Meantone: Difference between revisions

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== Tunings ==

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 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]]

* [[2/7-comma meantone]] ~~–~~ with eigenmonzo [[25/24]]

* [[2/7-comma meantone]] – with eigenmonzo [[25/24]]

* [[Quarter-comma meantone|1/4-comma meantone]] ~~–~~ with eigenmonzo [[5/4]]

* [[Quarter-comma meantone|1/4-comma meantone]] – with eigenmonzo [[5/4]]

* [[1/5-comma meantone]] ~~–~~ with eigenmonzo [[15/8]]

* [[1/5-comma meantone]] – with eigenmonzo [[15/8]]

* [[1/6-comma meantone]] ~~–~~ with eigenmonzo [[45/32]]

* [[1/6-comma meantone]] – with eigenmonzo [[45/32]]

* [[Ratwolf|Ratwolf tuning]]

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=== Prime-optimized tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 5-limit ~~Prime~~-~~Optimized Tunings~~

|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings

|-

! rowspan="2" |

! colspan="2" | Euclidean

|-

! ~~Unskewed~~

! Constrained

! ~~Skewed~~

! Constrained & skewed

|-

! Equilateral

| CEE: ~3/2 = 696.~~895¢~~ (4/17-comma)

| CEE: ~3/2 = 696.895{{c}} (4/17-comma)

| CSEE: ~3/2 = 696.~~453¢~~ (11/43-comma)

| CSEE: ~3/2 = 696.453{{c}} (11/43-comma)

|-

! Tenney

| CTE: ~3/2 = 697.~~214¢~~

| CTE: ~3/2 = 697.214{{c}}

| CWE: ~3/2 = 696.~~651¢~~

| CWE: ~3/2 = 696.651{{c}}

|-

! Benedetti, Wilson

| CBE: ~3/2 = 697.~~374¢~~ (36/169-comma)

| CBE: ~3/2 = 697.374{{c}} (36/169-comma)

| CSBE: ~3/2 = 696.~~787¢~~ (31/129-comma)

| CSBE: ~3/2 = 696.787{{c}} (31/129-comma)

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit ~~Prime~~-~~Optimized Tunings~~

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings

|-

! rowspan="2" |

! colspan="2" | Euclidean

|-

! ~~Unskewed~~

! Constrained

! ~~Skewed~~

! Constrained & skewed

|-

! Equilateral

| CEE: ~3/2 = 696.~~884¢~~

| CEE: ~3/2 = 696.884{{c}}

| CSEE: ~3/2 = 696.~~725¢~~

| CSEE: ~3/2 = 696.725{{c}}

|-

! Tenney

| CTE: ~3/2 = 696.~~952¢~~

| CTE: ~3/2 = 696.952{{c}}

| CWE: ~3/2 = 696.~~656¢~~

| CWE: ~3/2 = 696.656{{c}}

|-

! Benedetti, Wilson

| CBE: ~3/2 = 697.~~015¢~~

| CBE: ~3/2 = 697.015{{c}}

| CSBE: ~3/2 = 696.~~631¢~~

| CSBE: ~3/2 = 696.631{{c}}

|}

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{| class="wikitable center-all left-4"

|-

! Edo ~~Generator~~

! Edo generator

! [[Eigenmonzo|Eigenmonzo (~~Unchanged~~-interval)]]*

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! Generator (¢)

! Generator (¢)

! Comments

|-

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| [[27/20]]

| 680.449

| Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)

| Full comma (syntonic comma; from here onwards ''comma'' without an adjective refers to syntonic comma)

|-

| '''[[7edo|4\7]]'''

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|-

|

| {{nowrap|''f''4 + 2''f'' ~~−~~ 8 {{=}} 0}}

| {{nowrap|''f''4 + 2''f'' − 8 {{=}} 0}}

| 697.278

| 1–3–5 equal-beating tuning ([[DR]] 3:4:5), virtually 5/23 comma

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| [[3/14-comma meantone|3/14 comma]]

|-

| {{nowrap|(√(10) ~~−~~ 2)\2}}

| {{nowrap|(√(10) − 2)\2}}

|

| 697.367

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|

| '''700.000'''

| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone]] ~~(the difference is too small to appear in the digits provided here)~~

| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone|1/11 comma]]†

|-

|

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| '''Upper bound of 5-odd-limit diamond monotone'''

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

† The difference is too small to appear in the digits provided here

=== Formula for ''n''-comma meantone ===

The [[generator]] ''g'' of ''n''-comma meantone, where ''n'' is a fraction (like 1/5, 2/9, etc.), can be found by

~~<math>\displaystyle~~ g = g_J - ng_c~~</math>~~

$$ g = g_J - ng_c $$

where {{nowrap|''g''''J'' {{=}} 701.955001}} cents is the size of the just perfect fifth, and ''g''c = 21.506290 cents is the size of the syntonic comma.

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Conversely, ''n'' can be found by

~~<math>\displaystyle~~ n = (g_J - g)/g_c~~</math>~~

$$ n = (g_J - g)/g_c $$

=== Other tunings ===

* [[DKW theory|DKW]] (2.3.5): ~2 = ~~1\1~~, ~3/2 = 696.353

* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}

== Music ==

@@ Line 196: / Line 196: @@
 == Tunings ==
-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 eigenmonzo (unchanged-interval) tunings
-* [[1/2-comma meantone]] &ndash; with eigenmonzo [[10/9]]
+* [[1/2-comma meantone]] – with eigenmonzo [[10/9]]
-* [[1/3-comma meantone]] &ndash; with eigenmonzo [[5/3]]
+* [[1/3-comma meantone]] – with eigenmonzo [[5/3]]
-* [[2/7-comma meantone]] &ndash; with eigenmonzo [[25/24]]
+* [[2/7-comma meantone]] – with eigenmonzo [[25/24]]
-* [[Quarter-comma meantone|1/4-comma meantone]] &ndash; with eigenmonzo [[5/4]]
+* [[Quarter-comma meantone|1/4-comma meantone]] – with eigenmonzo [[5/4]]
-* [[1/5-comma meantone]] &ndash; with eigenmonzo [[15/8]]
+* [[1/5-comma meantone]] – with eigenmonzo [[15/8]]
-* [[1/6-comma meantone]] &ndash; with eigenmonzo [[45/32]]
+* [[1/6-comma meantone]] – with eigenmonzo [[45/32]]
 * [[Ratwolf|Ratwolf tuning]]
@@ Line 216: / Line 216: @@
 === Prime-optimized tunings ===
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 5-limit Prime-Optimized Tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings
 |-
 ! rowspan="2" |
 ! colspan="2" | Euclidean
 |-
-! Unskewed
+! Constrained
-! Skewed
+! Constrained & skewed
 |-
 ! Equilateral
-| CEE: ~3/2 = 696.895¢<br>(4/17-comma)
+| CEE: ~3/2 = 696.895{{c}}<br>(4/17-comma)
-| CSEE: ~3/2 = 696.453¢<br>(11/43-comma)
+| CSEE: ~3/2 = 696.453{{c}}<br>(11/43-comma)
 |-
 ! Tenney
-| CTE: ~3/2 = 697.214¢
+| CTE: ~3/2 = 697.214{{c}}
-| CWE: ~3/2 = 696.651¢
+| CWE: ~3/2 = 696.651{{c}}
 |-
 ! Benedetti, <br>Wilson
-| CBE: ~3/2 = 697.374¢<br>(36/169-comma)
+| CBE: ~3/2 = 697.374{{c}}<br>(36/169-comma)
-| CSBE: ~3/2 = 696.787¢<br>(31/129-comma)
+| CSBE: ~3/2 = 696.787{{c}}<br>(31/129-comma)
 |}
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-Optimized Tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
 |-
 ! rowspan="2" |
 ! colspan="2" | Euclidean
 |-
-! Unskewed
+! Constrained
-! Skewed
+! Constrained & skewed
 |-
 ! Equilateral
-| CEE: ~3/2 = 696.884¢
+| CEE: ~3/2 = 696.884{{c}}
-| CSEE: ~3/2 = 696.725¢
+| CSEE: ~3/2 = 696.725{{c}}
 |-
 ! Tenney
-| CTE: ~3/2 = 696.952¢
+| CTE: ~3/2 = 696.952{{c}}
-| CWE: ~3/2 = 696.656¢
+| CWE: ~3/2 = 696.656{{c}}
 |-
 ! Benedetti, <br>Wilson
-| CBE: ~3/2 = 697.015¢
+| CBE: ~3/2 = 697.015{{c}}
-| CSBE: ~3/2 = 696.631¢
+| CSBE: ~3/2 = 696.631{{c}}
 |}
@@ Line 264: / Line 264: @@
 {| class="wikitable center-all left-4"
 |-
-! Edo<br />Generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
-! Generator<br />(¢)
+! Generator<br>(¢)
 ! Comments
 |-
@@ Line 272: / Line 272: @@
 | [[27/20]]
 | 680.449
-| Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)
+| Full comma (syntonic comma; from here onwards ''comma'' without an adjective refers to syntonic comma)
 |-
 | '''[[7edo|4\7]]'''
@@ Line 510: / Line 510: @@
 |-
 |
-| {{nowrap|''f''<sup>4</sup> + 2''f'' &minus; 8 {{=}} 0}}
+| {{nowrap|''f''<sup>4</sup> + 2''f'' − 8 {{=}} 0}}
 | 697.278
 | 1–3–5 equal-beating tuning ([[DR]] 3:4:5), virtually 5/23 comma
@@ Line 529: / Line 529: @@
 | [[3/14-comma meantone|3/14 comma]]
 |-
-| {{nowrap|(√(10) &minus; 2)\2}}
+| {{nowrap|(√(10) − 2)\2}}
 |
 | 697.367
@@ Line 647: / Line 647: @@
 |
 | '''700.000'''
-| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone]] (the difference is too small to appear in the digits provided here)
+| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone|1/11 comma]]†
 |-
 |
@@ Line 669: / Line 669: @@
 | '''Upper bound of 5-odd-limit diamond monotone'''
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
+† The difference is too small to appear in the digits provided here
 === Formula for ''n''-comma meantone ===
 The [[generator]] ''g'' of ''n''-comma meantone, where ''n'' is a fraction (like 1/5, 2/9, etc.), can be found by
-<math>\displaystyle g = g_J - ng_c</math>
+$$ g = g_J - ng_c $$
 where {{nowrap|''g''<sub>''J''</sub> {{=}} 701.955001}} cents is the size of the just perfect fifth, and ''g''<sub>c</sub> = 21.506290 cents is the size of the syntonic comma.
@@ Line 680: / Line 682: @@
 Conversely, ''n'' can be found by
-<math>\displaystyle n = (g_J - g)/g_c</math>
+$$ n = (g_J - g)/g_c $$
 === Other tunings ===
-* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~3/2 = 696.353
+* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}
 == Music ==