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 as a 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 and No-fives_subgroup_temperaments#Hemif for more technical data.
Interval chain
In the following table, odd harmonics 1–21 are labeled in bold.
| # | Cents* | Approximate Ratios | ups and downs
notation ** | |
|---|---|---|---|---|
| 7-limit | 13-limit Extension | |||
| 0 | 0.0 | 1/1 | P1 | |
| 1 | 351.4 | 49/40, 60/49 | 11/9, 16/13, 27/22, 39/32 | ~3 = ^m3 = vM3 |
| 2 | 702.9 | 3/2 | P5 | |
| 3 | 1054.3 | 90/49 | 11/6, 24/13 | ~7 = ^m7 = vM7 |
| 4 | 205.8 | 9/8 | M2 | |
| 5 | 557.2 | 112/81 | 11/8, 18/13 | ~4 = ^4 = vA4 |
| 6 | 908.7 | 27/16 | 22/13 | M6 |
| 7 | 60.1 | 28/27 | 33/32, 27/26 | ^1 = \m2 |
| 8 | 411.6 | 80/63, 81/64 | 14/11, 33/26 | M3 |
| 9 | 763.0 | 14/9 | ^5 = \m6 | |
| 10 | 1114.5 | 40/21 | 21/11 | M7 |
| 11 | 265.9 | 7/6 | ^M2 = \m3 | |
| 12 | 617.4 | 10/7 | A4 = \~5 | |
| 13 | 968.8 | 7/4 | ^M6 = \m7 | |
| 14 | 120.2 | 15/14 | 14/13 | A1 = \~2 |
| 15 | 471.7 | 21/16 | ^M3 = \4 | |
| 16 | 823.1 | 45/28 | 21/13 | A5 = \~6 |
| 17 | 1174.6 | 63/32, 160/81 | ^M7 = \8 | |
| 18 | 326.0 | 98/81, 135/112 | 40/33 | A2 = \~3 |
| 19 | 677.5 | 40/27 | ^A4 = \5 | |
| 20 | 1028.9 | 49/27 | 20/11 | A6 = \~7 |
| 21 | 180.4 | 10/9 | ^A1 = \M2 | |
| 22 | 531.8 | 49/36 | 15/11 | A3 = \~4 |
| 23 | 883.3 | 5/3 | ^A5 = \M6 | |
| 24 | 34.7 | 49/48, 50/49 | 40/39, 45/44, 55/54, 65/64 | A7 - P8 = -d2 = ^\1 |
| 25 | 386.2 | 5/4 | ^A2 = \M3 | |
| 26 | 737.6 | 49/32 | 20/13 | AA4 = ^\5 |
| 27 | 1089.1 | 15/8 | ^A6 = \M7 | |
| 28 | 240.5 | 147/128 | 15/13 | AA1= ^\2 |
| 29 | 591.9 | 45/32 | ^A3 = \A4 | |
* In 7-limit CTE tuning, generator = 351.445¢, P5 = 702.89¢, c = 2.89¢
** Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ - 8.5c
Notation
Hemififths can be notated in neutral chain-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–A 
). 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 one or more additional modules of accidentals such as arrows or +/- signs to represent the comma steps. There are two notable comma steps:
- The syntonic~septimal comma (-17 gensteps, semidiminished second);
- The Pythagorean comma (+24 gensteps, inverse diminished second).
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).
| 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–Ft |
| 13/8 | Neutral sixth | C–Ad |
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).
| 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–Ft |
| 13/8 | Neutral sixth | C–Ad |
Chords
Scales
Tunings
Tuning spectrum
| 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 | Very close to argent temperament with neutral intervals (351.47186 cents) | |
| 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