58edo: Difference between revisions

{{ED intro}}

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 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 [[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]].

58edo contains [[2edo]] and [[29edo]] as subsets.

! Approximate ratios*

| 0.0

| 20.7

| [[56/55]], [[64/63]], [[81/80]], [[91/90]], [[105/104]]

| 41.4

| [[36/35]], [[40/39]], [[45/44]], [[49/48]], [[50/49]], [[55/54]]

| 62.1

| 82.8

| [[21/20]], [[22/21]], ''[[25/24]]''

| 103.4

| 124.1

| [[14/13]], [[15/14]]

| 144.8

| 165.5

| 186.2

| 206.9

| 227.6

| 248.3

| 269.0

| 289.7

| 310.3

| 331.0

| [[17/14]], [[40/33]]

| 351.7

| 372.4

| [[21/17]], [[26/21]]

| 393.1

| 413.8

| 434.5

| 455.2

| 475.9

| 496.6

| 517.2

| 537.9

| 558.6

| 579.3

| 600.0

| 620.7

| 641.4

| 662.1

| 682.8

| 703.4

| 724.1

| 744.8

| [[17/11]], [[20/13]], [[26/17]]

| 765.5

| 786.2

| 806.9

| 827.6

| [[21/13]], [[34/21]]

| 848.3

| 869.0

| [[28/17]], [[33/20]]

| 889.7

| 910.3

| [[17/10]], [[22/13]]

| 931.0

| 951.7

| 972.4

| 993.1

| 1013.8

| 1034.5

| 1055.2

| 1075.9

| 1096.6

| [[15/8]], [[17/9]], [[32/17]]

| 1117.2

| [[21/11]], [[40/21]], ''[[48/25]]''

| 1137.9

| [[25/13]], [[27/14]], [[52/27]], [[64/33]]

| 1158.6

| [[35/18]], [[39/20]], [[49/25]], [[88/45]], [[96/49]], [[108/55]]

| 1179.3

| [[55/28]], [[63/32]], [[160/81]], [[180/91]], [[208/105]]

| 1200.0

 

 

 

 

 

 

=== Sagittal notation ===

==== Evo flavor ====

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 662 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 120 80 230 106 [[55/54]]

default [[File:58-EDO_Evo_Sagittal.svg]]

 

 

{| class="wikitable center-all right-2 center-3 mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)

! Note names<br />on D

| 0.0

| 41.4

|-

| 103.4

| 496.6

| 58

| 1200.0

| D

== Approximation to JI ==

! rowspan="2" | [[Comma list]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

| 2048/2025, [[1594323/1562500]]

| {{Mapping| 58 92 135 }}

| −1.29

| {{Mapping| 58 92 135 163 }}

| −1.29

| {{Mapping| 58 92 135 163 201 }}

| −1.45

| {{Mapping| 58 92 135 163 201 215 }}

| −1.56

| {{Mapping| 58 92 135 163 201 215 237 }}

| −1.28

* 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 [[72edo|72]].  

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods<br>per 8ve

! Generator*

! Cents*

! Associated<br>ratio*

| 62.1

| 227.6

| 269.0

| 310.3

| 351.7

| 393.1

| 475.9

| 558.6

| 62.1

| 20.7

| [[Bicommatic]]

| 186.2

| 351.7<br>(248.3)

| 434.5<br>(165.5)

| 496.6<br>(103.4)

| 517.2<br>(82.8)

| 393.1<br>(20.7)

; [[150ed6]]  

{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}

 

; [[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]]

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=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}

 

* 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=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}

 

* Step size: 20.646{{c}}, octave size: 1197.45{{c}}

* [https://www.youtube.com/watch?v=0373hBH87LY ''Fruitbats in Formation''] (2023)

 

 

 

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