1553edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1553}}
{{ED intro}}
 
== Theory ==
== Theory ==
1553et tempers out 200120949/200000000 in the 7-limit; 759375/758912, 2359296/2358125, 369140625/369098752 and 102487/102400 in the 11-limit; 200000/199927, 34034175/34027136, 59535/59488, 105644/105625, 8859375/8859136, 75000000/74942413, 40656/40625, 1716/1715, 196625/196608 and 823875/823543 in the 13-limit.
1553edo is only [[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.13 [[subgroup]], where it notably tempers out [[4096/4095]] and 140625/140608.  
===Subsets and supersets===
 
1553edo is the 245th [[prime edo]].
=== Odd harmonics ===
===Odd harmonics===
{{Harmonics in equal|1553}}
{{Harmonics in equal|1553}}


==Regular temperament properties==
=== Subsets and supersets ===
1553edo is the 245th [[prime edo]]. 3106edo, which doubles it, provides a good correction to the harmonic 3.
 
== 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" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{monzo|-2461 1553}}
! [[TE simple badness|Relative]] (%)
|{{val|1553 2461}}
| 0.1089
| 0.1089
| 14.09
|-
|-
|2.3.5
| 2.9
|1224440064/1220703125, {{monzo|-201 99 19}}
| {{monzo| 4923 -1553 }}
|{{val|1553 2461 3606}}
| {{mapping| 1553 4923 }}
| 0.0675
| −0.0130
| 0.1064
| 0.0130
| 13.77
| 1.68
|-
|-
|2.3.5.7
| 2.9.5
|200120949/200000000, 2579890176/2573571875, 23066015625/23018340352
| {{monzo| 93 -33 5 }}, {{monzo| -36 -26 51 }}
|{{val|1553 2461 3606 4360}}
| {{mapping| 1553 4923 3606 }}
| 0.0384
| −0.0137
| 0.1051
| 0.0106
| 13.60
| 1.38
|-
|-
|2.3.5.7.11
| 2.9.5.7
|24057/24010, 759375/758912, 2359296/2358125, 992436543/991232000
| {{monzo| -5 5 5 -8 }}, {{monzo| 2 -10 14 -1 }}, {{monzo| 37 1 -4 -11 }}
|{{val|1553 2461 3606 4360 5372}}
| {{mapping| 1553 4923 3606 4360 }}
| 0.0529
| −0.0225
| 0.0984
| 0.0178
| 12.73
| 2.31
|-
|-
|2.3.5.7.11.13
| 2.9.5.7.13
|729/728, 1716/1715, 40656/40625, 196625/196608, 14085981/14080000
| 4096/4095, 140625/140608, 28829034/28824005, {{monzo| 4 10 -9 0 -4 }}
|{{val|1553 2461 3606 4360 5372 5747}}
| {{mapping| 1553 4923 3606 4360 5372 }}
| 0.0366
| −0.0271
| 0.0970
| 0.0184
| 12.55
| 2.38
|-
|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
|}
|}
== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=gdxwRJSLyvw ''Stumbling Over Mystery''] (2023)
[[Category:Listen]]

Latest revision as of 13:11, 21 February 2025

← 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

1553 equal divisions of the octave (abbreviated 1553edo or 1553ed2), also called 1553-tone equal temperament (1553tet) or 1553 equal temperament (1553et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1553 equal parts of about 0.773 ¢ each. Each step represents a frequency ratio of 21/1553, or the 1553rd root of 2.

Theory

1553edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.13 subgroup, where it notably tempers out 4096/4095 and 140625/140608.

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

Music

Francium
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