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→Tunings: +some notable prime-based tunings since my optimizer is working so well
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== Tunings ==
== Tunings ==
Common meantone tunings can be classified into eigenmonzo tunings, EDO tunings, prime-optimized tunings and others. In [[eigenmonzo]] tunings such as the [[quarter-comma meantone]], a certain interval is tuned pure and certain others are equally off. EDO tunings like [[31edo]] have rational size relationship between steps, and happen to send an additional comma to unison. Prime-optimized tunings include TE, POTE, CTE, etc. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.  
Common meantone tunings can be classified into eigenmonzo tunings, edo tunings, prime-optimized tunings and others. In [[eigenmonzo]] tunings such as the [[quarter-comma meantone]], a certain interval is tuned pure and certain others are equally off. Edo tunings like [[31edo]] have rational size relationship between steps, and happen to send an additional comma to unison. Prime-optimized tunings are optimized for all intervals. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.  


; Eigenmonzo tunings
; Notable eigenmonzo tunings
* [[1/2 syntonic comma meantone]] – with eigenmonzo [[10/9]]
* [[1/2 syntonic comma meantone]] – with eigenmonzo [[10/9]]
* [[1/3 syntonic comma meantone]] – with eigenmonzo [[5/3]]
* [[1/3 syntonic comma meantone]] – with eigenmonzo [[5/3]]
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* [[1/5 syntonic comma meantone]] – with eigenmonzo [[15/8]]
* [[1/5 syntonic comma meantone]] – with eigenmonzo [[15/8]]
* [[1/6 syntonic comma meantone]] – with eigenmonzo [[45/32]]
* [[1/6 syntonic comma meantone]] – with eigenmonzo [[45/32]]
* Ratwolf tuning
* [[Ratwolf|Ratwolf tuning]]
 
; Prime-optimized tunings
* ~3/2 = 696.2387¢ – 5-limit POTE tuning
* ~3/2 = 697.2143¢ – 5-limit CTE tuning
* ~3/2 = 696.4949¢ – 7-limit POTE tuning
* ~3/2 = 696.9521¢ – 7-limit CTE tuning


; Other optimized tunings
; Other optimized tunings
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* [[Lucy tuning]]
* [[Lucy tuning]]
* Equal beating tuning
* Equal beating tuning
=== Prime-optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 5-limit Prime-Optimized Tunings
|-
! Weight-skew\Order !! Euclidean
|-
| Tenney || CTE<br>~3/2 = 697.2143¢
|-
| Tenney-Weil || CTWE<br>~3/2 = 696.6512¢
|-
| Frobenius || CFE<br>~3/2 = 696.8947¢<br>Eigenmonzo basis: 2.1875 (4/17-comma tuning)
|-
| Frobenius-Weil || CFWE<br>~3/2 = 696.4534¢<br>Eigenmonzo basis: 2.48828125/3 (11/43-comma tuning)
|-
| Benedetti || CBE<br>~3/2 = 697.3738¢<br>Eigenmonzo basis: 2.{{monzo| 0 25 36 }} (36/169-comma tuning)
|-
| Benedetti-Weil || CBWE<br>~3/2 = 696.7868¢<br>Eigenmonzo basis: 2.{{monzo| 0 5 31 }} (31/129-comma tuning)
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 7-limit Prime-Optimized Tunings
|-
! Weight-skew\Order !! Euclidean
|-
| Tenney || CTE<br>~3/2 = 696.9521¢
|-
| Tenney-Weil || CTWE<br>~3/2 = 696.6562¢
|-
| Frobenius || CFE<br>~3/2 = 696.8843¢<br>Eigenmonzo basis: 2.{{monzo| 0 1 4 10 }}
|-
| Frobenius-Weil || CFWE<br>~3/2 = 696.7248¢<br>Eigenmonzo basis: 2.4117715/9
|-
| Benedetti || CBE<br>~3/2 = 697.0147¢<br>Eigenmonzo basis: 2.{{monzo| 0 1225 1764 2250 }}
|-
| Benedetti-Weil || CBWE<br>~3/2 = 696.6306¢<br>Eigenmonzo basis: 2.{{monzo| 0 -3290 3171 7215 }}
|}


=== Tuning spectrum ===
=== Tuning spectrum ===
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