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== Theory ==
== Theory ==


58edo 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-limits]]. It is the smallest [[EDO|equal temperament]] which is [[consistent]] through the 17-odd limit, and is also the smallest uniquely consistent in the 11-odd limit (the first et to map the entire 11-limit [[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]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[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]].
58edo 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]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest uniquely consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-limit [[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]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]] and [[thuja]] [[Regular temperament|temperament]]s, and supplies the [[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-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[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]].
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]].
 
{{Primes in edo|58}}


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
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|}
|}


== Just approximation ==
== Regular temperament properties ==
=== Selected just intervals ===
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-all"
! rowspan="2" | Subgroup
! colspan="2" |
! rowspan="2" | [[Comma list]]
! prime 2
! rowspan="2" | [[Mapping]]
! prime 3
! rowspan="2" | Optimal<br>8ve stretch (¢)
! prime 5
! colspan="2" | Tuning error
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|-
|-
! rowspan="2" |Error
! [[TE error|Absolute]] (¢)
! absolute (¢)
! [[TE simple badness|Relative]] (%)
| 0.00
| +1.59
| +6.79
| +3.59
| +7.30
| +7.75
| -1.51
| -7.86
| -7.58
|-
|-
! relative (%)
| 2.3.5
| 0.0
| 2048/2025, 1594323/1562500
| +7.2
| [{{val| 58 92 135 }}]
| +32.8
| -1.29
| +17.3
| 1.22
| +35.3
| 5.89
| +37.4
| -7.3
| -38.0
| -36.7
|}
 
=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 58et.  
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 11-limit
! 13-limit
! 17-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5.7
| -0.47
| 126/125, 1728/1715, 2048/2025
| -1.29
| [{{val| 58 92 135 163 }}]
| -1.29
| -1.29
| 1.05
| 5.10
|-
| 2.3.5.7.11
| 126/125, 176/175, 243/242, 896/891
| [{{val| 58 92 135 163 201 }}]
| -1.45
| -1.45
| 1.00
| 4.83
|-
| 2.3.5.7.11.13
| 126/125, 144/143, 176/175, 196/195, 364/363
| [{{val| 58 92 135 163 201 215 }}]
| -1.56
| -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
| [{{val| 58 92 135 163 201 215 237 }}]
| -1.28
| -1.28
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.47
| 1.22
| 1.05
| 1.00
| 0.94
| 1.10
| 1.10
|-
! [[TE simple badness|relative]] (%)
| 2.28
| 5.89
| 5.10
| 4.83
| 4.56
| 5.33
| 5.33
|}
|}


* 58et has a lower relative error than any previous ETs in the 13-limit. The next ET that does better in this subgroup is 72.  
58et is lower in relative error than any previous equal temperaments in the 13-limit, and the next ET that does better in this subgroup is 72.  


== Rank two temperaments ==
=== Rank-2 temperaments ===


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
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