Hemififths: Difference between revisions

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The '''hemififths''' temperament tempers out the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the name suggests, uses a neutral-third generator. '''Hemif''' is the no-5 subgroup version of hemififths.  
'''Hemififths''' is the [[temperament]] [[tempering out]] the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the name suggests, uses a neutral-third generator. '''Hemif''' is the no-5 subgroup version of hemififths. It is supported by [[41edo|41-]], [[58edo|58-]], and [[99edo|99et]].  


See [[Breedsmic temperaments #Hemififths]] for more technical data.
See [[Breedsmic temperaments #Hemififths]] for more technical data.
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|}
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki>*</nowiki> in 7-limit POTE tuning
== Notation ==
Hemififths can be notated in [[neutral circle-of-fifths notation]], in which case 5/4 is represented by a sesqui-augmented second (C-D#+), and 7/4 by a semi-augmented sixth (C-A+). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C-F+), and 13/8 by the neutral sixth (C-Ad). This, of course, defies the tradition of tertian harmony. The just major triad on C is C-D#+-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step. There are two solutions:
# let an arrow represent a bend by the syntonic~septimal comma (17 gensteps, semidiminished second);
# let an arrow represent a bend by the Pythagorean comma (24 gensteps, negative diminished second).
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma.
{| class="wikitable center-1 center-3"
|+Hemififths nomenclature<br>for selected intervals
! Ratio
! Nominal
! Example
|-
| 3/2
| Perfect fifth
| C-G
|-
| 5/4
| Down major third
| C-vE
|-
| 7/4
| Down minor seventh
| C-vBb
|-
| 11/8
| Semi-augmented fourth
| C-F+
|-
| 13/8
| Neutral sixth
| C-Ad
|}
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma.
{| class="wikitable center-1 center-3"
|+Hemififths nomenclature<br>for selected intervals
! Ratio
! Nominal
! Example
|-
| 3/2
| Perfect fifth
| C-G
|-
| 5/4
| Up neutral third
| C-^Ed
|-
| 7/4
| Up semidiminished seventh
| C-^Bdb
|-
| 11/8
| Semi-augmented fourth
| C-F+
|-
| 13/8
| Neutral sixth
| C-Ad
|}


== Chords ==
== Chords ==
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Gencom: [2 11/9; 144/143 196/195 243/242 364/363]
Gencom: [2 11/9; 144/143 196/195 243/242 364/363]


Gencom mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
Gencom mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


{| class="wikitable center-all"
{| class="wikitable center-1 center-2"
|-
|-
! ET<br>generator
! Edo<br>generator
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! generator<br>(¢)
! Generator<br>(¢)
! comments
! Comments
|-
|-
|  
|  

Revision as of 13:14, 16 September 2023

Hemififths is the temperament tempering out the breedsma, 2401/2400, and the hemifamity comma, 5120/5103, and as the name suggests, uses a neutral-third generator. Hemif is the no-5 subgroup version of hemififths. It is supported by 41-, 58-, and 99et.

See Breedsmic temperaments #Hemififths for more technical data.

Interval chain

In the following table, prime harmonics are labeled in bold.

# Cents* Approximate Ratios
7-limit 13-limit Extension
0 0.000 1/1
1 351.477 49/40, 60/49 11/9, 16/13, 27/22, 39/32
2 702.955 3/2
3 1054.432 90/49 11/6, 24/13
4 205.910 9/8
5 557.387 112/81 11/8, 18/13
6 908.865 27/16 22/13
7 60.342 28/27 33/32, 27/26
8 411.819 81/64, 80/63 14/11, 33/26
9 763.297 14/9
10 1114.774 40/21 21/11
11 266.252 7/6
12 617.729 10/7
13 969.206 7/4
14 120.684 15/14 14/13
15 472.161 21/16
16 823.639 45/28 21/13
17 1175.116 63/32, 160/81
18 326.594 98/81, 135/112 40/33
19 678.071 40/27
20 1029.549 49/27 20/11
21 181.026 10/9
22 532.503 49/36 15/11
23 883.981 5/3
24 35.458 49/48, 50/49 45/44, 55/54
25 386.936 5/4

* in 7-limit POTE tuning

Notation

Hemififths can be notated in neutral circle-of-fifths notation, in which case 5/4 is represented by a sesqui-augmented second (C-D#+), and 7/4 by a semi-augmented sixth (C-A+). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C-F+), and 13/8 by the neutral sixth (C-Ad). This, of course, defies the tradition of tertian harmony. The just major triad on C is C-D#+-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step. There are two solutions:

  1. let an arrow represent a bend by the syntonic~septimal comma (17 gensteps, semidiminished second);
  2. let an arrow represent a bend by the Pythagorean comma (24 gensteps, negative diminished second).

Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma.

Hemififths nomenclature
for selected intervals
Ratio Nominal Example
3/2 Perfect fifth C-G
5/4 Down major third C-vE
7/4 Down minor seventh C-vBb
11/8 Semi-augmented fourth C-F+
13/8 Neutral sixth C-Ad

Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma.

Hemififths nomenclature
for selected intervals
Ratio Nominal Example
3/2 Perfect fifth C-G
5/4 Up neutral third C-^Ed
7/4 Up semidiminished seventh C-^Bdb
11/8 Semi-augmented fourth C-F+
13/8 Neutral sixth C-Ad

Chords

Main article: Chords of hemififths

Scales

Tuning spectrum

Gencom: [2 11/9; 144/143 196/195 243/242 364/363]

Gencom mapping: [⟨1 1 -5 -1 2 4], ⟨0 2 25 13 5 -1]]

Edo
generator 		Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
11/9 347.408
12/11 349.788
7\24 350.000
11/8 350.264
4/3 350.978
12\41 351.220
15/14 351.389
16/15 351.417
41\140 351.429
8/7 351.448 7-, 9- and 11-odd-limit hemif minimax
5/4 351.453 5-, 7-, 9- and 11-odd-limit minimax
7/5 351.457
6/5 351.494
29\99 351.515
7/6 351.534
10/9 351.543
9/7 351.657
15/11 351.680
15/13 351.705 15-odd-limit minimax
17\58 351.724
11/10 351.750
13/10 351.761 13-odd-limit minimax
13/11 351.798 13- and 15-odd-limit hemif minimax
22\75 352.000
14/13 352.021
14/11 352.188
18/13 352.676
13/12 353.809
16/13 359.472
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