1361edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1361}}
{{ED intro}}


== Theory ==
== Theory ==
Line 13: Line 13:
== 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]]
! 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]] (%)
Line 25: Line 26:
| {{monzo| -2157 1361 }}
| {{monzo| -2157 1361 }}
| {{mapping| 1361 2157 }}
| {{mapping| 1361 2157 }}
| 0.0373
| +0.0373
| 0.0373
| 0.0373
| 4.23
| 4.23
Line 32: Line 33:
| {{monzo| -31 43 -16 }}, {{monzo| -67 -9 35 }}
| {{monzo| -31 43 -16 }}, {{monzo| -67 -9 35 }}
| {{mapping| 1361 2157 3160 }}
| {{mapping| 1361 2157 3160 }}
| 0.0431
| +0.0431
| 0.0315
| 0.0315
| 3.57
| 3.57
Line 39: Line 40:
| 703125/702464, 14348907/14336000, {{monzo| 37 2 -4 -11 }}
| 703125/702464, 14348907/14336000, {{monzo| 37 2 -4 -11 }}
| {{mapping| 1361 2157 3160 3821 }}
| {{mapping| 1361 2157 3160 3821 }}
| 0.0174
| +0.0174
| 0.0522
| 0.0522
| 5.92
| 5.92
Line 46: Line 47:
| 3025/3024, 759375/758912, 14348907/14336000, 262766592/262609375
| 3025/3024, 759375/758912, 14348907/14336000, 262766592/262609375
| {{mapping| 1361 2157 3160 3821 4708 }}
| {{mapping| 1361 2157 3160 3821 4708 }}
| 0.0285
| +0.0285
| 0.0517
| 0.0517
| 5.86
| 5.86
Line 53: Line 54:
| 1716/1715, 3025/3024, 4225/4224, 91125/91091, 131107977/131072000
| 1716/1715, 3025/3024, 4225/4224, 91125/91091, 131107977/131072000
| {{mapping| 1361 2157 3160 3821 4708 5036 }}
| {{mapping| 1361 2157 3160 3821 4708 5036 }}
| 0.0356
| +0.0356
| 0.0498
| 0.0498
| 5.65
| 5.65
Line 60: Line 61:
| 1716/1715, 2500/2499, 3025/3024, 4225/4224, 57375/57344, 3536379/3536000
| 1716/1715, 2500/2499, 3025/3024, 4225/4224, 57375/57344, 3536379/3536000
| {{mapping| 1361 2157 3160 3821 4708 5036 5563 }}
| {{mapping| 1361 2157 3160 3821 4708 5036 5563 }}
| 0.0317
| +0.0317
| 0.0471
| 0.0471
| 5.34
| 5.34
|}
|}
== Music ==
; [[Francium]]
* "The Nights Get Better and Better." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/11VP8L05cT6pDNQX8SJ0u6 Spotify] | [https://francium223.bandcamp.com/track/the-nights-get-better-and-better Bandcamp] | [https://www.youtube.com/watch?v=WmjKnu0OzLQ Bandcamp] – in Beattic, 1361edo tuning

Latest revision as of 12:05, 23 February 2025

← 1360edo 1361edo 1362edo →
Prime factorization 1361 (prime)
Step size 0.881705 ¢ 
Fifth 796\1361 (701.837 ¢)
Semitones (A1:m2) 128:103 (112.9 ¢ : 90.82 ¢)
Consistency limit 17
Distinct consistency limit 17

1361 equal divisions of the octave (abbreviated 1361edo or 1361ed2), also called 1361-tone equal temperament (1361tet) or 1361 equal temperament (1361et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1361 equal parts of about 0.882 ¢ each. Each step represents a frequency ratio of 21/1361, or the 1361st root of 2.

Theory

1361edo is consistent in the 17-odd-limit. As an equal temperament, it tempers out 3025/3024, 1716/1715, 2500/2499, 4225/4224, 57375/57344 and 3536379/3536000. Using the 2.3.5.13.17.43 subgroup, it tempers out 2925/2924. It supports galaxy and geb.

Prime harmonics

Approximation of prime harmonics in 1361edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.118 -0.127 +0.167 -0.253 -0.263 -0.033 -0.379 +0.381 +0.254 +0.299
Relative (%) +0.0 -13.4 -14.4 +19.0 -28.6 -29.8 -3.7 -42.9 +43.2 +28.8 +33.9
Steps
(reduced) 		1361
(0) 		2157
(796) 		3160
(438) 		3821
(1099) 		4708
(625) 		5036
(953) 		5563
(119) 		5781
(337) 		6157
(713) 		6612
(1168) 		6743
(1299)

Subsets and supersets

1361edo is the 218th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-2157 1361⟩ [⟨1361 2157]] +0.0373 0.0373 4.23
2.3.5 [-31 43 -16⟩, [-67 -9 35⟩ [⟨1361 2157 3160]] +0.0431 0.0315 3.57
2.3.5.7 703125/702464, 14348907/14336000, [37 2 -4 -11⟩ [⟨1361 2157 3160 3821]] +0.0174 0.0522 5.92
2.3.5.7.11 3025/3024, 759375/758912, 14348907/14336000, 262766592/262609375 [⟨1361 2157 3160 3821 4708]] +0.0285 0.0517 5.86
2.3.5.7.11.13 1716/1715, 3025/3024, 4225/4224, 91125/91091, 131107977/131072000 [⟨1361 2157 3160 3821 4708 5036]] +0.0356 0.0498 5.65
2.3.5.7.11.13.17 1716/1715, 2500/2499, 3025/3024, 4225/4224, 57375/57344, 3536379/3536000 [⟨1361 2157 3160 3821 4708 5036 5563]] +0.0317 0.0471 5.34

Music

Francium
  • "The Nights Get Better and Better." from Random Sentences (2025) – Spotify | Bandcamp | Bandcamp – in Beattic, 1361edo tuning
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