Hemifamity family: Difference between revisions

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The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma, or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vBb).

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* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just

: {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.7/3

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3

* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp

: {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.9/7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7

{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}

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* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5.11/9

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}

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Minimax tuning:

* 13-odd-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.9/5.13/9

* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9

* 15-odd-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.5/3.13/9

* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9

Line 147:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.11/7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7

{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}

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* 13- and 15-odd-limit

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11.13/7

: unchanged-interval (eigenmonzo) basis: 2.11.13/7

{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }}*

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* 17-odd-limit

: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11.13/7

: unchanged-interval (eigenmonzo) basis: 2.11.13/7

Line 221:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5.11/5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5

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* 13- and 15-odd-limit

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.7/5.11/5

: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}

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* [[11-odd-limit]]:

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7.11/9

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Hemifamity family| ]]

[[Category:Hemifamity| ]]

[[Category:Rank 3]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma, or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.
+The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.
 It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vBb).
@@ Line 24: / Line 24: @@
 * [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just
 : {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7/3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
 * [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp
 : {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
 {{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}
@@ Line 86: / Line 86: @@
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/9
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
 {{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}
@@ Line 104: / Line 104: @@
 Minimax tuning:
-* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.13/9
+* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
-* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/9
+* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9
 {{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}
@@ Line 147: / Line 147: @@
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.11/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
 {{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
@@ Line 167: / Line 167: @@
 * 13- and 15-odd-limit
 : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11.13/7
+: unchanged-interval (eigenmonzo) basis: 2.11.13/7
 {{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }}*
@@ Line 200: / Line 200: @@
 * 17-odd-limit
 : [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11.13/7
+: unchanged-interval (eigenmonzo) basis: 2.11.13/7
 {{Optimal ET sequence|legend=0| 58, 94, 111, 152f, 205, 263df }}
@@ Line 221: / Line 221: @@
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
 {{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}
@@ Line 245: / Line 245: @@
 * 13- and 15-odd-limit
 : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.7/5.11/5
+: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
 {{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}
@@ Line 293: / Line 293: @@
 * [[11-odd-limit]]:
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7.11/9
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9
 {{Optimal ET sequence|legend=1| 41, 87, 111, 152, 239, 391 }}
@@ Line 330: / Line 330: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Hemifamity family| ]] <!-- main article -->
 [[Category:Hemifamity| ]] <!-- key article -->
 [[Category:Rank 3]]
 [[Category:Listen]]