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! colspan="2" | nearest edomapping
! colspan="2" | Nearest edomapping
| 53
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|-
! colspan="2" | [[fifthspan]]
! colspan="2" | [[Fifthspan]]
| 0
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| +1
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[[File:53-edo spiral.png|702x702px]]
[[File:53-edo spiral.png|702x702px]]


== Just Approximation ==
== JI approximation ==
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.


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One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.
One notable property of 53EDO is that it offers good approximations for both just and Pythagorean major thirds.


The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.


=== Selected just intervals by error ===
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 53edo. Octave-reduced prime harmonics are '''bolded'''; inconsistent intervals are in ''italic''.
The following table shows how [[15-odd-limit intervals]] are represented in 53edo. Octave-reduced prime harmonics are '''bolded'''; inconsistent intervals are in ''italic''.
{| class="wikitable center-all"
{| class="wikitable center-all"
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|}
|}


=== Temperament measures ===
== Regular temperament properties ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 53et.
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-all"
! rowspan="2" | Subgroup
! colspan="2" |
! rowspan="2" | [[Comma list]]
! 3-limit
! rowspan="2" | [[Mapping]]
! 5-limit
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 7-limit
! colspan="2" | Tuning error
! 11-limit
|-
! 13-limit
! [[TE error|Absolute]] (¢)
! 2.3.5.7.13.19
! [[TE simple badness|Relative]] (%)
! 2.3.5.7.13.19.23
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3
| {{monzo| -84 53 }}
| [{{val| 53 84 }}]
| +0.022
| +0.022
| 0.022
| 0.10
|-
| 2.3.5
| 15625/15552, 32805/32768
| [{{val| 53 84 123 }}]
| +0.216
| +0.216
| 0.276
| 1.22
|-
| 2.3.5.7
| 225/224, 1728/1715, 3125/3087
| [{{val| 53 84 123 149 }}]
| -0.262
| -0.262
| 0.861
| 3.81
|-
| 2.3.5.7.11
| 99/98, 121/120, 176/175, 2200/2187
| [{{val| 53 84 123 149 183 }}]
| +0.248
| +0.248
| 1.279
| 5.64
|-
| 2.3.5.7.11.13
| 99/98, 121/120, 169/168, 176/175, 275/273
| [{{val| 53 84 123 149 183 196 }}]
| +0.332
| +0.332
| +0.075
| -0.115
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.022
| 0.276
| 0.861
| 1.279
| 1.183
| 1.183
| 0.850
| 5.22
| 0.915
|-
|-
! [[TE simple badness|relative]] (%)
| 2.3.5.7.11.13.19
| 0.10
| 99/98, 121/120, 169/168, 176/175, 209/208, 275/273
| 1.22
| [{{val| 53 84 123 149 183 196 225 }}]
| 3.81
| +0.391
| 5.64
| 1.105
| 5.22
| 4.88
| 3.75
| 4.04
|}
|}


* 53et has a lower relative error than any previous ETs in the 3-, 5-, and 13-limit. The next ET that does better in these subgroups is 306, 118, and 58, respectively.  
53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next ETs better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next ET better in either subgroup is 130.  
* 53et is prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups. The next ET that does better in either case is 130.  


== Linear temperaments ==
=== Linear temperaments ===
* [[List of edo-distinct 53et rank two temperaments]]
* [[List of edo-distinct 53et rank two temperaments]]
* [[Schismic-Mercator equivalence continuum]]
* [[Schismic-Mercator equivalence continuum]]
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[[Category:53edo| ]] <!-- main article: sort under ' ' on category page as usual -->
[[Category:53edo| ]] <!-- main article: sort under ' ' on category page as usual -->
[[Category:Equal divisions of the octave]]
[[Category:Amity]]
[[Category:Amity]]
[[Category:Athene]]
[[Category:Athene]]
[[Category:Big brother]]
[[Category:Big brother]]
[[Category:Equal divisions of the octave]]
[[Category:Hanson]]
[[Category:Hanson]]
[[Category:Kleismic]]
[[Category:Island]]
[[Category:Island]]
[[Category:Kleismic]]
[[Category:Listen]]
[[Category:Marvel]]
[[Category:Marvel]]
[[Category:Orson]]
[[Category:Orwell]]
[[Category:Orwell]]
[[Category:Schismic]]
[[Category:Pythagorean]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Schismic]]
[[Category:Semicomma]]
[[Category:Zeta]]
[[Category:Zeta]]
[[Category:Listen]]
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