1553edo: 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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|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]] (%)
|-
|-
|2.3
| 2.9
|{{monzo|-2461 1553}}
| {{monzo| 4923 -1553 }}
|{{val|1553 2461}}
| {{val| 1553 4923 }}
| 0.1089
| -0.0130
| 0.1089
| 0.0130
| 14.09
| 1.68
|-
|-
|2.3.5
| 2.9.5
|1224440064/1220703125, {{monzo|-201 99 19}}
| {{monzo| 93 -33 5 }}, {{monzo| -36 -26 51 }}
|{{val|1553 2461 3606}}
| {{val| 1553 4923 3606 }}
| 0.0675
| -0.0137
| 0.1064
| 0.0106
| 13.77
| 1.38
|-
|-
|2.3.5.7
| 2.9.5.7
|200120949/200000000, 2579890176/2573571875, 23066015625/23018340352
| {{monzo| -5 5 5 -8 }}, {{monzo| 2 -10 14 -1 }}, {{monzo| 37 1 -4 -11 }}
|{{val|1553 2461 3606 4360}}
| {{val| 1553 4923 3606 4360 }}
| 0.0384
| -0.0225
| 0.1051
| 0.0178
| 13.60
| 2.31
|-
|-
|2.3.5.7.11
| 2.9.5.7.13
|24057/24010, 759375/758912, 2359296/2358125, 992436543/991232000
| 4096/4095, 140625/140608, 28829034/28824005, {{monzo| 4 10 -9 0 -4 }}
|{{val|1553 2461 3606 4360 5372}}
| {{val| 1553 4923 3606 4360 5372 }}
| 0.0529
| -0.0271
| 0.0984
| 0.0184
| 12.73
| 2.38
|-
|2.3.5.7.11.13
|729/728, 1716/1715, 40656/40625, 196625/196608, 14085981/14080000
|{{val|1553 2461 3606 4360 5372 5747}}
| 0.0366
| 0.0970
| 12.55
|-
|2.3.5.7.11.13.17
|729/728, 1716/1715, 1089/1088, 14400/14399, 27648/27625, 14085981/14080000
|{{val|1553 2461 3606 4360 5372 5747 6348}}
| 0.0267
| 0.0930
| 12.04
|-
|2.3.5.7.11.13.17.19
|729/728, 1716/1715, 1089/1088, 5985/5984, 4200/4199, 21888/21875, 287469/287375
|{{val|1553 2461 3606 4360 5372 5747 6348 6597}}
| 0.0241
| 0.0872
| 11.29
|}
|}

Revision as of 04:39, 18 April 2023

← 1552edo 1553edo 1554edo →
Prime factorization 1553 (prime)
Step size 0.772698 ¢ 
Fifth 908\1553 (701.61 ¢)
Semitones (A1:m2) 144:119 (111.3 ¢ : 91.95 ¢)
Dual sharp fifth 909\1553 (702.382 ¢)
Dual flat fifth 908\1553 (701.61 ¢)
Dual major 2nd 264\1553 (203.992 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

1553edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Assume the patent val, 1553et tempers out 200120949/200000000 in the 7-limit; 102487/102400, 759375/758912, 2359296/2358125, and 369140625/369098752 in the 11-limit; 1716/1715, 40656/40625, 59535/59488, 105644/105625, 196625/196608, 200000/199927, 823875/823543, 8859375/8859136, 34034175/34027136, and 75000000/74942413, in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 1553edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.345 +0.035 +0.137 +0.082 -0.384 +0.168 -0.310 +0.132 -0.024 -0.208 -0.071
Relative (%) -44.7 +4.6 +17.8 +10.6 -49.7 +21.7 -40.1 +17.0 -3.1 -26.9 -9.2
Steps
(reduced) 		2461
(908) 		3606
(500) 		4360
(1254) 		4923
(264) 		5372
(713) 		5747
(1088) 		6067
(1408) 		6348
(136) 		6597
(385) 		6821
(609) 		7025
(813)

Subsets and supersets

1553edo is the 245th prime edo. 3106edo, which doubles it, provides a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.9 [4923 -1553⟩ ⟨1553 4923] -0.0130 0.0130 1.68
2.9.5 [93 -33 5⟩, [-36 -26 51⟩ ⟨1553 4923 3606] -0.0137 0.0106 1.38
2.9.5.7 [-5 5 5 -8⟩, [2 -10 14 -1⟩, [37 1 -4 -11⟩ ⟨1553 4923 3606 4360] -0.0225 0.0178 2.31
2.9.5.7.13 4096/4095, 140625/140608, 28829034/28824005, [4 10 -9 0 -4⟩ ⟨1553 4923 3606 4360 5372] -0.0271 0.0184 2.38
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