32768, 91125/91091, 557568/557375
| {{mapping|743 2355 1725 2086 2570 2749}}
| {{mapping|743 2355 1725 2086 2570 2749}}
| 0.0898
| +0.0898
| 0.0957
| 0.0957
| 5.93
| 5.93
Line 60: Line 61:
| 1089/1088, 67392/67375, 35750/35721, 14400/14399, 61965/61952, 75735/75712
| 1089/1088, 67392/67375, 35750/35721, 14400/14399, 61965/61952, 75735/75712
| {{mapping|743 2355 1725 2086 2570 2749 3037}}
| {{mapping|743 2355 1725 2086 2570 2749 3037}}
| 0.0761
| +0.0761
| 0.0947
| 0.0947
| 5.86
| 5.86
|}
|}
== Music ==
; [[Francium]]
* "focus" from ''wiloliquy'' (2025) – [https://open.spotify.com/track/4dAsGgZMItvNjGeDBGFicY Spotify] | [https://francium223.bandcamp.com/track/focus Bandcamp] | [https://www.youtube.com/watch?v=qN2UjaCQZno YouTube]

Latest revision as of 12:24, 3 March 2025

← 742edo 743edo 744edo →
Prime factorization 743 (prime)
Step size 1.61507 ¢ 
Fifth 435\743 (702.557 ¢)
Semitones (A1:m2) 73:54 (117.9 ¢ : 87.21 ¢)
Dual sharp fifth 435\743 (702.557 ¢)
Dual flat fifth 434\743 (700.942 ¢)
Dual major 2nd 126\743 (203.499 ¢)
Consistency limit 3
Distinct consistency limit 3

743 equal divisions of the octave (abbreviated 743edo or 743ed2), also called 743-tone equal temperament (743tet) or 743 equal temperament (743et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 743 equal parts of about 1.62 ¢ each. Each step represents a frequency ratio of 21/743, or the 743rd root of 2.

Theory

743et is only consistent to the 3-odd-limit and its harmonic 3 is about halfway its steps. It can be used in the 2.9.5.7.11.13.17.19.23.29 subgroup, tempering out 2025/2024, 1089/1088, 67392/67375, 35750/35721, 14400/14399, 15840/15827, 3520/3519, 1521/1520 and 2465/2464.

Odd harmonics

Approximation of odd harmonics in 743edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.602 -0.311 +0.219 -0.411 -0.578 -0.689 +0.291 +0.024 -0.339 -0.794 -0.011
Relative (%) +37.3 -19.3 +13.5 -25.4 -35.8 -42.7 +18.0 +1.5 -21.0 -49.2 -0.7
Steps
(reduced) 		1178
(435) 		1725
(239) 		2086
(600) 		2355
(126) 		2570
(341) 		2749
(520) 		2903
(674) 		3037
(65) 		3156
(184) 		3263
(291) 		3361
(389)

Subsets and supersets

743edo is the 132nd prime edo. 1486edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [-2355 743⟩ [⟨743 2355]] +0.0648 0.0648 4.01
2.9.5 32805/32768, [-120 -83 165⟩ [⟨743 2355 1725]] +0.0878 0.0621 3.85
2.9.5.7 32805/32768, 30517578125/30474952704, 247669456896/247165842875 [⟨743 2355 1725 2086]] +0.0464 0.0897 5.55
2.9.5.7.11 32805/32768, 1890625/1889568, 539055/537824, 102487/102400 [⟨743 2355 1725 2086 2570]] +0.0705 0.0936 5.80
2.9.5.7.11.13 67392/67375, 35750/35721, 32805/32768, 91125/91091, 557568/557375 [⟨743 2355 1725 2086 2570 2749]] +0.0898 0.0957 5.93
2.9.5.7.11.13.17 1089/1088, 67392/67375, 35750/35721, 14400/14399, 61965/61952, 75735/75712 [⟨743 2355 1725 2086 2570 2749 3037]] +0.0761 0.0947 5.86

Music

Francium
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