58edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|58 equal temperament}}
{{Wikipedia|58 equal temperament}}
{{EDO intro|58}}
{{ED intro}}


== Theory ==
== Theory ==
58edo is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest distinctly consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-odd-limit [[tonality diamond]] to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]].  
58edo is a strong system in the [[11-limit|11-]], [[13-limit|13-]] and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest distinctly consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-odd-limit [[tonality diamond]] to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]].  


While the [[17/1|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. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent fifth with [[29edo]].
While the [[17/1|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. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent perfect fifth with [[29edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth and no [[5edo]] or [[7edo]] fifths.  


As an equal temperament, 58et 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]]. It [[support]]s [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]], [[thuja]] [[regular temperament|temperament]]s plus a number of [[gravity family]] extensions, and supplies the [[optimal patent val]] for the 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-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].
As an equal temperament, 58et [[tempering out|tempers out]] [[2048/2025]] in the [[5-limit]]; [[126/125]], [[1728/1715]], and [[5120/5103]] in the [[7-limit]]; [[176/175]], [[243/242]], [[441/440]], [[540/539]], and [[896/891]] in the 11-limit; [[144/143]], [[351/350]], [[364/363]] in the 13-limit. It [[support]]s [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]], [[thuja]] [[regular temperament|temperament]]s plus a number of [[gravity family]] [[extension]]s, and supplies the [[optimal patent val]] for the 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-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].


Of all edos which map the syntonic comma ([[81/80]]) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1{{cent}} narrower than the just interval.
Of all edos which map the syntonic comma ([[81/80]]) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1{{cent}} narrower than the just interval.
Line 16: Line 16:


=== Subsets and supersets ===
=== Subsets and supersets ===
58edo contains [[2edo]] and [[29edo]] as subsets.  
58edo contains [[2edo]] and [[29edo]] as subsets.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4""
{| class="wikitable center-all right-2 left-3 left-4""
|-
|-
! #
! #
! Cents
! Cents
! Approximate ratios
! Approximate ratios*
! [[Ups and downs notation]]
! [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.00
| 0.0
| [[1/1]]
| [[1/1]]
| {{UDnote|step=0}}
| {{UDnote|step=0}}
|-
|-
| 1
| 1
| 20.69
| 20.7
| [[56/55]], [[64/63]], [[81/80]], [[128/125]]
| [[56/55]], [[64/63]], [[81/80]], [[91/90]], [[105/104]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 41.38
| 41.4
| [[36/35]], [[49/48]], [[50/49]], [[55/54]]
| [[36/35]], [[40/39]], [[45/44]], [[49/48]], [[50/49]], [[55/54]]
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 62.07
| 62.1
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 82.76
| 82.8
| [[25/24]], [[21/20]], [[22/21]]
| [[21/20]], [[22/21]], ''[[25/24]]''
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 103.45
| 103.4
| [[16/15]], [[17/16]], [[18/17]]
| [[16/15]], [[17/16]], [[18/17]]
| {{UDnote|step=5}}
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 124.14
| 124.1
| [[14/13]], [[15/14]], [[27/25]]
| [[14/13]], [[15/14]]
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 144.83
| 144.8
| [[12/11]], [[13/12]]
| [[12/11]], [[13/12]]
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 165.52
| 165.5
| [[11/10]]
| [[11/10]]
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
| 9
| 9
| 186.21
| 186.2
| [[10/9]]
| [[10/9]]
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 206.90
| 206.9
| [[9/8]], [[17/15]]
| [[9/8]], [[17/15]]
| {{UDnote|step=10}}
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 227.59
| 227.6
| [[8/7]]
| [[8/7]]
| {{UDnote|step=11}}
| {{UDnote|step=11}}
|-
|-
| 12
| 12
| 248.28
| 248.3
| [[15/13]]
| [[15/13]]
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|-
|-
| 13
| 13
| 268.97
| 269.0
| [[7/6]]
| [[7/6]]
| {{UDnote|step=13}}
| {{UDnote|step=13}}
|-
|-
| 14
| 14
| 289.66
| 289.7
| [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|-
|-
| 15
| 15
| 310.34
| 310.3
| [[6/5]]
| [[6/5]]
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|-
|-
| 16
| 16
| 331.03
| 331.0
| [[17/14]]
| [[17/14]], [[40/33]]
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
| 17
| 17
| 351.72
| 351.7
| [[11/9]], [[16/13]]
| [[11/9]], [[16/13]]
| {{UDnote|step=17}}
| {{UDnote|step=17}}
|-
|-
| 18
| 18
| 372.41
| 372.4
| [[21/17]]
| [[21/17]], [[26/21]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 393.10
| 393.1
| [[5/4]]
| [[5/4]]
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|-
|-
| 20
| 20
| 413.79
| 413.8
| [[14/11]]
| [[14/11]]
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|-
|-
| 21
| 21
| 434.48
| 434.5
| [[9/7]]
| [[9/7]]
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|-
|-
| 22
| 22
| 455.17
| 455.2
| [[13/10]], [[17/13]], [[22/17]]
| [[13/10]], [[17/13]], [[22/17]]
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 475.86
| 475.9
| [[21/16]]
| [[21/16]]
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 496.55
| 496.6
| [[4/3]]
| [[4/3]]
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
| 25
| 25
| 517.24
| 517.2
| [[27/20]]
| [[27/20]]
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|-
|-
| 26
| 26
| 537.93
| 537.9
| [[15/11]]
| [[15/11]]
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|-
|-
| 27
| 27
| 558.62
| 558.6
| [[11/8]], [[18/13]]
| [[11/8]], [[18/13]]
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|-
|-
| 28
| 28
| 579.31
| 579.3
| [[7/5]]
| [[7/5]]
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|-
|-
| 29
| 29
| 600.00
| 600.0
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
| {{UDnote|step=29}}
| {{UDnote|step=29}}
|-
|-
| 30
| 30
| 620.69
| 620.7
| [[10/7]]
| [[10/7]]
| {{UDnote|step=30}}
| {{UDnote|step=30}}
|-
|-
| 31
| 31
| 641.38
| 641.4
| [[13/9]], [[16/11]]
| [[13/9]], [[16/11]]
| {{UDnote|step=31}}
| {{UDnote|step=31}}
|-
|-
| 32
| 32
| 662.07
| 662.1
| [[22/15]]
| [[22/15]]
| {{UDnote|step=32}}
| {{UDnote|step=32}}
|-
|-
| 33
| 33
| 682.76
| 682.8
| [[40/27]]
| [[40/27]]
| {{UDnote|step=33}}
| {{UDnote|step=33}}
|-
|-
| 34
| 34
| 703.45
| 703.4
| [[3/2]]
| [[3/2]]
| {{UDnote|step=34}}
| {{UDnote|step=34}}
|-
|-
| 35
| 35
| 724.14
| 724.1
| [[32/21]]
| [[32/21]]
| {{UDnote|step=35}}
| {{UDnote|step=35}}
|-
|-
| 36
| 36
| 744.83
| 744.8
| [[20/13]], [[26/17]], [[17/11]]
| [[17/11]], [[20/13]], [[26/17]]
| {{UDnote|step=36}}
| {{UDnote|step=36}}
|-
|-
| 37
| 37
| 765.52
| 765.5
| [[14/9]]
| [[14/9]]
| {{UDnote|step=37}}
| {{UDnote|step=37}}
|-
|-
| 38
| 38
| 786.21
| 786.2
| [[11/7]]
| [[11/7]]
| {{UDnote|step=38}}
| {{UDnote|step=38}}
|-
|-
| 39
| 39
| 806.90
| 806.9
| [[8/5]]
| [[8/5]]
| {{UDnote|step=39}}
| {{UDnote|step=39}}
|-
|-
| 40
| 40
| 827.59
| 827.6
| [[34/21]]
| [[21/13]], [[34/21]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|-
|-
| 41
| 41
| 848.28
| 848.3
| [[13/8]], [[18/11]]
| [[13/8]], [[18/11]]
| {{UDnote|step=41}}
| {{UDnote|step=41}}
|-
|-
| 42
| 42
| 868.97
| 869.0
| [[28/17]]
| [[28/17]], [[33/20]]
| {{UDnote|step=42}}
| {{UDnote|step=42}}
|-
|-
| 43
| 43
| 889.66
| 889.7
| [[5/3]]
| [[5/3]]
| {{UDnote|step=43}}
| {{UDnote|step=43}}
|-
|-
| 44
| 44
| 910.34
| 910.3
| [[22/13]], [[17/10]]
| [[17/10]], [[22/13]]
| {{UDnote|step=44}}
| {{UDnote|step=44}}
|-
|-
| 45
| 45
| 931.03
| 931.0
| [[12/7]]
| [[12/7]]
| {{UDnote|step=45}}
| {{UDnote|step=45}}
|-
|-
| 46
| 46
| 951.72
| 951.7
| [[26/15]]
| [[26/15]]
| {{UDnote|step=46}}
| {{UDnote|step=46}}
|-
|-
| 47
| 47
| 972.41
| 972.4
| [[7/4]]
| [[7/4]]
| {{UDnote|step=47}}
| {{UDnote|step=47}}
|-
|-
| 48
| 48
| 993.10
| 993.1
| [[16/9]], [[30/17]]
| [[16/9]], [[30/17]]
| {{UDnote|step=48}}
| {{UDnote|step=48}}
|-
|-
| 49
| 49
| 1013.79
| 1013.8
| [[9/5]]
| [[9/5]]
| {{UDnote|step=49}}
| {{UDnote|step=49}}
|-
|-
| 50
| 50
| 1034.48
| 1034.5
| [[20/11]]
| [[20/11]]
| {{UDnote|step=50}}
| {{UDnote|step=50}}
|-
|-
| 51
| 51
| 1055.17
| 1055.2
| [[11/6]], [[24/13]]
| [[11/6]], [[24/13]]
| {{UDnote|step=51}}
| {{UDnote|step=51}}
|-
|-
| 52
| 52
| 1075.86
| 1075.9
| [[13/7]], [[28/15]]
| [[13/7]], [[28/15]]
| {{UDnote|step=52}}
| {{UDnote|step=52}}
|-
|-
| 53
| 53
| 1096.55
| 1096.6
| [[15/8]], [[32/17]], [[17/9]]
| [[15/8]], [[17/9]], [[32/17]]
| {{UDnote|step=53}}
| {{UDnote|step=53}}
|-
|-
| 54
| 54
| 1117.24
| 1117.2
| [[48/25]], [[40/21]], [[21/11]]
| [[21/11]], [[40/21]], ''[[48/25]]''
| {{UDnote|step=54}}
| {{UDnote|step=54}}
|-
|-
| 55
| 55
| 1137.93
| 1137.9
| [[25/13]], [[52/27]], [[27/14]], [[64/33]]
| [[25/13]], [[27/14]], [[52/27]], [[64/33]]
| {{UDnote|step=55}}
| {{UDnote|step=55}}
|-
|-
| 56
| 56
| 1158.62
| 1158.6
| [[35/18]], [[96/49]], [[49/25]], [[108/55]]
| [[35/18]], [[39/20]], [[49/25]], [[88/45]], [[96/49]], [[108/55]]
| {{UDnote|step=56}}
| {{UDnote|step=56}}
|-
|-
| 57
| 57
| 1179.31
| 1179.3
| [[55/28]], [[63/32]], [[160/81]], [[125/64]]
| [[55/28]], [[63/32]], [[160/81]], [[180/91]], [[208/105]]
| {{UDnote|step=57}}
| {{UDnote|step=57}}
|-
|-
| 58
| 58
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
| {{UDnote|step=58}}
| {{UDnote|step=58}}
|}
|}
<nowiki/>* As a 17-limit temperament, inconsistently mapped intervals in ''italic''


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Ups and downs notation ===
In 58edo, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used to fill in the gaps.
58edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}


Alternatively, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.
{{Sharpness-sharp6}}
{{Sharpness-sharp6}}


If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}
{{Sharpness-sharp6-qt}}


Line 494: Line 499:
=== Interval mappings ===
=== Interval mappings ===
{{15-odd-limit|58}}
{{15-odd-limit|58}}
=== Zeta peak index ===
{| class="wikitable center-all"
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[289zpi]]
| 58.0667185533159
| 20.6658827964969
| 7.814035
| 1.358357
| 18.056292
| 58edo
| 1198.62120219682
| 16
| 12
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 539: Line 514:
| 2.3.5
| 2.3.5
| 2048/2025, [[1594323/1562500]]
| 2048/2025, [[1594323/1562500]]
| {{mapping| 58 92 135 }}
| {{Mapping| 58 92 135 }}
| −1.29
| −1.29
| 1.22
| 1.22
Line 546: Line 521:
| 2.3.5.7
| 2.3.5.7
| 126/125, 1728/1715, 2048/2025
| 126/125, 1728/1715, 2048/2025
| {{mapping| 58 92 135 163 }}
| {{Mapping| 58 92 135 163 }}
| −1.29
| −1.29
| 1.05
| 1.05
Line 553: Line 528:
| 2.3.5.7.11
| 2.3.5.7.11
| 126/125, 176/175, 243/242, 896/891
| 126/125, 176/175, 243/242, 896/891
| {{mapping| 58 92 135 163 201 }}
| {{Mapping| 58 92 135 163 201 }}
| −1.45
| −1.45
| 1.00
| 1.00
Line 560: Line 535:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 126/125, 144/143, 176/175, 196/195, 364/363
| 126/125, 144/143, 176/175, 196/195, 364/363
| {{mapping| 58 92 135 163 201 215 }}
| {{Mapping| 58 92 135 163 201 215 }}
| −1.56
| −1.56
| 0.94
| 0.94
Line 567: Line 542:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 126/125, 136/135, 144/143, 176/175, 196/195, 364/363
| 126/125, 136/135, 144/143, 176/175, 196/195, 364/363
| {{mapping| 58 92 135 163 201 215 237 }}
| {{Mapping| 58 92 135 163 201 215 237 }}
| −1.28
| −1.28
| 1.10
| 1.10
Line 586: Line 561:
| 1
| 1
| 3\58
| 3\58
| 62.07
| 62.1
| 28/27
| 28/27
| [[Unicorn]] / alicorn / qilin
| [[Unicorn]] / alicorn / qilin
Line 592: Line 567:
| 1
| 1
| 11\58
| 11\58
| 227.59
| 227.6
| 8/7
| 8/7
| [[Gorgik]]
| [[Gorgik]]
Line 598: Line 573:
| 1
| 1
| 13\58
| 13\58
| 268.97
| 269.0
| 7/6
| 7/6
| [[Infraorwell]]
| [[Infraorwell]]
Line 604: Line 579:
| 1
| 1
| 15\58
| 15\58
| 310.34
| 310.3
| 6/5
| 6/5
| [[Myna]]
| [[Myna]]
Line 610: Line 585:
| 1
| 1
| 17\58
| 17\58
| 351.72
| 351.7
| 49/40
| 49/40
| [[Hemififths]]
| [[Hemififths]]
Line 616: Line 591:
| 1
| 1
| 19\58
| 19\58
| 393.10
| 393.1
| 64/51
| 64/51
| [[Emmthird]]
| [[Emmthird]]
Line 622: Line 597:
| 1
| 1
| 23\58
| 23\58
| 475.86
| 475.9
| 21/16
| 21/16
| [[Buzzard]] / [[subfourth]]
| [[Buzzard]] / [[subfourth]]
Line 628: Line 603:
| 1
| 1
| 27\58
| 27\58
| 558.62
| 558.6
| 11/8
| 11/8
| [[Thuja]]
| [[Thuja]]
Line 634: Line 609:
| 2
| 2
| 3\58
| 3\58
| 62.07
| 62.1
| 28/27
| 28/27
| [[Monocerus]]
| [[Monocerus]]
Line 640: Line 615:
| 2
| 2
| 1\58
| 1\58
| 20.69
| 20.7
| 81/80
| 81/80
| [[Bicommatic]]
| [[Bicommatic]]
Line 646: Line 621:
| 2
| 2
| 9\58
| 9\58
| 186.21
| 186.2
| 10/9
| 10/9
| [[Secant]]
| [[Secant]]
Line 652: Line 627:
| 2
| 2
| 17\58<br>(12\58)
| 17\58<br>(12\58)
| 351.72<br>(248.28)
| 351.7<br>(248.3)
| 11/9<br>(15/13)
| 11/9<br>(15/13)
| [[Sruti]]
| [[Sruti]]
Line 658: Line 633:
| 2
| 2
| 21\58<br>(8\58)
| 21\58<br>(8\58)
| 434.48<br>(165.52)
| 434.5<br>(165.5)
| 9/7<br>(11/10)
| 9/7<br>(11/10)
| [[Echidna]]
| [[Echidna]]
Line 664: Line 639:
| 2
| 2
| 24\58<br>(5\58)
| 24\58<br>(5\58)
| 496.55<br>(103.45)
| 496.6<br>(103.4)
| 4/3<br>(17/16)
| 4/3<br>(17/16)
| [[Diaschismic]]
| [[Diaschismic]]
Line 670: Line 645:
| 2
| 2
| 25\58<br>(4\58)
| 25\58<br>(4\58)
| 517.24<br>(82.76)
| 517.2<br>(82.8)
| 27/20<br>(21/20)
| 27/20<br>(21/20)
| [[Harry]]
| [[Harry]]
Line 676: Line 651:
| 29
| 29
| 19\58<br>(1\58)
| 19\58<br>(1\58)
| 393.10<br>(20.69)
| 393.1<br>(20.7)
| 5/4<br>(91/90)
| 5/4<br>(91/90)
| [[Mystery]]
| [[Mystery]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
 
58et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).
 
== Octave stretch or compression ==
58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].
 
What follows is a comparison of stretched- and compressed-octave 58edo tunings.


58et can also be detempered to [[semihemi]] ({{nowrap|58 &amp; 140}}), [[supers]] ({{nowrap|58 &amp; 152}}), [[condor]] ({{nowrap|58 &amp; 159}}), and [[eagle]] ({{nowrap|58 &amp; 212}}).
; [[zpi|288zpi]]  
* Step size: 20.736{{c}}, octave size: 1202.69{{c}}
Stretching the octave of 58edo by around 2.5{{c}} results in improved primes 11, 13, 19 and 23, but worse primes 2, 3, 5, 7 and 17. This approximates all harmonics up to 16 within 9.98{{c}}. The tuning 288zpi does this.
{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}
{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}
 
; 58edo
* Step size: 20.690{{c}}, octave size: 1200.00{{c}}
Pure-octaves 58edo approximates all harmonics up to 16 within 8.28{{c}}.
{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}
 
; [[150ed6]]
* Step size: 20.680{{c}}, octave size: 1199.42{{c}}
Compressing the octave of 58edo by around half a cent results in improved primes 3, 5, 7, 11 and 13 but a worse prime 2. This approximates all harmonics up to 16 within 6.02{{c}}. The tuning 150ed6 does this.
{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}
 
; [[92edt]]  
* Step size: 20.673{{c}}, octave size: 1199.06{{c}}
Compressing the octave of 58edo by around 1{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 4.60{{c}}. The tuning 92edt does this.
{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}
{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}
 
; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]
* Step size: 20.666{{c}}, octave size: 1198.63{{c}}
Compressing the octave of 58edo by just under 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 5.49{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.
{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}
{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}
 
; [[WE|58et, 13-limit WE tuning]]
* Step size: 20.663{{c}}, octave size: 1198.45{{c}}
Compressing the octave of 58edo by just over 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 6.18{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}
{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}


== Scales ==
== Scales ==
Line 698: Line 714:
== Music ==
== Music ==
; [[Jeff Brown]]
; [[Jeff Brown]]
* [https://www.youtube.com/watch?v=0373hBH87LY ''Fruitbats in Formation'']
* [https://www.youtube.com/watch?v=0373hBH87LY ''Fruitbats in Formation''] (2023)
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/4J4MNno-4PA ''58edo improv''] (2025)
* [https://www.youtube.com/shorts/7gkRyld5OU8 ''Waltz in 58edo''] (2025)


; [[Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=XXMjoUxVfLs ''We Wish You A Larry Christmas''] (2024) – larry in 58edo
* [https://www.youtube.com/watch?v=XXMjoUxVfLs ''We Wish You A Larry Christmas''] (2024) – in larry, 58edo tuning


; [[Cam Taylor]]
; [[Cam Taylor]]
* [https://youtu.be/Keclakcqie8 58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone]
* [https://www.youtube.com/watch?v=Keclakcqie8 ''58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone''] (2021)


[[Category:Buzzard]]
[[Category:Buzzard]]
[[Category:Diaschismic]]
[[Category:Diaschismic]]
[[Category:Harry]]
[[Category:Harry]]
[[Category:Harry Partch]]
[[Category:Hemififths]]
[[Category:Hemififths]]
[[Category:Listen]]
[[Category:Myna]]
[[Category:Myna]]
[[Category:Mystery]]
[[Category:Mystery]]
[[Category:Harry Partch]]
[[Category:Listen]]

Latest revision as of 23:47, 26 August 2025

← 57edo 58edo 59edo →
Prime factorization 2 × 29
Step size 20.6897 ¢ 
Fifth 34\58 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 6:4 (124.1 ¢ : 82.76 ¢)
Consistency limit 17
Distinct consistency limit 11
English Wikipedia has an article on:

58 equal divisions of the octave (abbreviated 58edo or 58ed2), also called 58-tone equal temperament (58tet) or 58 equal temperament (58et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 58 equal parts of about 20.7 ¢ each. Each step represents a frequency ratio of 21/58, or the 58th root of 2.

Theory

58edo is a strong system in the 11-, 13- and 17-limit. It is the smallest edo which is consistent through the 17-odd-limit, and is also the smallest distinctly consistent in the 11-odd-limit (the first equal temperament to map the entire 11-odd-limit tonality diamond to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note Genesis scale of Harry Partch.

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. Since 58 = 2 × 29, 58edo shares the same excellent perfect fifth with 29edo. It is the last edo to have exactly one diatonic perfect fifth and no 5edo or 7edo fifths.

As an equal temperament, 58et tempers out 2048/2025 in the 5-limit; 126/125, 1728/1715, and 5120/5103 in the 7-limit; 176/175, 243/242, 441/440, 540/539, and 896/891 in the 11-limit; 144/143, 351/350, 364/363 in the 13-limit. It supports hemififths, myna, diaschismic, harry, mystery, buzzard, thuja temperaments plus a number of gravity family extensions, and supplies the optimal patent val for the 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-3 temperaments thrush, bluebird, aplonis and jofur.

Of all edos which map the syntonic comma (81/80) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1 ¢ narrower than the just interval.

Prime harmonics

Approximation of prime harmonics in 58edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.49 +6.79 +3.59 +7.30 +7.75 -1.51 -7.86 -7.58 +4.91 -7.10
Relative (%) +0.0 +7.2 +32.8 +17.3 +35.3 +37.4 -7.3 -38.0 -36.7 +23.7 -34.3
Steps
(reduced) 		58
(0) 		92
(34) 		135
(19) 		163
(47) 		201
(27) 		215
(41) 		237
(5) 		246
(14) 		262
(30) 		282
(50) 		287
(55)

Subsets and supersets

58edo contains 2edo and 29edo as subsets.

Intervals

# Cents Approximate ratios* Ups and downs notation
0 0.0 1/1 D
1 20.7 56/55, 64/63, 81/80, 91/90, 105/104 ^D, v3E♭
2 41.4 36/35, 40/39, 45/44, 49/48, 50/49, 55/54 ^^D, vvE♭
3 62.1 26/25, 27/26, 28/27, 33/32 ^3D, vE♭
4 82.8 21/20, 22/21, 25/24 vvD♯, E♭
5 103.4 16/15, 17/16, 18/17 vD♯, ^E♭
6 124.1 14/13, 15/14 D♯, ^^E♭
7 144.8 12/11, 13/12 ^D♯, v3E
8 165.5 11/10 ^^D♯, vvE
9 186.2 10/9 ^3D♯, vE
10 206.9 9/8, 17/15 E
11 227.6 8/7 ^E, v3F
12 248.3 15/13 ^^E, vvF
13 269.0 7/6 ^3E, vF
14 289.7 13/11, 20/17 F
15 310.3 6/5 ^F, v3G♭
16 331.0 17/14, 40/33 ^^F, vvG♭
17 351.7 11/9, 16/13 ^3F, vG♭
18 372.4 21/17, 26/21 vvF♯, G♭
19 393.1 5/4 vF♯, ^G♭
20 413.8 14/11 F♯, ^^G♭
21 434.5 9/7 ^F♯, v3G
22 455.2 13/10, 17/13, 22/17 ^^F♯, vvG
23 475.9 21/16 ^3F♯, vG
24 496.6 4/3 G
25 517.2 27/20 ^G, v3A♭
26 537.9 15/11 ^^G, vvA♭
27 558.6 11/8, 18/13 ^3G, vA♭
28 579.3 7/5 vvG♯, A♭
29 600.0 17/12, 24/17 vG♯, ^A♭
30 620.7 10/7 G♯, ^^A♭
31 641.4 13/9, 16/11 ^G♯, v3A
32 662.1 22/15 ^^G♯, vvA
33 682.8 40/27 ^3G♯, vA
34 703.4 3/2 A
35 724.1 32/21 ^A, v3B♭
36 744.8 17/11, 20/13, 26/17 ^^A, vvB♭
37 765.5 14/9 ^3A, vB♭
38 786.2 11/7 vvA♯, B♭
39 806.9 8/5 vA♯, ^B♭
40 827.6 21/13, 34/21 A♯, ^^B♭
41 848.3 13/8, 18/11 ^A♯, v3B
42 869.0 28/17, 33/20 ^^A♯, vvB
43 889.7 5/3 ^3A♯, vB
44 910.3 17/10, 22/13 B
45 931.0 12/7 ^B, v3C
46 951.7 26/15 ^^B, vvC
47 972.4 7/4 ^3B, vC
48 993.1 16/9, 30/17 C
49 1013.8 9/5 ^C, v3D♭
50 1034.5 20/11 ^^C, vvD♭
51 1055.2 11/6, 24/13 ^3C, vD♭
52 1075.9 13/7, 28/15 vvC♯, D♭
53 1096.6 15/8, 17/9, 32/17 vC♯, ^D♭
54 1117.2 21/11, 40/21, 48/25 C♯, ^^D♭
55 1137.9 25/13, 27/14, 52/27, 64/33 ^C♯, v3D
56 1158.6 35/18, 39/20, 49/25, 88/45, 96/49, 108/55 ^^C♯, vvD
57 1179.3 55/28, 63/32, 160/81, 180/91, 208/105 ^3C♯, vD
58 1200.0 2/1 D

* As a 17-limit temperament, inconsistently mapped intervals in italic

Notation

Ups and downs notation

58edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Alternatively, a combination of quarter tone accidentals and arrow accidentals from Helmholtz–Ellis notation can be used.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8055/5433/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8055/5433/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8055/5433/32

Hemipyth notation

Hemipyth notation for 58edo (SW3-style)
# Cents Note names
on D
0 0.0 D
2 41.4 α𝄳
5 103.4 α
7 144.8 E𝄳
10 206.9 E
12 248.3 β𝄳
14 289.7 F
15 310.3 β
17 351.7 F‡
19 393.1 γ
22 455.2 γ‡
24 496.6 G
27 558.6 G‡
29 600.0 δ
31 641.4 A𝄳
34 703.4 A
36 744.8 ε𝄳
39 806.9 ε
41 848.3 B𝄳
43 889.7 ζ
44 910.3 B
46 951.7 ζ‡
48 993.1 C
51 1055.2 C‡
53 1096.6 η
56 1158.6 η‡
58 1200.0 D

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 58edo. Prime harmonics are in bold.

As 58edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 58edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/11, 22/13 0.445 2.2
11/10, 20/11 0.513 2.5
15/13, 26/15 0.535 2.6
9/7, 14/9 0.601 2.9
13/10, 20/13 0.958 4.6
15/11, 22/15 0.980 4.7
3/2, 4/3 1.493 7.2
7/6, 12/7 2.095 10.1
9/8, 16/9 2.987 14.4
7/5, 10/7 3.202 15.5
7/4, 8/7 3.588 17.3
11/7, 14/11 3.715 18.0
9/5, 10/9 3.803 18.4
13/7, 14/13 4.160 20.1
11/9, 18/11 4.316 20.9
15/14, 28/15 4.695 22.7
13/9, 18/13 4.762 23.0
5/3, 6/5 5.296 25.6
11/6, 12/11 5.809 28.1
13/12, 24/13 6.255 30.2
5/4, 8/5 6.790 32.8
11/8, 16/11 7.303 35.3
13/8, 16/13 7.748 37.4
15/8, 16/15 8.283 40.0

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 2048/2025, 1594323/1562500 [⟨58 92 135]] −1.29 1.22 5.89
2.3.5.7 126/125, 1728/1715, 2048/2025 [⟨58 92 135 163]] −1.29 1.05 5.10
2.3.5.7.11 126/125, 176/175, 243/242, 896/891 [⟨58 92 135 163 201]] −1.45 1.00 4.83
2.3.5.7.11.13 126/125, 144/143, 176/175, 196/195, 364/363 [⟨58 92 135 163 201 215]] −1.56 0.94 4.56
2.3.5.7.11.13.17 126/125, 136/135, 144/143, 176/175, 196/195, 364/363 [⟨58 92 135 163 201 215 237]] −1.28 1.10 5.33
  • 58et has a lower relative error than any previous equal temperaments in the 13-limit, and the next equal temperament that does better in this subgroup is 72.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 3\58 62.1 28/27 Unicorn / alicorn / qilin
1 11\58 227.6 8/7 Gorgik
1 13\58 269.0 7/6 Infraorwell
1 15\58 310.3 6/5 Myna
1 17\58 351.7 49/40 Hemififths
1 19\58 393.1 64/51 Emmthird
1 23\58 475.9 21/16 Buzzard / subfourth
1 27\58 558.6 11/8 Thuja
2 3\58 62.1 28/27 Monocerus
2 1\58 20.7 81/80 Bicommatic
2 9\58 186.2 10/9 Secant
2 17\58
(12\58) 		351.7
(248.3) 		11/9
(15/13) 		Sruti
2 21\58
(8\58) 		434.5
(165.5) 		9/7
(11/10) 		Echidna
2 24\58
(5\58) 		496.6
(103.4) 		4/3
(17/16) 		Diaschismic
2 25\58
(4\58) 		517.2
(82.8) 		27/20
(21/20) 		Harry
29 19\58
(1\58) 		393.1
(20.7) 		5/4
(91/90) 		Mystery

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

58et can also be detempered to semihemi (58 & 140), supers (58 & 152), condor (58 & 159), and eagle (58 & 212).

Octave stretch or compression

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 92edt or 150ed6.

What follows is a comparison of stretched- and compressed-octave 58edo tunings.

288zpi
  • Step size: 20.736 ¢, octave size: 1202.69 ¢

Stretching the octave of 58edo by around 2.5 ¢ results in improved primes 11, 13, 19 and 23, but worse primes 2, 3, 5, 7 and 17. This approximates all harmonics up to 16 within 9.98 ¢. The tuning 288zpi does this.

Approximation of harmonics in 288zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.69 +5.76 +5.38 -7.69 +8.44 -9.59 +8.06 -9.22 -5.00 -4.12 -9.60
Relative (%) +13.0 +27.8 +25.9 -37.1 +40.7 -46.3 +38.9 -44.5 -24.1 -19.9 -46.3
Step 58 92 116 134 150 162 174 183 192 200 207
Approximation of harmonics in 288zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.02 -6.91 -1.93 -9.98 +9.48 -6.53 +3.54 -2.31 -3.84 -1.43 +4.56 -6.92
Relative (%) -14.6 -33.3 -9.3 -48.1 +45.7 -31.5 +17.1 -11.2 -18.5 -6.9 +22.0 -33.3
Step 214 220 226 231 237 241 246 250 254 258 262 265
58edo
  • Step size: 20.690 ¢, octave size: 1200.00 ¢

Pure-octaves 58edo approximates all harmonics up to 16 within 8.28 ¢.

Approximation of harmonics in 58edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 +1.49 +0.00 +6.79 +1.49 +3.59 +0.00 +2.99 +6.79 +7.30 +1.49
Relative (%) +0.0 +7.2 +0.0 +32.8 +7.2 +17.3 +0.0 +14.4 +32.8 +35.3 +7.2
Steps
(reduced) 		58
(0) 		92
(34) 		116
(0) 		135
(19) 		150
(34) 		163
(47) 		174
(0) 		184
(10) 		193
(19) 		201
(27) 		208
(34)
Approximation of harmonics in 58edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.75 +3.59 +8.28 +0.00 -1.51 +2.99 -7.86 +6.79 +5.08 +7.30 -7.58 +1.49
Relative (%) +37.4 +17.3 +40.0 +0.0 -7.3 +14.4 -38.0 +32.8 +24.6 +35.3 -36.7 +7.2
Steps
(reduced) 		215
(41) 		221
(47) 		227
(53) 		232
(0) 		237
(5) 		242
(10) 		246
(14) 		251
(19) 		255
(23) 		259
(27) 		262
(30) 		266
(34)
150ed6
  • Step size: 20.680 ¢, octave size: 1199.42 ¢

Compressing the octave of 58edo by around half a cent results in improved primes 3, 5, 7, 11 and 13 but a worse prime 2. This approximates all harmonics up to 16 within 6.02 ¢. The tuning 150ed6 does this.

Approximation of harmonics in 150ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.58 +0.58 -1.15 +5.45 +0.00 +1.97 -1.73 +1.15 +4.87 +5.30 -0.58
Relative (%) -2.8 +2.8 -5.6 +26.3 +0.0 +9.5 -8.4 +5.6 +23.5 +25.6 -2.8
Steps
(reduced) 		58
(58) 		92
(92) 		116
(116) 		135
(135) 		150
(0) 		163
(13) 		174
(24) 		184
(34) 		193
(43) 		201
(51) 		208
(58)
Approximation of harmonics in 150ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.61 +1.39 +6.02 -2.31 -3.87 +0.58 -10.31 +4.29 +2.54 +4.72 -10.19 -1.15
Relative (%) +27.1 +6.7 +29.1 -11.2 -18.7 +2.8 -49.8 +20.7 +12.3 +22.8 -49.3 -5.6
Steps
(reduced) 		215
(65) 		221
(71) 		227
(77) 		232
(82) 		237
(87) 		242
(92) 		246
(96) 		251
(101) 		255
(105) 		259
(109) 		262
(112) 		266
(116)
92edt
  • Step size: 20.673 ¢, octave size: 1199.06 ¢

Compressing the octave of 58edo by around 1 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 4.60 ¢. The tuning 92edt does this.

Approximation of harmonics in 92edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.94 +0.00 -1.88 +4.60 -0.94 +0.94 -2.82 +0.00 +3.66 +4.04 -1.88
Relative (%) -4.6 +0.0 -9.1 +22.2 -4.6 +4.6 -13.7 +0.0 +17.7 +19.5 -9.1
Steps
(reduced) 		58
(58) 		92
(0) 		116
(24) 		135
(43) 		150
(58) 		163
(71) 		174
(82) 		184
(0) 		193
(9) 		201
(17) 		208
(24)
Approximation of harmonics in 92edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.26 +0.00 +4.60 -3.77 -5.35 -0.94 +8.82 +2.72 +0.94 +3.10 +8.84 -2.82
Relative (%) +20.6 +0.0 +22.2 -18.2 -25.9 -4.6 +42.7 +13.1 +4.6 +15.0 +42.7 -13.7
Steps
(reduced) 		215
(31) 		221
(37) 		227
(43) 		232
(48) 		237
(53) 		242
(58) 		247
(63) 		251
(67) 		255
(71) 		259
(75) 		263
(79) 		266
(82)
289zpi / 58et, 7-limit WE tuning
  • Step size: 20.666 ¢, octave size: 1198.63 ¢

Compressing the octave of 58edo by just under 1.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 5.49 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06 ¢.

Approximation of harmonics in 289zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.37 -0.68 -2.74 +3.60 -2.06 -0.27 -4.12 -1.37 +2.22 +2.55 -3.43
Relative (%) -6.6 -3.3 -13.3 +17.4 -9.9 -1.3 -19.9 -6.6 +10.8 +12.3 -16.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in 289zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.66 -1.64 +2.91 -5.49 -7.11 -2.74 +6.99 +0.85 -0.95 +1.18 +6.88 -4.80
Relative (%) +12.9 -7.9 +14.1 -26.6 -34.4 -13.2 +33.8 +4.1 -4.6 +5.7 +33.3 -23.2
Step 215 221 227 232 237 242 247 251 255 259 263 266
58et, 13-limit WE tuning
  • Step size: 20.663 ¢, octave size: 1198.45 ¢

Compressing the octave of 58edo by just over 1.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 6.18 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 58et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.55 -0.96 -3.09 +3.19 -2.51 -0.76 -4.64 -1.92 +1.65 +1.95 -4.05
Relative (%) -7.5 -4.6 -15.0 +15.4 -12.1 -3.7 -22.4 -9.3 +8.0 +9.4 -19.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in 58et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.02 -2.30 +2.23 -6.18 -7.82 -3.46 +6.25 +0.10 -1.72 +0.40 +6.09 -5.60
Relative (%) +9.8 -11.1 +10.8 -29.9 -37.9 -16.8 +30.2 +0.5 -8.3 +1.9 +29.5 -27.1
Step 215 221 227 232 237 242 247 251 255 259 263 266

Scales

Instruments

Music

Jeff Brown
Bryan Deister
Francium
Cam Taylor
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