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'''Septimal meantone''' or '''7-limit meantone''' is a natural extension of meantone which also addresses septimal intervals including but not limited to [[7/4]], [[7/5]], and [[7/6]]. By extending the [[circle of fifths]], consonant septimal intervals start to appear. For example, 7/4 is represented by an augmented sixth and is notably present in the augmented sixth chord; it can also be seen as a diesis-flat minor seventh.  
'''Septimal meantone''' or '''7-limit meantone''' is a natural extension of meantone which also addresses septimal intervals including but not limited to [[7/4]], [[7/5]], and [[7/6]]. By extending the [[circle of fifths]], consonant septimal intervals start to appear. For example, 7/4 is represented by an augmented sixth and is notably present in the augmented sixth chord; it can also be seen as a diesis-flat minor seventh.  


See [[meantone vs meanpop]] for a comparison of undecimal (11-limit) extensions.
See [[huygens vs meanpop]] for a comparison of undecimal (11-limit) extensions.


=== Other septimal extensions ===
=== Other septimal extensions ===
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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.  
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 unchanged-interval (eigenmonzo) 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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* [[Lucy tuning]]
* [[Lucy tuning]]
* Equal beating tunings
* Equal beating tunings
* 5-limit [[DKW theory|DKW]]: ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}


=== Prime-optimized tunings ===
=== Prime-optimized tunings ===
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|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Constrained
! Constrained
! Constrained & skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~3/2 = 696.895{{c}}<br>(4/17-comma)
| CEE: ~3/2 = 696.8947{{c}}<br>(4/17 comma)
| CSEE: ~3/2 = 696.453{{c}}<br>(11/43-comma)
| CSEE: ~3/2 = 696.4534{{c}}<br>(11/43 comma)
| POEE: ~3/2 = 695.2311{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~3/2 = 697.214{{c}}
| CTE: ~3/2 = 697.2143{{c}}
| CWE: ~3/2 = 696.651{{c}}
| CWE: ~3/2 = 696.6512{{c}}
| POTE: ~3/2 = 696.2387{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~3/2 = 697.374{{c}}<br>(36/169-comma)
| CBE: ~3/2 = 697.3738{{c}}<br>(36/169 comma)
| CSBE: ~3/2 = 696.787{{c}}<br>(31/129-comma)
| CSBE: ~3/2 = 696.7868{{c}}<br>(31/129 comma)
| POBE: ~3/2 = 696.2984{{c}}
|}
|}


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|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Constrained
! Constrained
! Constrained & skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~3/2 = 696.884{{c}}
| CEE: ~3/2 = 696.8843{{c}}
| CSEE: ~3/2 = 696.725{{c}}
| CSEE: ~3/2 = 696.7248{{c}}
| POEE: ~3/2 = 696.4375{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~3/2 = 696.952{{c}}
| CTE: ~3/2 = 696.9521{{c}}
| CWE: ~3/2 = 696.656{{c}}
| CWE: ~3/2 = 696.6562{{c}}
| POTE: ~3/2 = 696.4949{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~3/2 = 697.015{{c}}
| CBE: ~3/2 = 697.0147{{c}}
| CSBE: ~3/2 = 696.631{{c}}
| CSBE: ~3/2 = 696.6306{{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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| [[9/5]]
| [[9/5]]
| 691.202
| 691.202
| '''Lower bound of 9-odd-limit diamond tradeoff'''<br />[[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9  
| '''Lower bound of 9-odd-limit diamond tradeoff'''<br>[[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9  
|-
|-
| [[59edo|34\59]]
| [[59edo|34\59]]
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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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$$ n = (g_J - g)/g_c $$
$$ n = (g_J - g)/g_c $$
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}


== Music ==
== Music ==
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