Meantone family: Difference between revisions

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The [[7/4]] of septimal meantone is the augmented sixth (C-A#), and other septimal intervals are [[7/6]], the augmented second (C-D#), [[7/5]], the augmented fourth (C-F#), and [[21/16]], the augmented third (C-E#). Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

The [[7/4]] in septimal meantone is the augmented sixth (C-A#), and other septimal intervals are [[7/6]], the augmented second (C-D#), [[7/5]], the augmented fourth (C-F#), and [[21/16]], the augmented third (C-E#). Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

[[Subgroup]]: 2.3.5.7

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[[Minimax tuning]]:

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}

: {{~~Monzo~~ list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}

: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}

: [[Eigenmonzo basis|~~Eigenmonzo~~ (unchanged-interval) basis]]: 2.5

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5

[[Tuning ranges]]:

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[[Badness]]: 0.013707

=== Undecimal meantone aka ~~Huygens~~ ===

=== Undecimal meantone aka huygens ===

Undecimal meantone maps the [[11/8]] to the double augmented third (C-Ex), and tridecimal meantone maps the [[13/8]] to the double augmented fifth (C-Gx). Note 13/11 is conflated with 6/5, represented by the minor third. Note also 11/10~13/12 is the double augmented unison; 12/11, double diminished third; and 14/13, minor second.

Subgroup: 2.3.5.7.11

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Minimax tuning:

* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}

: [{{~~Monzo~~| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]

: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.11/9

: eigenmonzo (unchanged-interval) basis: 2.11/9

Tuning ranges:

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Minimax tuning:

* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.11/9

: eigenmonzo (unchanged-interval) basis: 2.11/9

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===== Meantonic =====

Dubbed ''meantonic'' here, this extension maps the 17/16 to the triple augmented seventh (C-Bx#) octave reduced, and 19/16 to the quadruple augmented unison (C-Cxx). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.

Subgroup: 2.3.5.7.11.13.17

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===== Meantoid =====

Dubbed ''meantoid'' here, this extension maps 17/16~19/18 to the augmented unison (C-C#) and 19/16 to the augmented second (C-D#). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverse, so genuine septendecimal and undevicesimal harmony cannot be expected.

Subgroup: 2.3.5.7.11.13.17

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===== Huygens =====

Dubbed ''huygens'' here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C-Db), and 19/16 to the minor third (C-Eb), suitable for a system generated by a mildly tempered fifth.

Subgroup: 2.3.5.7.11.13.17

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==== Grosstone ====

Grosstone maps 13/8 to the double diminished seventh (C-Bbbb).

Subgroup: 2.3.5.7.11.13

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Minimax tuning:

* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}

: ~~Eigenmonzo~~ basis (unchanged-interval basis): 2.13/7

: eigenmonzo basis (unchanged-interval basis): 2.13/7

Tuning ranges:

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==== Meridetone ====

Meridetone maps the 13/8 to the quadruple augmented fourth (C-Fxx).

Subgroup: 2.3.5.7.11.13

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Minimax tuning:

* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.13/9

: eigenmonzo (unchanged-interval) basis: 2.13/9

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=== Meanpop ===

Meanpop maps the 11/8 to the double diminished fifth (C-Gbb), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-Gx). Note also 11/10 is the double diminished third; 12/11~13/12, double augmented unison; and 14/13, minor second.

Subgroup: 2.3.5.7.11

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Minimax tuning:

* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}

: [{{~~Monzo~~| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.5

: eigenmonzo (unchanged-interval) basis: 2.5

Tuning ranges:

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Minimax tuning:

* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}

: ~~Eigenmonzo~~ (unchanged-interval) basis: 2.13/11

: eigenmonzo (unchanged-interval) basis: 2.13/11

Tuning ranges:

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=== Meanenneadecal ===

Meanenneadecal maps the 11/8 to the augmented fourth (C-F#), and tridecimal meanenneadecal still maps the 13/8 to the double augmented fifth (C-Gx). Note also 11/10 is the major second; 12/11~14/13, minor second; and 13/12, double augmented unison.

Subgroup: 2.3.5.7.11

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=== Meanundeci ===

Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C-F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C-Ab).

Subgroup: 2.3.5.7.11

@@ Line 80: / Line 80: @@
 {{Wikipedia| Septimal meantone temperament }}
-The [[7/4]] of septimal meantone is the augmented sixth (C-A#), and other septimal intervals are [[7/6]], the augmented second (C-D#), [[7/5]], the augmented fourth (C-F#), and [[21/16]], the augmented third (C-E#). Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
+The [[7/4]] in septimal meantone is the augmented sixth (C-A#), and other septimal intervals are [[7/6]], the augmented second (C-D#), [[7/5]], the augmented fourth (C-F#), and [[21/16]], the augmented third (C-E#). Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
 [[Subgroup]]: 2.3.5.7
@@ Line 94: / Line 94: @@
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}
-: {{Monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
+: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
-: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
 [[Tuning ranges]]:
@@ Line 110: / Line 110: @@
 [[Badness]]: 0.013707
-=== Undecimal meantone aka Huygens ===
+=== Undecimal meantone aka huygens ===
 {{See also| Meantone vs meanpop }}
+Undecimal meantone maps the [[11/8]] to the double augmented third (C-Ex), and tridecimal meantone maps the [[13/8]] to the double augmented fifth (C-Gx). Note 13/11 is conflated with 6/5, represented by the minor third. Note also 11/10~13/12 is the double augmented unison; 12/11, double diminished third; and 14/13, minor second.
 Subgroup: 2.3.5.7.11
@@ Line 123: / Line 125: @@
 Minimax tuning:
 * 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
-: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
+: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
-: Eigenmonzo (unchanged-interval) basis: 2.11/9
+: eigenmonzo (unchanged-interval) basis: 2.11/9
 Tuning ranges:
@@ Line 151: / Line 153: @@
 Minimax tuning:
 * [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
-: Eigenmonzo (unchanged-interval) basis: 2.11/9
+: eigenmonzo (unchanged-interval) basis: 2.11/9
 {{Optimal ET sequence|legend=1| 12f, 19e, 31 }}
@@ Line 158: / Line 160: @@
 ===== Meantonic =====
+Dubbed ''meantonic'' here, this extension maps the 17/16 to the triple augmented seventh (C-Bx#) octave reduced, and 19/16 to the quadruple augmented unison (C-Cxx). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.
 Subgroup: 2.3.5.7.11.13.17
@@ Line 184: / Line 188: @@
 ===== Meantoid =====
+Dubbed ''meantoid'' here, this extension maps 17/16~19/18 to the augmented unison (C-C#) and 19/16 to the augmented second (C-D#). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverse, so genuine septendecimal and undevicesimal harmony cannot be expected.
 Subgroup: 2.3.5.7.11.13.17
@@ Line 210: / Line 216: @@
 ===== Huygens =====
+Dubbed ''huygens'' here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C-Db), and 19/16 to the minor third (C-Eb), suitable for a system generated by a mildly tempered fifth.
 Subgroup: 2.3.5.7.11.13.17
@@ Line 236: / Line 244: @@
 ==== Grosstone ====
+Grosstone maps 13/8 to the double diminished seventh (C-Bbbb).
 Subgroup: 2.3.5.7.11.13
@@ Line 246: / Line 256: @@
 Minimax tuning:
 * 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
-: Eigenmonzo basis (unchanged-interval basis): 2.13/7
+: eigenmonzo basis (unchanged-interval basis): 2.13/7
 Tuning ranges:
@@ Line 284: / Line 294: @@
 ==== Meridetone ====
+Meridetone maps the 13/8 to the quadruple augmented fourth (C-Fxx).
 Subgroup: 2.3.5.7.11.13
@@ Line 294: / Line 306: @@
 Minimax tuning:
 * 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
-: Eigenmonzo (unchanged-interval) basis: 2.13/9
+: eigenmonzo (unchanged-interval) basis: 2.13/9
 {{Optimal ET sequence|legend=1| 12f, 31f, 43 }}
@@ Line 462: / Line 474: @@
 === Meanpop ===
 {{See also| Meantone vs meanpop }}
+Meanpop maps the 11/8 to the double diminished fifth (C-Gbb), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-Gx). Note also 11/10 is the double diminished third; 12/11~13/12, double augmented unison; and 14/13, minor second.
 Subgroup: 2.3.5.7.11
@@ Line 475: / Line 489: @@
 Minimax tuning:
 * 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
-: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
-: Eigenmonzo (unchanged-interval) basis: 2.5
+: eigenmonzo (unchanged-interval) basis: 2.5
 Tuning ranges:
@@ Line 504: / Line 518: @@
 Minimax tuning:
 * 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
-: Eigenmonzo (unchanged-interval) basis: 2.13/11
+: eigenmonzo (unchanged-interval) basis: 2.13/11
 Tuning ranges:
@@ Line 637: / Line 651: @@
 === Meanenneadecal ===
+Meanenneadecal maps the 11/8 to the augmented fourth (C-F#), and tridecimal meanenneadecal still maps the 13/8 to the double augmented fifth (C-Gx). Note also 11/10 is the major second; 12/11~14/13, minor second; and 13/12, double augmented unison.
 Subgroup: 2.3.5.7.11
@@ Line 915: / Line 931: @@
 === Meanundeci ===
+Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C-F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C-Ab).
 Subgroup: 2.3.5.7.11