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== Theory ==
The ''58 equal temperament'', often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the [[Octave|octave]] into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents. It tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]], [[13-limit|13]] and [[17-limit|17-limit]]s. It is the smallest [[EDO|equal temperament]] which is [[consistent|consistent]] through the 17-limit, and is also the first et to map the entire 11-limit [[Tonality_diamond|tonality diamond]] to distinct scale steps, and hence the first et which can define a version of the famous 43-note [[Harry_Partch_related_scales|Genesis scale]] of [[Harry_Partch|Harry Partch]]. It supports [[Hemififths|hemififths]], [[Myna|myna]], [[Diaschismic|diaschismic]], [[Harry|harry]], [[Hemifamity_temperaments#Mystery|mystery]], [[Hemifamity_temperaments#Buzzard|buzzard]] and [[Starling_temperaments#Thuja|thuja]] [[Regular_Temperaments|temperament]]s, and supplies the [[Optimal_patent_val|optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank three temperaments [[Starling_family#Thrush|thrush]], [[Starling_family#Thrush-Bluebird|bluebird]], [[Starling_family#Aplonis|aplonis]] and [[Breed_family#Jove, aka Wonder-Jofur|jofur]].
The ''58 equal temperament'', often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the [[Octave|octave]] into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents. It tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]], [[13-limit|13]] and [[17-limit|17-limit]]s. It is the smallest [[EDO|equal temperament]] which is [[consistent|consistent]] through the 17-limit, and is also the first et to map the entire 11-limit [[Tonality_diamond|tonality diamond]] to distinct scale steps, and hence the first et which can define a version of the famous 43-note [[Harry_Partch_related_scales|Genesis scale]] of [[Harry_Partch|Harry Partch]]. It supports [[Hemififths|hemififths]], [[Myna|myna]], [[Diaschismic|diaschismic]], [[Harry|harry]], [[Hemifamity_temperaments#Mystery|mystery]], [[Hemifamity_temperaments#Buzzard|buzzard]] and [[Starling_temperaments#Thuja|thuja]] [[Regular_Temperaments|temperament]]s, and supplies the [[Optimal_patent_val|optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank three temperaments [[Starling_family#Thrush|thrush]], [[Starling_family#Thrush-Bluebird|bluebird]], [[Starling_family#Aplonis|aplonis]] and [[Breed_family#Jove, aka Wonder-Jofur|jofur]].


While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2*29, and 58 shares the same excellent fifth with [[29edo|29edo]].
While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2*29, and 58 shares the same excellent fifth with [[29edo|29edo]].


==Scales==
== Intervals ==
[[hemif7|hemif7]]
 
[[hemif10|hemif10]]
 
[[hemif17|hemif17]]
 
==Intervals==


{| class="wikitable"
{| class="wikitable center-all right-2 left-3"
|-
|-
!| Degree
!| #
!| Cents
!| Cents
!| Approximate Ratios
!| Approximate Ratios
Line 20: Line 15:
| 0
| 0
|0.00
|0.00
| | [[1/1]]
| [[1/1]]
|-
|-
| | 1
| 1
| | 20.69
| 20.69
| | [[56/55]], [[64/63]], [[81/80]], [[128/125]]
| [[56/55]], [[64/63]], [[81/80]], [[128/125]]
|-
|-
| | 2
| 2
| | 41.38
| 41.38
| | [[36/35]], [[49/48]], [[50/49]], [[55/54]]
| [[36/35]], [[49/48]], [[50/49]], [[55/54]]
|-
|-
| | 3
| 3
| | 62.07
| 62.07
| | [[26/25]], [[27/26]], [[28/27]], [[33/32]]
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
|-
|-
| | 4
| 4
| | 82.76
| 82.76
| | [[25/24]], [[21/20]], [[22/21]]
| [[25/24]], [[21/20]], [[22/21]]
|-
|-
| | 5
| 5
| | 103.45
| 103.45
| | [[16/15]], [[17/16]], [[18/17]]
| [[16/15]], [[17/16]], [[18/17]]
|-
|-
| | 6
| 6
| | 124.14
| 124.14
| | [[14/13]], [[15/14]], [[27/25]]
| [[14/13]], [[15/14]], [[27/25]]
|-
|-
| | 7
| 7
| | 144.83
| 144.83
| | [[12/11]], [[13/12]]
| [[12/11]], [[13/12]]
|-
|-
| | 8
| 8
| | 165.52
| 165.52
| | [[11/10]]
| [[11/10]]
|-
|-
| | 9
| 9
| | 186.21
| 186.21
| | [[10/9]]
| [[10/9]]
|-
|-
| | 10
| 10
| | 206.9
| 206.90
| | [[9/8]], [[17/15]]
| [[9/8]], [[17/15]]
|-
|-
| | 11
| 11
| | 227.59
| 227.59
| | [[8/7]]
| [[8/7]]
|-
|-
| | 12
| 12
| | 248.28
| 248.28
| | [[15/13]]
| [[15/13]]
|-
|-
| | 13
| 13
| | 268.97
| 268.97
| | [[7/6]]
| [[7/6]]
|-
|-
| | 14
| 14
| | 289.655
| 289.66
| | [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
|-
|-
| | 15
| 15
| | 310.345
| 310.34
| | [[6/5]]
| [[6/5]]
|-
|-
| | 16
| 16
| | 331.035
| 331.03
| | [[17/14]]
| [[17/14]]
|-
|-
| | 17
| 17
| | 351.72
| 351.72
| | [[11/9]], [[16/13]]
| [[11/9]], [[16/13]]
|-
|-
| | 18
| 18
| | 372.41
| 372.41
| | [[21/17]]
| [[21/17]]
|-
|-
| | 19
| 19
| | 393.1
| 393.10
| | [[5/4]]
| [[5/4]]
|-
|-
| | 20
| 20
| | 413.79
| 413.79
| | [[14/11]]
| [[14/11]]
|-
|-
| | 21
| 21
| | 434.48
| 434.48
| | [[9/7]]
| [[9/7]]
|-
|-
| | 22
| 22
| | 455.17
| 455.17
| | [[13/10]], [[17/13]], [[22/17]]
| [[13/10]], [[17/13]], [[22/17]]
|-
|-
| | 23
| 23
| | 475.86
| 475.86
| | [[21/16]]
| [[21/16]]
|-
|-
| | 24
| 24
| | 496.55
| 496.55
| | [[4/3]]
| [[4/3]]
|-
|-
| | 25
| 25
| | 517.24
| 517.24
| | [[27/20]]
| [[27/20]]
|-
|-
| | 26
| 26
| | 537.93
| 537.93
| | [[15/11]]
| [[15/11]]
|-
|-
| | 27
| 27
| | 558.62
| 558.62
| | [[11/8]], [[18/13]]
| [[11/8]], [[18/13]]
|-
|-
| | 28
| 28
| | 579.31
| 579.31
| | [[7/5]]
| [[7/5]]
|-
|-
| | 29
| 29
| | 600
| 600.00
| | [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
|-
|-
| | 30
| 30
| | 620.69
| 620.69
| | [[10/7]]
| [[10/7]]
|-
|-
| | 31
| 31
| | 641.38
| 641.38
| | [[13/9]], [[16/11]]
| [[13/9]], [[16/11]]
|-
|-
| | 32
| 32
| | 662.07
| 662.07
| | [[22/15]]
| [[22/15]]
|-
|-
| | 33
| 33
| | 682.76
| 682.76
| | [[40/27]]
| [[40/27]]
|-
|-
| | 34
| 34
| | 703.45
| 703.45
| | [[3/2]]
| [[3/2]]
|-
|-
| | 35
| 35
| | 724.14
| 724.14
| | [[32/21]]
| [[32/21]]
|-
|-
| | 36
| 36
| | 744.83
| 744.83
| | [[20/13]], [[26/17]], [[17/11]]
| [[20/13]], [[26/17]], [[17/11]]
|-
|-
| | 37
| 37
| | 765.52
| 765.52
| | [[14/9]]
| [[14/9]]
|-
|-
| | 38
| 38
| | 786.21
| 786.21
| | [[11/7]]
| [[11/7]]
|-
|-
| | 39
| 39
| | 806.9
| 806.90
| | [[8/5]]
| [[8/5]]
|-
|-
| | 40
| 40
| | 827.59
| 827.59
| | [[34/21]]
| [[34/21]]
|-
|-
| | 41
| 41
| | 848.28
| 848.28
| | [[13/8]], [[18/11]]
| [[13/8]], [[18/11]]
|-
|-
| | 42
| 42
| | 868.97
| 868.97
| | [[28/17]]
| [[28/17]]
|-
|-
| | 43
| 43
| | 889.655
| 889.66
| | [[5/3]]
| [[5/3]]
|-
|-
| | 44
| 44
| | 910.345
| 910.34
| | [[22/13]], [[17/10]]
| [[22/13]], [[17/10]]
|-
|-
| | 45
| 45
| | 931.035
| 931.03
| | [[12/7]]
| [[12/7]]
|-
|-
| | 46
| 46
| | 951.72
| 951.72
| | [[26/15]]
| [[26/15]]
|-
|-
| | 47
| 47
| | 972.41
| 972.41
| | [[7/4]]
| [[7/4]]
|-
|-
| | 48
| 48
| | 993.1
| 993.10
| | [[16/9]]
| [[16/9]], [[30/17]]
|-
|-
| | 49
| 49
| | 1013.79
| 1013.79
| | [[9/5]]
| [[9/5]]
|-
|-
| | 50
| 50
| | 1034.48
| 1034.48
| | [[20/11]]
| [[20/11]]
|-
|-
| | 51
| 51
| | 1055.17
| 1055.17
| | [[11/6]], [[24/13]]
| [[11/6]], [[24/13]]
|-
|-
| | 52
| 52
| | 1075.86
| 1075.86
| | [[13/7]], [[28/15]]
| [[13/7]], [[28/15]]
|-
|-
| | 53
| 53
| | 1096.55
| 1096.55
| | [[15/8]], [[32/17]], [[17/9]]
| [[15/8]], [[32/17]], [[17/9]]
|-
|-
| | 54
| 54
| | 1117.24
| 1117.24
| | [[48/25]], [[40/21]], [[21/11]]
| [[48/25]], [[40/21]], [[21/11]]
|-
|-
| | 55
| 55
| | 1137.93
| 1137.93
| | [[25/13]], [[52/27]], [[27/14]], [[64/33]]
| [[25/13]], [[52/27]], [[27/14]], [[64/33]]
|-
|-
| | 56
| 56
| | 1158.62
| 1158.62
| | [[35/18]], [[96/49]], [[49/25]], [[108/55]]
| [[35/18]], [[96/49]], [[49/25]], [[108/55]]
|-
|-
| | 57
| 57
| | 1179.31
| 1179.31
| | [[55/28]], [[63/32]], [[160/81]], [[125/64]]
| [[55/28]], [[63/32]], [[160/81]], [[125/64]]
|-
|-
| 58
| 58
Line 254: Line 249:
| [[2/1]]
| [[2/1]]
|}
|}
===Approximations to prime harmonics===


{| class="wikitable" style="text-align:center;"
=== Approximations to prime harmonics ===
!
 
!prime 2
{| class="wikitable center-all"
!prime 3
! colspan="2" |
!prime 5
! prime 2
!prime 7
! prime 3
!prime 11
! prime 5
!prime 13
! prime 7
!prime 17
! prime 11
!prime 19
! prime 13
!prime 23
! prime 17
!prime 29
! prime 19
!prime 31
! prime 23
|-
! rowspan="2" |Error
!absolute (¢)
| 0.0
| +1.59
| +6.79
| +3.59
| +7.30
| +7.75
| -1.51
| -7.86
| -7.58
|-
|-
!error
!relative (%)
|0.0¢
| 0.0
| +1.593¢
| +7.2
| +6.790¢
| +32.8
| +3.588¢
| +17.34
| +7.303¢
| +35.3
| +7.748¢
| +37.4
| -1.507¢
| -7.3
| -7.858¢
| -38.0
| -7.585¢
| -36.7
| +4.906¢
| -7.105¢
|}
|}


==Rank two temperaments==
== Rank two temperaments ==


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Period
! Period
! | Generator
! Generator
! | Name
! Name
|-
|-
| | 1\1
| 1\1
| | 1\58
| 1\58
| |  
|  
|-
|-
| |  
|  
| | 3\58
| 3\58
| |  
|  
|-
|-
| |  
|  
| | 5\58
| 5\58
| |  
|  
|-
|-
| |  
|  
| | 7\58
| 7\58
| |  
|  
|-
|-
| |  
|  
| | 9\58
| 9\58
| |  
|  
|-
|-
| |  
|  
| | 11\58
| 11\58
| | Gorgik
| Gorgik
|-
|-
| |  
|  
| | 13\58
| 13\58
| |  
|  
|-
|-
| |  
|  
| | 15\58
| 15\58
| | Myna
| Myna
|-
|-
| |  
|  
| | 17\58
| 17\58
| | Hemififths
| Hemififths
|-
|-
| |  
|  
| | 19\58
| 19\58
| |  
|  
|-
|-
| |  
|  
| | 21\58
| 21\58
| |  
|  
|-
|-
| |  
|  
| | 23\58
| 23\58
| | Buzzard
| Buzzard
|-
|-
| |  
|  
| | 25\58
| 25\58
| |  
|  
|-
|-
| |  
|  
| | 27\58
| 27\58
| | Thuja
| Thuja
|-
|-
| | 1\2
| 1\2
| | 1\58
| 1\58
| |  
|  
|-
|-
| |  
|  
| | 2\58
| 2\58
| |  
|  
|-
|-
| |  
|  
| | 3\58
| 3\58
| |  
|  
|-
|-
| |  
|  
| | 4\58
| 4\58
| | Harry
| Harry
|-
|-
| |  
|  
| | 5\58
| 5\58
| | Srutal/Diaschismic
| Srutal/Diaschismic
|-
|-
| |  
|  
| | 6\58
| 6\58
| |  
|  
|-
|-
| |  
|  
| | 7\58
| 7\58
| |  
|  
|-
|-
| |  
|  
| | 8\58
| 8\58
| | Echidna, Supers
| Echidna, Supers
|-
|-
| |  
|  
| | 9\58
| 9\58
| | Secant
| Secant
|-
|-
| |  
|  
| | 10\58
| 10\58
| |  
|  
|-
|-
| |  
|  
| | 11\58
| 11\58
| |  
|  
|-
|-
| |  
|  
| | 12\58
| 12\58
| | Sruti
| Sruti
|-
|-
| |  
|  
| | 13\58
| 13\58
| |  
|  
|-
|-
| |  
|  
| | 14\58
| 14\58
| |  
|  
|-
|-
| | 1\29
| 1\29
| | 1\58
| 1\58
| | Mystery
| Mystery
|}
|}


== Scales ==
[[hemif7]]
[[hemif10]]


[[hemif17]]


[[Category:58edo]]
[[Category:58edo]]
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