Hemifamity family: Difference between revisions

Line 1:

The '''hemifamity family''' of [[~~Rank~~-3 temperament|rank-3]] [[temperament]]s [[~~Tempering~~ out|tempers out]] [[5120/5103]] = {{monzo| 10 -6 1 -1 }}.

The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out|tempers out]] [[5120/5103]] = {{monzo| 10 -6 1 -1 }}. 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]].

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).

== Hemifamity ==

Line 35:

Line 37:

* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]

* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]

=== Overview to extensions ===

==== 11- and 13-limit extensions ====

Strong extensions of hemifamity are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] at the down diminished fifth (C-vGb); laka, up augmented third (C-^E#); akea, double-up fourth (C-^^F); lono, triple-down augmented fourth (C-v3F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third, tempering out [[352/351]], [[847/845]], and [[2080/2079]].

==== Subgroup extensions ====

A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.

=== Counterpyth ===

Developed analogous to [[parapyth]], counterpyth is an extension of hemifamity with an even milder fifth, as it finds [[19/15]] at the major third (C-E) and [[19/10]] at the major seventh (C-B). Notice the factorization 5120/5103 = ([[400/399]])([[1216/1215]]). Other important ratios are [[21/19]] at the diminished third (C-Ebb) and [[19/14]] at the augmented third (C-E#).

It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.

Subgroup: 2.3.5.7.19

Comma list: 400/399, 1216/1215

Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.6411, ~5/4 = 385.4452

Optimal ET sequence: {{Optimal ET sequence| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}

Badness: 0.212 × 10-3

== Pele ==

Line 78:

Line 104:

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

{{Optimal ET sequence~~|legend=1~~| 29, 41, 46, 58, 87, 145, 232 }}

Optimal ET sequence: {{Optimal ET sequence| 29, 41, 46, 58, 87, 145, 232 }}

Badness: 0.703 × 10-3

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363

Mapping: {{mapping| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.5544, ~5/4 = 387.9654

Optimal ET sequence: {{Optimal ET sequence| 29, 41, 46, 58, 87, 99ef, 145 }}

Badness: 0.930 × 10-3

== Laka ==

@@ Line 1: / Line 1: @@
-The '''hemifamity family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] [[5120/5103]] = {{monzo| 10 -6 1 -1 }}.
+The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out|tempers out]] [[5120/5103]] = {{monzo| 10 -6 1 -1 }}. 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]].
+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).
 == Hemifamity ==
@@ Line 35: / Line 37: @@
 * [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
 * [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]
+=== Overview to extensions ===
+==== 11- and 13-limit extensions ====
+Strong extensions of hemifamity are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] at the down diminished fifth (C-vGb); laka, up augmented third (C-^E#); akea, double-up fourth (C-^^F); lono, triple-down augmented fourth (C-v<sup>3</sup>F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third, tempering out [[352/351]], [[847/845]], and [[2080/2079]].
+==== Subgroup extensions ====
+A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.
+=== Counterpyth ===
+Developed analogous to [[parapyth]], counterpyth is an extension of hemifamity with an even milder fifth, as it finds [[19/15]] at the major third (C-E) and [[19/10]] at the major seventh (C-B). Notice the factorization 5120/5103 = ([[400/399]])([[1216/1215]]). Other important ratios are [[21/19]] at the diminished third (C-Ebb) and [[19/14]] at the augmented third (C-E#).
+It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.
+Subgroup: 2.3.5.7.19
+Comma list: 400/399, 1216/1215
+Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.6411, ~5/4 = 385.4452
+Optimal ET sequence: {{Optimal ET sequence| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
+Badness: 0.212 × 10<sup>-3</sup>
 == Pele ==
@@ Line 78: / Line 104: @@
 * 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/9
-{{Optimal ET sequence|legend=1| 29, 41, 46, 58, 87, 145, 232 }}
+Optimal ET sequence: {{Optimal ET sequence| 29, 41, 46, 58, 87, 145, 232 }}
 Badness: 0.703 × 10<sup>-3</sup>
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 196/195, 256/255, 352/351, 364/363
+Mapping: {{mapping| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.5544, ~5/4 = 387.9654
+Optimal ET sequence: {{Optimal ET sequence| 29, 41, 46, 58, 87, 99ef, 145 }}
+Badness: 0.930 × 10<sup>-3</sup>
 == Laka ==