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.

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-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 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

• Hemif7
• Hemif10
• Hemif17

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