Hemififths
- This page is about the regular temperament. For the irrational interval of a hemififth, see Sqrt(3/2).
Hemififths is a temperament that uses a neutral third as a generator, just as the name suggests. A stack of 13 generators represents 7/4 and a stack of 25 generators represents 5/4, tempering out the breedsma, 2401/2400, and the argent comma, 5120/5103.
| Hemififths |
243/242, 441/440, 896/891 (11-limit);
144/143, 196/195, 243/242, 364/363 (13-limit)
13-limit 21-odd-limit: 7.77 ¢
13-limit 21-odd-limit: 41 notes
It extends fairly naturally to the 11- and 13-limit by treating the generator as 11/9~16/13. This lowers the overall accuracy, but supplies more harmonic resources. The no-5 subgroup restriction, called hemif, is also notable. Possible tunings include 41-, 58-, and 99edo (using the 99ef val in the 13-limit).
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 and their inversions are labeled in bold.
| # | Cents* | Approximate ratios | |
|---|---|---|---|
| 7-limit | 13-limit extension | ||
| 0 | 0.0 | 1/1 | |
| 1 | 351.5 | 49/40, 60/49 | 11/9, 16/13, 27/22, 39/32 |
| 2 | 702.9 | 3/2 | |
| 3 | 1054.4 | 90/49 | 11/6, 24/13 |
| 4 | 205.9 | 9/8 | |
| 5 | 557.3 | 112/81 | 11/8, 18/13 |
| 6 | 908.8 | 27/16 | 22/13 |
| 7 | 60.3 | 28/27 | 33/32, 27/26 |
| 8 | 411.7 | 80/63, 81/64 | 14/11, 33/26 |
| 9 | 763.2 | 14/9 | |
| 10 | 1114.7 | 40/21 | 21/11 |
| 11 | 266.1 | 7/6 | |
| 12 | 617.6 | 10/7 | |
| 13 | 969.1 | 7/4 | |
| 14 | 120.5 | 15/14 | 14/13 |
| 15 | 472.0 | 21/16 | |
| 16 | 823.5 | 45/28 | 21/13 |
| 17 | 1174.9 | 63/32, 160/81 | 55/28, 65/33, 77/39 |
| 18 | 326.4 | 98/81, 135/112 | 40/33 |
| 19 | 677.9 | 40/27 | |
| 20 | 1029.3 | 49/27 | 20/11 |
| 21 | 180.8 | 10/9 | |
| 22 | 532.3 | 49/36 | 15/11 |
| 23 | 883.7 | 5/3 | |
| 24 | 35.2 | 49/48, 50/49 | 40/39, 45/44, 55/54, 65/64 |
| 25 | 386.7 | 5/4 | |
| 26 | 738.1 | 49/32 | 20/13 |
| 27 | 1089.6 | 15/8 | |
| 28 | 241.1 | 147/128 | 15/13 |
| 29 | 592.5 | 45/32 | |
* In 7-limit CWE tuning, octave reduced
As a detemperament of 17et
Hemififths is very naturally considered as a detemperament of the 17 equal temperament. The diagram on the right shows a 58-tone detempered scale, with a generator range of -28 to +29. 58 is the largest number of tones for a mos where intervals in the 17 categories do not overlap. Each category may be further divided into "sub", "plain" and "super" qualities, separated by the commatic step of -17 generator steps, which represents 56/55, 64/63, 66/65, 78/77, 81/80, 91/90, 99/98, 121/120, and 169/168. Combining this division with the minor, neutral, and major qualities of the 17 equal temperament, hemififths gives us at least nine qualities for each diatonic category: subminor, minor, supraminor, subneutral, neutral, supraneutral, submajor, major, and supermajor.
Notice also the little interval between the largest of a category and the smallest of the next. This interval separates supraminor from subneutral and supraneutral from submajor, and spans 41 generator steps. 41edo tempers it out so that it conflates supraminor with subneutral and supraneutral with submajor, whereas 58edo exaggerates it to the size of the comma. 99edo tunes it to one half the size of the commatic step, which can be seen as a good compromise.
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, as the just major triad on C is C–D –G, for example, so one may want to adopt one or more additional modules of accidentals such as arrows or +/- signs to represent the commatic steps (-17 generator steps, a semidiminished second).
Below is tabulated how to notate each prime harmonic with an arrow representing a commatic step (thus ↑C = D).
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C–G |
| 5/4 | Down major third | C–↓E |
| 7/4 | Down minor seventh | C–↓B♭ |
| 11/8 | Semi-augmented fourth | C–F
|
| 13/8 | Neutral sixth | C–A
|
Ups and downs notation
In Kite's ups and downs notation, the equivalences are vvA1 and v\m2. Let c be the amount by which the fifth exceeds 7\12, then ^1 = 50 ¢ + 3.5c and /1 = 50 ¢ − 8.5c. For 7-limit CWE tuning, c = 2.934 ¢.
| # | Cents* | Ups and downs notation |
Associated ratios |
|---|---|---|---|
| 0 | 0.0 | P1 | 1/1 |
| 1 | 351.5 | ~3 = ^m3 = vM3 | 11/9~16/13 |
| 2 | 702.9 | P5 | 3/2 |
| 3 | 1054.4 | ~7 = ^m7 = vM7 | 11/6~24/13 |
| 4 | 205.9 | M2 | 9/8 |
| 5 | 557.3 | ~4 = ^4 = vA4 | 11/8~18/13 |
| 6 | 908.8 | M6 | 22/13~27/16 |
| 7 | 60.3 | ^1 = \m2 | 27/26~33/32 |
| 8 | 411.7 | M3 | 14/11~33/26 |
| 9 | 763.2 | ^5 = \m6 | 14/9 |
| 10 | 1114.7 | M7 | 21/11~40/21 |
| 11 | 266.1 | ^M2 = \m3 | 7/6 |
| 12 | 617.6 | A4 = \~5 | 10/7 |
| 13 | 969.1 | ^M6 = \m7 | 7/4 |
| 14 | 120.5 | A1 = \~2 | 14/13~15/14 |
| 15 | 472.0 | ^M3 = \4 | 21/16 |
| 16 | 823.5 | A5 = \~6 | 21/13 |
| 17 | 1174.9 | ^M7 = \8 | 63/32~160/81 |
| 18 | 326.4 | A2 = \~3 | 40/33 |
| 19 | 677.9 | ^A4 = \5 | 40/27 |
| 20 | 1029.3 | A6 = \~7 | 20/11 |
| 21 | 180.8 | ^A1 = \M2 | 10/9 |
| 22 | 532.3 | A3 = \~4 | 15/11 |
| 23 | 883.7 | ^A5 = \M6 | 5/3 |
| 24 | 35.2 | A7 - P8 = -d2 = ^\1 | 49/48~50/49 |
| 25 | 386.7 | ^A2 = \M3 | 5/4 |
| 26 | 738.1 | AA4 = ^\5 | 20/13 |
| 27 | 1089.6 | ^A6 = \M7 | 15/8 |
| 28 | 241.1 | AA1= ^\2 | 15/13 |
| 29 | 592.5 | ^A3 = \A4 | 45/32 |
* In 7-limit CWE tuning, octave reduced
Chords and harmony
Scales
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~49/40 = 351.4464 ¢ | CSEE: ~49/40 = 351.4671 ¢ | POEE: ~49/40 = 351.4774 ¢ |
| Tenney | CTE: ~49/40 = 351.4492 ¢ | CWE: ~49/40 = 351.4639 ¢ | POTE: ~49/40 = 351.4834 ¢ |
| Benedetti, Wilson |
CBE: ~49/40 = 351.4447 ¢ | CSBE: ~49/40 = 351.4675 ¢ | POBE: ~49/40 = 351.4787 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~11/9 = 351.4230 ¢ | CSEE: ~11/9 = 351.5800 ¢ | POEE: ~11/9 = 351.6627 ¢ |
| Tenney | CTE: ~11/9 = 351.4331 ¢ | CWE: ~11/9 = 351.5438 ¢ | POTE: ~11/9 = 351.5734 ¢ |
| Benedetti, Wilson |
CBE: ~11/9 = 351.4380 ¢ | CSBE: ~11/9 = 351.5144 ¢ | POBE: ~11/9 = 351.5243 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 11/9 | 347.408 | ||
| 11/6 | 349.788 | ||
| 7\24 | 350.000 | 24c val, 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 | 140ef val | |
| 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 tuning with neutral intervals (351.47186 cents) | |
| 49/48 | 351.487 | ||
| 5/3 | 351.494 | ||
| 29\99 | 351.515 | 99ef val | |
| 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 and 13-limit 21-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 | 75ce val | |
| 13/7 | 352.021 | ||
| 11/7 | 352.188 | ||
| 13/9 | 352.676 | ||
| 5\17 | 352.941 | 17c val, 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