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Meantone: Difference between revisions

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→Tunings: + destretched tunings
→Tunings: move the target tunings that don't correspond to significant eigenintervals out of the tuning spectra
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== Tunings ==
== Tunings ==
Common meantone tunings can be classified into [[target tuning]]s, edo tunings, prime-optimized tunings and others. In target 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.  
Common meantone tunings can be classified into [[eigenmonzo|eigenmonzo (unchanged-interval)]] 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.  


; Notable target tunings
; Notable eigenmonzo (unchanged-interval) tunings
* [[1/2-comma meantone]] – with eigenmonzo [[10/9]]
* [[1/2-comma meantone]] – with eigenmonzo [[10/9]]
* [[1/3-comma meantone]] – with eigenmonzo [[5/3]]
* [[1/3-comma meantone]] – with eigenmonzo [[5/3]]
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|-
|-
! Equilateral
! Equilateral
| CEE: ~3/2 = 696.8947{{c}}<br>(4/17-comma)
| CEE: ~3/2 = 696.8947{{c}}<br>(4/17 comma)
| CSEE: ~3/2 = 696.4534{{c}}<br>(11/43-comma)
| CSEE: ~3/2 = 696.4534{{c}}<br>(11/43 comma)
| POEE: ~3/2 = 695.2311{{c}}
| POEE: ~3/2 = 695.2311{{c}}
|-
|-
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|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~3/2 = 697.3738{{c}}<br>(36/169-comma)
| CBE: ~3/2 = 697.3738{{c}}<br>(36/169 comma)
| CSBE: ~3/2 = 696.7868{{c}}<br>(31/129-comma)
| CSBE: ~3/2 = 696.7868{{c}}<br>(31/129 comma)
| POBE: ~3/2 = 696.2984{{c}}
| POBE: ~3/2 = 696.2984{{c}}
|}
|}
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| CSBE: ~3/2 = 696.6306{{c}}
| CSBE: ~3/2 = 696.6306{{c}}
| POBE: ~3/2 = 696.4596{{c}}
| POBE: ~3/2 = 696.4596{{c}}
|}
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings
|-
! Target
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~3/2 = 696.578{{c}}
| 5/4
|-
| 7-odd-limit
| ~3/2 = 696.578{{c}}
| 5/4
|-
| 9-odd-limit
| ~3/2 = 696.578{{c}}
| 5/4
|}
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings
|-
! Target
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~3/2 = 696.165{{c}}<br>(7/26 comma)
| {{Monzo| -13 -2 7 }}
|-
| 7-odd-limit
| ~3/2 = 696.648{{c}}
| {{Monzo| -55 -11 1 25 }}
|-
| 9-odd-limit
| ~3/2 = 696.436{{c}}
| {{Monzo| 19 9 -1 -11 }}
|}
|}


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|-
|-
|
|
| {{nowrap|''f''<sup>4</sup> &minus; 2''f'' &minus; 2 {{=}} 0}}
| {{nowrap| ''f''<sup>4</sup> − 2''f'' − 2 {{=}} 0 }}
| 695.630
| 695.630
| 1–3–5 equal-beating tuning, Wilson's "metameantone" ([[DR]] 4:5:6), virtually 5/17 comma
| 1–3–5 equal-beating tuning, Wilson's "metameantone" ([[DR]] 4:5:6), virtually 5/17 comma
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| [[15/14]]
| [[15/14]]
| 696.111
| 696.111
|
|  
|-
|-
|
|  
| [[78125/73728]]
| [[78125/73728]]
| 696.165
| 696.165
| [[7/26-comma meantone|7/26 comma]], [[5-odd-limit]] least squares
| [[7/26-comma meantone|7/26 comma]], 5-odd-limit least squares
|-
|-
| {{nowrap|(8 &minus; φ)\11}}
| {{nowrap| (8 − φ)\11 }}
|
|
| 696.214
| 696.214
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| 696.399
| 696.399
|
|
|-
|
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 9-odd-limit least squares
|-
|-
|
|
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| 696.626
| 696.626
|
|
|-
|
| {{monzo| -55 -11 1 25 }}
| 696.648
| [[7-odd-limit]] least squares
|-
|-
| [[31edo|18\31]]
| [[31edo|18\31]]
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| [[1875/1024]]
| [[1875/1024]]
| 696.895
| 696.895
| [[4/17-comma meantone|4/17 comma]]; 2.3.5 [[CEE]] tuning
| [[4/17-comma meantone|4/17 comma]]; 5-limit [[CEE]] tuning
|-
|-
|
|
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