1637edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1637}}
{{ED intro}}


== Theory ==
== Theory ==
1637edo is [[consistent]] to the [[7-odd-limit]], but the error of its [[harmonic]] [[3/1|3]] is quite large. Using the 2.9.5.7.11.13.17.19.23 [[subgroup]], it tempers out [[4096/4095]], 67392/67375, [[14400/14399]], [[6175/6174]], [[11016/11011]], [[1863/1862]], [[3060/3059]] and 152361/152320. In the 2.5.11.17.23.43 subgroup it tempers out [[10880/10879]].  
1637edo is [[consistent]] to the [[7-odd-limit]], but the error of its [[harmonic]] [[3/1|3]] is quite large. Using the 2.9.5.7.11.13.17.19.23 [[subgroup]], it tempers out [[4096/4095]], 67392/67375, [[14400/14399]], [[6175/6174]], [[11016/11011]], [[1863/1862]], [[3060/3059]] and 152361/152320. In the 2.5.11.17.23.43 subgroup it tempers out [[4301/4300]] and [[10880/10879]].  


=== Odd harmonics ===
=== Odd harmonics ===
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" |[[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
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| {{monzo|-7 -2 13 -6}}, {{monzo|-24 12 0 -5}}, {{monzo|22 5 -3 -11}}
| {{monzo|-7 -2 13 -6}}, {{monzo|-24 12 0 -5}}, {{monzo|22 5 -3 -11}}
| {{mapping|1637 5189 3801 4596}}
| {{mapping|1637 5189 3801 4596}}
| -0.0141
| −0.0141
| 0.0488
| 0.0488
| 6.66
| 6.66
Line 46: Line 47:
| 2359296/2358125, 820125/819896, 50014503/50000000, 275653125/275365888
| 2359296/2358125, 820125/819896, 50014503/50000000, 275653125/275365888
| {{mapping|1637 5189 3801 4596 5663}}
| {{mapping|1637 5189 3801 4596 5663}}
| -0.0075
| −0.0075
| 0.0456
| 0.0456
| 6.22
| 6.22
Line 53: Line 54:
| 4096/4095, 67392/67375, 3720087/3718000, 225000/224939, 6125625/6117748
| 4096/4095, 67392/67375, 3720087/3718000, 225000/224939, 6125625/6117748
| {{mapping|1637 5189 3801 4596 5663 6058}}
| {{mapping|1637 5189 3801 4596 5663 6058}}
| -0.0188
| −0.0188
| 0.0487
| 0.0487
| 6.64
| 6.64

Latest revision as of 06:07, 24 February 2025

← 1636edo 1637edo 1638edo →
Prime factorization 1637 (prime)
Step size 0.733048 ¢ 
Fifth 958\1637 (702.26 ¢)
Semitones (A1:m2) 158:121 (115.8 ¢ : 88.7 ¢)
Dual sharp fifth 958\1637 (702.26 ¢)
Dual flat fifth 957\1637 (701.527 ¢)
Dual major 2nd 278\1637 (203.787 ¢)
Consistency limit 7
Distinct consistency limit 7

1637 equal divisions of the octave (abbreviated 1637edo or 1637ed2), also called 1637-tone equal temperament (1637tet) or 1637 equal temperament (1637et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1637 equal parts of about 0.733 ¢ each. Each step represents a frequency ratio of 21/1637, or the 1637th root of 2.

Theory

1637edo is consistent to the 7-odd-limit, but the error of its harmonic 3 is quite large. Using the 2.9.5.7.11.13.17.19.23 subgroup, it tempers out 4096/4095, 67392/67375, 14400/14399, 6175/6174, 11016/11011, 1863/1862, 3060/3059 and 152361/152320. In the 2.5.11.17.23.43 subgroup it tempers out 4301/4300 and 10880/10879.

Odd harmonics

Approximation of odd harmonics in 1637edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.305 +0.003 +0.264 -0.123 -0.066 +0.279 +0.308 -0.130 +0.105 -0.164 -0.052
Relative (%) +41.6 +0.4 +36.0 -16.7 -9.0 +38.0 +42.0 -17.7 +14.3 -22.4 -7.1
Steps
(reduced) 		2595
(958) 		3801
(527) 		4596
(1322) 		5189
(278) 		5663
(752) 		6058
(1147) 		6396
(1485) 		6691
(143) 		6954
(406) 		7190
(642) 		7405
(857)

Subsets and supersets

1637edo is the 259th prime edo. 3274edo, 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 [-5189 1637⟩ [⟨1637 5189]] +0.0193 0.0193 2.63
2.9.5 [-53 5 16⟩, [-56 77 -81⟩ [⟨1637 5189 3801]] +0.0125 0.0185 2.52
2.9.5.7 [-7 -2 13 -6⟩, [-24 12 0 -5⟩, [22 5 -3 -11⟩ [⟨1637 5189 3801 4596]] −0.0141 0.0488 6.66
2.9.5.7.11 2359296/2358125, 820125/819896, 50014503/50000000, 275653125/275365888 [⟨1637 5189 3801 4596 5663]] −0.0075 0.0456 6.22
2.9.5.7.11.13 4096/4095, 67392/67375, 3720087/3718000, 225000/224939, 6125625/6117748 [⟨1637 5189 3801 4596 5663 6058]] −0.0188 0.0487 6.64

Music

Francium
  • "Kasivo" from The Scallop Disco Accident (2025) – Spotify | Bandcamp | YouTube – in Kaguyic, 1637edo tuning
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