Neutral temperaments

Neutral temperaments are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the dicot temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.

10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot Archy temperament.

Notation

Neutral temperaments may be notated with neutral chain-of-fifths notation.

Note	24edo	Notation	2.3...	5 (Dicot)	11 (Rastmatic)	13 (Namo)	13-limit (no-fives)	13-limit
A	0	P1	1/1
At	50	sA1		81/80	33/32
Bb	100	m2	256/243
Bd	150	n2		10/9, 16/15	12/11
B	200	M2	9/8				8/7	11/10
C	300	m3	32/27				7/6
Ct	350	n3		5/4, 6/5	11/9	16/13
C#	400	M3	81/64				9/7
Dd	450	sd4					14/11
D	500	P4	4/3
Dt	550	sA4		27/20	11/8	18/13		10/7
Ed	650	sd5		40/27	16/11	13/9		7/5
E	700	P5	3/2
Et	750	sA5					11/7
F	800	m6	128/81				14/9
Ft	850	n6		5/3, 8/5	18/11	13/8
F#	900	M6	27/16				12/7
G	1000	m7	16/9				7/4	20/11
Gt	1050	n7		15/8, 9/5	11/6
G#	1100	M7	243/128
Ad	1150	sd8		160/81	64/33
A	1200	P8	2/1

These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:

	0\7	1\7	2\7	3\7	4\7	5\7	6\7	7\7
0\10	P1	m2	d3
1\10	sA1	n2	sd3
2\10	A1	M2	m3	d4
3\10		sA2	n3	sd4
4\10		A2	M3	P4	d5	d6
5\10			sA3	sA4	sd5	sd6
6\10			A3	A4	P5	m6	d7
7\10					sA5	n6	sd7
8\10					A5	M6	m7	d8
9\10						sA6	n7	sd8
10\10						A6	M7	P8

Rastmatic

Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22.

Etymology

Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam Rast which utilizes a scale with several neutral intervals.

Hemififths

Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which equates 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an aberschismic temperament.

Dicot

Dicot^[a], not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out 25/24, the Dicot comma. The provided edo join also tempers out 45/44 and 64/63 in the 11-limit, representing the extension Dichotic and also tempering out 55/54. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.

Dicot makes 4:5:6 equidistant, suggesting the simplified structure of tertian harmony, the same way Semaphore does for chthonic harmony. As a result, the temperament archetype dicot is named after it.

Etymology

Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name Dicot would also inspire Tetracot, Alphatricot, and by extension the ploidacot temperament archetype naming system as a whole.

Tuning considerations

A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.

Namo

Namo, Intertridecimal, or Harmoneutral is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.

It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.

Patent vals

List of patent vals

EDO	Mappings supported	Generator tuning	3/2 tuning
10	5, 13, 11	360.0c	720.0c
37	13	356.8c	713.5c
27	13	355.6c	711.1c
71	13	354.9c	709.9c
44	13	354.5c	709.1c
61	13	354.1c	708.2c
78	13	353.8c	707.7c
95	13	353.7c	707.4c
17	5, 13, 11	352.9c	705.9c
75	13	352.0c	704.0c
58	13, 11	351.7c	703.4c
41	13, 11	351.2c	702.4c
147	11	351.0c	702.0c
106	11	350.9c	701.9c
171	11	350.9c	701.8c
65	13, 11	350.8c	701.5c
219	11	350.7c	701.4c
154	11	350.6c	701.3c
243	11	350.62c	701.23c
332	11	350.60c	701.20c
89	11	350.56c	701.12c
380	11	350.53c	701.05c
291	11	350.52c	701.03c
202	11	350.50c	700.99c
517	11	350.48c	700.97c
315	11	350.48c	700.95c
428	11	350.47c	700.93c
541	11	350.46c	700.92c
113	11	350.44c	700.88c
476	11	350.42c	700.84c
363	11	350.41c	700.83c
250	11	350.40c	700.80c
387	11	350.39c	700.78c
137	11	350.36c	700.73c
435	11	350.34c	700.69c
298	11	350.34c	700.67c
459	11	350.33c	700.65c
161	11	350.31c	700.62c
346	11	350.29c	700.58c
185	11	350.27c	700.54c
394	11	350.25c	700.51c
209	11	350.24c	700.48c
233	11	350.21c	700.43c
257	11	350.19c	700.39c
281	11	350.18c	700.36c
305	11	350.16c	700.33c
329	11	350.15c	700.30c
353	11	350.14c	700.28c
24	13, 11	350.00c	700.00c
247	11	349.80c	699.60c
223	11	349.78c	699.55c
199	11	349.7c	699.5c
175	11	349.7c	699.4c
151	11	349.7c	699.3c
127	11	349.6c	699.2c
103	11	349.5c	699.0c
79	11	349.4c	698.7c
55	13, 11	349.1c	698.2c
31	13, 11	348.4c	696.8c
38	13, 11	347.4c	694.7c
45	13	346.7c	693.3c
7	5, 13, 11	342.9c	685.7c

Footnotes

[a] The name Interpental has been proposed, however it currently is used by 43 & 53, a weak extension of Buzzard.