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 41-, 58-, and 99et.
Hemififths was named by Gene Ward Smith in 2004^{[1]}.
See Breedsmic temperaments #Hemififths for more technical data.
Interval chain
In the following table, odd harmonics 1–21 are labeled in bold.
# | Cents* | Approximate Ratios | |
---|---|---|---|
7-limit | 13-limit Extension | ||
0 | 0.0 | 1/1 | |
1 | 351.4 | 49/40, 60/49 | 11/9, 16/13, 27/22, 39/32 |
2 | 702.9 | 3/2 | |
3 | 1054.3 | 90/49 | 11/6, 24/13 |
4 | 205.8 | 9/8 | |
5 | 557.2 | 112/81 | 11/8, 18/13 |
6 | 908.7 | 27/16 | 22/13 |
7 | 60.1 | 28/27 | 33/32, 27/26 |
8 | 411.6 | 80/63, 81/64 | 14/11, 33/26 |
9 | 763.0 | 14/9 | |
10 | 1114.5 | 40/21 | 21/11 |
11 | 265.9 | 7/6 | |
12 | 617.4 | 10/7 | |
13 | 968.8 | 7/4 | |
14 | 120.2 | 15/14 | 14/13 |
15 | 471.7 | 21/16 | |
16 | 823.1 | 45/28 | 21/13 |
17 | 1174.6 | 63/32, 160/81 | |
18 | 326.0 | 98/81, 135/112 | 40/33 |
19 | 677.5 | 40/27 | |
20 | 1028.9 | 49/27 | 20/11 |
21 | 180.4 | 10/9 | |
22 | 531.8 | 49/36 | 15/11 |
23 | 883.3 | 5/3 | |
24 | 34.7 | 49/48, 50/49 | 40/39, 45/44, 55/54, 65/64 |
25 | 386.2 | 5/4 | |
26 | 737.6 | 49/32 | 20/13 |
27 | 1089.1 | 15/8 | |
28 | 240.5 | 147/128 | 15/13 |
29 | 591.9 | 45/32 |
* in 7-limit CTE 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.
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.
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
Scales
Tunings
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 | ||
11/6 | 349.788 | ||
7\24 | 350.000 | Lower bound of 7- and 9-odd-limit diamond monotone | |
11/8 | 350.264 | ||
3/2 | 350.978 | ||
12\41 | 351.220 | Lower bound of 11- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
21/16 | 351.385 | ||
15/14 | 351.389 | ||
15/8 | 351.417 | ||
41\140 | 351.429 | ||
7/4 | 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 | ||
25/24 | 351.472 | ||
49/48 | 351.487 | ||
5/3 | 351.494 | ||
29\99 | 351.515 | ||
7/6 | 351.534 | ||
9/5 | 351.543 | ||
21/20 | 351.553 | ||
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 | |
21/13 | 351.891 | ||
21/11 | 351.946 | ||
22\75 | 352.000 | ||
13/7 | 352.021 | ||
11/7 | 352.188 | ||
13/9 | 352.676 | ||
5\17 | 352.941 | Upper bound of 7- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
13/12 | 353.809 | ||
13/8 | 359.472 |
* besides the octave