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== Theory ==
== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.
In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).


=== Prime harmonics ===
=== Prime harmonics ===
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| 2.3
| 2.3
| {{monzo| -163 103 }}
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| {{mapping| 103 166 }}
| +0.923
| +0.923
| 0.924
| 0.924
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| 2.3.5
| 2.3.5
| 78732/78125, 34171875/33554432
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| {{mapping| 103 166 239 }}
| +0.881
| +0.881
| 0.757
| 0.757
Line 43: Line 43:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| {{mapping| 103 166 239 289 }}
| +0.824
| +0.824
| 0.663
| 0.663
Line 50: Line 50:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| {{mapping| 103 166 239 289 356 }}
| +0.876
| +0.876
| 0.602
| 0.602
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| +0.806
| 0.571
| 0.571
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| +0.694
| 0.595
| 0.595
| 5.10
| 5.10
|}
|}
Its closest [[zeta peak index]], [[596zpi]], stretches the octave by +0.739 cents. This expansion is both consistent and uniquely consistent within the 15-integer limit.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| 34.951
| 34.951
| 1990656/1953125
| 1990656/1953125
| [[Gammic]] (5-limit)
| [[Gammy]]
|-
|-
| 1
| 1
Line 188: Line 187:
| [[Neptune]]
| [[Neptune]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Music ==
== Music ==

Revision as of 09:04, 8 June 2024

← 102edo 103edo 104edo →
Prime factorization 103 (prime)
Step size 11.6505 ¢ 
Fifth 60\103 (699.029 ¢)
Semitones (A1:m2) 8:9 (93.2 ¢ : 104.9 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

In 103edo, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%). Its closest zeta peak index, 596zpi, stretches the octave by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good miracle tuning, especially for the 7-limit, and for benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.

Prime harmonics

Approximation of prime harmonics in 103edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -2.93 -1.85 -1.84 -3.75 -1.69 -0.10 +5.40 +0.85 -4.33 -3.29
Relative (%) +0.0 -25.1 -15.9 -15.8 -32.1 -14.5 -0.9 +46.3 +7.3 -37.2 -28.2
Steps
(reduced) 		103
(0) 		163
(60) 		239
(33) 		289
(83) 		356
(47) 		381
(72) 		421
(9) 		438
(26) 		466
(54) 		500
(88) 		510
(98)

Subsets and supersets

103edo is the 27th prime edo.

Intervals

Main article: Table of 103edo intervals

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-163 103⟩ [⟨103 166]] +0.923 0.924 7.92
2.3.5 78732/78125, 34171875/33554432 [⟨103 166 239]] +0.881 0.757 6.49
2.3.5.7 225/224, 1029/1024, 78732/78125 [⟨103 166 239 289]] +0.824 0.663 5.68
2.3.5.7.11 225/224, 243/242, 385/384, 43923/43750 [⟨103 166 239 289 356]] +0.876 0.602 5.16
2.3.5.7.11.13 225/224, 243/242, 351/350, 385/384, 847/845 [⟨103 166 239 289 356 381]] +0.806 0.571 4.90
2.3.5.7.11.13.17 225/224, 243/242, 273/272, 351/350, 375/374, 847/845 [⟨103 166 239 289 356 381 421]] +0.694 0.595 5.10

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 3\103 34.951 1990656/1953125 Gammy
1 5\103 58.252 27/26 Hemisecordite
1 9\103 104.854 17/16 Septendesemi
1 10\103 116.505 15/14~16/15 Miracle / benediction
1 16\103 186.408 10/9 Mintone
1 20\103 233.010 8/7 Slendric
1 21\103 244.660 15/13 Subsemifourth
1 26\103 303.013 25/21 Quinmite
1 31\103 361.165 16/13 Phicordial
1 37\103 431.06 77/60 Lockerbie
1 38\103 442.708 162/125 Sensei
1 39\103 454.369 13/10 Fibo
1 40\103 466.019 55/42 Hemiseptisix
1 42\103 489.320 65/49 Catafourth
1 45\103 524.272 65/48 Widefourth
1 47\103 547.573 11/8 Heinz
1 48\103 559.223 242/175 Tritriple
1 50\103 582.524 7/5 Neptune

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Francium
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