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.
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:
- 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
- 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 |