103edo: Difference between revisions
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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. | 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 192: | Line 192: | ||
== Zeta properties == | == Zeta properties == | ||
===Zeta peak index=== | === Zeta peak index === | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! colspan="3" | Tuning | ||
! | ! colspan="3" | Strength | ||
! colspan="2" | Closest EDO | |||
! colspan="2" | Integer limit | |||
!EDO | |||
! | |||
|- | |- | ||
|[[596zpi]] | ! ZPI | ||
|102.936629522070 | ! Steps per octave | ||
|11.6576577800491 | ! Step size (cents) | ||
|8.543510 | ! Height | ||
|1.340775 | ! Integral | ||
|18.270998 | ! Gap | ||
|103edo | ! EDO | ||
|1200.73875134506 | ! Octave (cents) | ||
|15 | ! Consistent | ||
|15 | ! Distinct | ||
|- | |||
| [[596zpi]] | |||
| 102.936629522070 | |||
| 11.6576577800491 | |||
| 8.543510 | |||
| 1.340775 | |||
| 18.270998 | |||
| 103edo | |||
| 1200.73875134506 | |||
| 15 | |||
| 15 | |||
|} | |} | ||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
Revision as of 02:43, 19 January 2025
| ← 102edo | 103edo | 104edo → |
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
| 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, following 101edo and before 107edo.
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
| 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 distinct
Zeta properties
Zeta peak index
| Tuning | Strength | Closest EDO | Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per octave | Step size (cents) | Height | Integral | Gap | EDO | Octave (cents) | Consistent | Distinct |
| 596zpi | 102.936629522070 | 11.6576577800491 | 8.543510 | 1.340775 | 18.270998 | 103edo | 1200.73875134506 | 15 | 15 |