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46edo: Difference between revisions

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| Prime factorization = 2 * 23
| Prime factorization = 2 * 23
| Subgroup = 2.3.5.7.11.13.17.23
| Subgroup = 2.3.5.7.11.13.17.23
| Step size = 26.087
| Step size = 26.087¢
| Fifth type = [[leapfrog]] 27\46 704.348¢
| Fifth type = [[leapfrog]] 27\46 = 704.348¢
| Major 2nd = 8\46 = 209¢
| Minor 2nd = 3\46 = 78¢
| Augmented 1sn = 5\46 = 130¢
| Common uses = neogothic
| Common uses = neogothic
| Important MOS =
| Important MOS =
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== Theory ==
== Theory ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|-
! rowspan="2" |Error
! absolute (¢)
| 0
|  +2.4
|  +5.0
|  -3.6
|  -3.5
|  -5.7
|  -0.6
|  -10.6
|  -2.1
|-
! [[Relative error|relative]] (%)
| 0
|  +9
|  +19
|  -14
|  -13
|  -22
|  -2
|  -40
|  -8
|-
! colspan="2" |[[nearest edomapping]]
|46
|27
|15
|37
|21
|32
|4
|11
|24
|-
! colspan="2" |[[Fifthspan]]
| 0
|  +1
|  +21
|  +15
|  +11
|  +8
|  -22
|  -3
|  +6
|}


46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank two temperaments]] it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[Target_tunings|minimax]] tuning for valentine temperament, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.
46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank two temperaments]] it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[Target_tunings|minimax]] tuning for valentine temperament, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.  


The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.
The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.
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== Just approximation ==
== Just approximation ==
=== Selected just intervals ===
=== Selected just intervals ===
{| class="wikitable center-all"
The following table shows how [[15-odd-limit intervals]] are represented in 46edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +2.4
| +5.0
| -3.6
| -3.5
| -5.7
| -0.61
| -10.6
| -2.1
|-
! relative (%)
| 0.0
| +9.2
| +19.1
| -13.8
| -13.4
| -22.0
| -2.3
| -40.5
| -8.4
|-
! colspan="2" |Fifthspan
| 0
| +1
| +21
| +15
| +11
| +8
| -22
| -3
| +6
|}
 
The following table shows how [[15-odd-limit intervals]] are represented in 46edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
 
{| class="wikitable center-all"
{| class="wikitable center-all"
|+Direct mapping (even if inconsistent)
|+Direct mapping (even if inconsistent)
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