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 hemifamity comma, 5120/5103.

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.

Hemififths was named by Gene Ward Smith in 2004.

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		Ups and downs notation**
		7-limit	13-limit extension
0	0.0	1/1		P1
1	351.5	49/40, 60/49	11/9, 16/13, 27/22, 39/32	~3 = ^m3 = vM3
2	702.9	3/2		P5
3	1054.4	90/49	11/6, 24/13	~7 = ^m7 = vM7
4	205.9	9/8		M2
5	557.3	112/81	11/8, 18/13	~4 = ^4 = vA4
6	908.8	27/16	22/13	M6
7	60.3	28/27	33/32, 27/26	^1 = \m2
8	411.7	80/63, 81/64	14/11, 33/26	M3
9	763.2	14/9		^5 = \m6
10	1114.7	40/21	21/11	M7
11	266.1	7/6		^M2 = \m3
12	617.6	10/7		A4 = \~5
13	969.1	7/4		^M6 = \m7
14	120.5	15/14	14/13	A1 = \~2
15	472.0	21/16		^M3 = \4
16	823.5	45/28	21/13	A5 = \~6
17	1174.9	63/32, 160/81	55/28, 65/33, 77/39	^M7 = \8
18	326.4	98/81, 135/112	40/33	A2 = \~3
19	677.9	40/27		^A4 = \5
20	1029.3	49/27	20/11	A6 = \~7
21	180.8	10/9		^A1 = \M2
22	532.3	49/36	15/11	A3 = \~4
23	883.7	5/3		^A5 = \M6
24	35.2	49/48, 50/49	40/39, 45/44, 55/54, 65/64	A7 - P8 = -d2 = ^\1
25	386.7	5/4		^A2 = \M3
26	738.1	49/32	20/13	AA4 = ^\5
27	1089.6	15/8		^A6 = \M7
28	241.1	147/128	15/13	AA1= ^\2
29	592.5	45/32		^A3 = \A4

* In 7-limit CWE tuning, generator = 351.467 ¢, P5 = 702.934 ¢ and c = 2.934 ¢

** Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ − 8.5c

As a detemperament of 17et

Hemififths as a 58-tone 17et detempering

Hemififths is very naturally considered as a detemperament of the 17 equal temperament. The table below shows a 58-tone detempered scale, with a generator range of -28 to +29. Each interval category of the 17 equal temperament is further divided into "sub", "plain" and "super" qualities, separated by -17 generator steps, which represents the syntonic~septimal comma; the "plain" type here consists of a 7L 10s scale in 8|8 mode. 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 comma between supraminor and subneutral, and between supraneutral and submajor. This interval 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 syntonic~septimal comma. 99edo tunes it to one half the size of the syntonic~septimal comma, which can be seen as a good compromise.

#	Interval category	"Double-Sub"			"Sub"			"Plain"			"Super"			"Double-super"
		Gen.	Cents*	Ratios	Gen.	Cents*	Ratios	Gen.	Cents*	Ratios	Gen.	Cents*	Ratios	Gen.	Cents*	Ratios
0	P1							0	0.0	1/1	-17	25.9	64/63~81/80
1	m2				24	35.2	49/48~50/49	7	60.3	28/27	-10	85.3	21/20~22/21	-27	110.4	16/15
2	n2				14	120.5	14/13~15/14	-3	145.6	12/11~13/12	-20	170.7	11/10
3	M2				21	180.8	10/9	4	205.9	9/8	-13	230.9	8/7
4	m3	28	241.1	15/13	11	266.1	7/6	-6	291.2	13/11	-23	316.3	6/5
5	n3				18	326.4	40/33	1	351.5	11/9~16/13	-16	376.5	26/21
6	M3				25	386.7	5/4	8	411.7	14/11	-9	436.8	9/7	-26	461.9	13/10
7	P4				15	472.0	21/16	-2	497.1	4/3	-19	522.1	27/20
8	sA4, d5				22	532.3	15/11	5	557.3	11/8~18/13	-12	582.4	7/5
9	sd5, A4	29	592.5	45/32	12	617.6	10/7	-5	642.7	13/9~16/11	-22	667.7	22/15
10	P5				19	677.9	40/27	2	702.9	3/2	-15	728.0	32/21
11	m6	26	738.1	20/13	9	763.2	14/9	-8	788.3	11/7	-25	813.3	8/5
12	n6				16	823.5	21/13	-1	848.5	13/8~18/11	-18	873.6	33/20
13	M6				23	883.7	5/3	6	908.8	22/13	-11	933.9	12/7	-28	958.9	26/15
14	m7				13	969.1	7/4	-4	994.1	16/9	-21	1019.2	9/5
15	n7				20	1029.3	20/11	3	1054.4	11/6~24/13	-14	1079.5	13/7~28/15
16	M7	27	1089.6	15/8	10	1114.7	21/11~28/15	-7	1139.7	27/14	-24	1164.8	39/20~49/25
17	P8				17	1174.9	55/28~63/32	0	1200.0	2/1

See the diagram on the right for an isomorphic version.

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:

1. The syntonic~septimal comma (-17 gensteps, semidiminished second);
2. 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).

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–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#).

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–Ft
13/8	Neutral sixth	C–Ad

Chords

Main article: Chords of hemififths

Scales

• Hemif7
• Hemif10
• Hemif17

Tunings

7-limit prime-optimized 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 ¢

13-limit prime-optimized tunings

	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	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 tuning 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