Diaschismic

Diaschismic; srutal archagall

Subgroups	2.3.5, 2.3.5.17
Comma basis	2048/2025 (2.3.5); 136/135, 256/255 (2.3.5.17)
Reduced mapping	⟨2; 1 -2 1]
Edo join	12 & 22
Generator (POTE)	~16/15 = 104.898 ¢
MOS scales	2L 8s, 10L 2s, 12L 10s
Ploidacot	diploid monocot
Pergen	(P8/2, P5)
Color name	Saguguti
Minimax error	(5-odd limit) 3.259 ¢; ((2.3.5.17) 25-odd limit) ??? ¢
Target scale size	(5-odd limit) 12 notes; ((2.3.5.17) 25-odd limit) 22 notes

Diaschismic, sometimes known as srutal in the 5-limit, is a half-octave temperament generated by a perfect fifth or that minus a half-octave period, which is a semitone representing 16/15. Two of these semitones give a whole tone of 9/8, so the diaschisma, 2048/2025, is tempered out, and we also have a whole tone plus a period represent 8/5. 9/8 splits in two very naturally into 17/16 × 18/17, and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out S16 = 256/255, S17 = 289/288, and their product 136/135, leading to a 2.3.5.17 subgroup extension called srutal archagall.

The canonical extension to the 7-limit lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate 8/7, tempering out the starling comma, 126/125, as well as the hemifamity comma, 5120/5103.

A stack of twelve perfect fifths octave reduced (a diesis), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent 16/11. Three more fifths on top of 16/11 give 16/13. Finally, since the whole tone has been split in two, each can be used to represent 17/16~18/17. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, 176/175, 196/195, and 256/255.

See Diaschismic family #Diaschismic and Diaschismic family #Septimal diaschismic for technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

#	Period 0		Period 1
	Cents*	Approximate ratios	Cents*	Approximate ratios
0	0.0	1/1	600.0	17/12, 24/17
1	703.9	3/2	103.9	16/15, 17/16, 18/17
2	207.7	9/8	807.7	8/5
3	911.6	17/10, 22/13	311.6	6/5
4	415.4	14/11	1015.4	9/5
5	1119.5	21/11, 40/21, 48/25	519.5	27/20
6	623.1	10/7	23.1	56/55, 64/63, 81/80
7	127.0	14/13, 15/14	727.0	32/21
8	830.8	21/13, 34/21	230.8	8/7
9	334.7	17/14, 40/33	934.7	12/7
10	1038.5	20/11	438.5	9/7
11	542.4	15/11	1142.4	27/14, 64/33
12	46.2	36/35, 40/39, 45/44, 50/49	646.2	16/11
13	750.1	17/11, 20/13	150.1	12/11
14	253.9	15/13	853.9	18/11
15	957.8	45/26, 68/39	357.8	16/13
16	461.6	17/13	1061.6	24/13
17	1165.5	51/26, 96/49, 108/55	565.5	18/13

* In 17-limit CWE tuning, octave-reduced

As a detemperament of 12et

Diaschismic as a 58-tone 12et detempering

Diaschismic is naturally considered as a detemperament of the 12 equal temperament. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 generator steps, which represent the syntonic comma. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.

The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.

Chords

Main article: Chords of diaschismic

Tunings

5-limit prime-optimized tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Equilateral	CEE: ~3/2 = 705.8655 ¢	CSEE: ~3/2 = 705.5568 ¢	POEE: ~3/2 = 704.9311 ¢
Tenney	CTE: ~3/2 = 705.1363 ¢	CWE: ~3/2 = 704.9585 ¢	POTE: ~3/2 = 704.8982 ¢
Benedetti, Wilson	CBE: ~3/2 = 704.8398 ¢	CSBE: ~3/2 = 704.7309 ¢	POBE: ~3/2 = 704.7977 ¢

7-limit prime-optimized tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Equilateral	CEE: ~3/2 = 704.0394 ¢	CSEE: ~3/2 = 703.8161 ¢	POEE: ~3/2 = 703.3785 ¢
Tenney	CTE: ~3/2 = 704.0493 ¢	CWE: ~3/2 = 703.7738 ¢	POTE: ~3/2 = 703.6809 ¢
Benedetti, Wilson	CBE: ~3/2 = 704.0589 ¢	CSBE: ~3/2 = 703.7520 ¢	POBE: ~3/2 = 703.7438 ¢

17-limit prime-optimized tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Equilateral	CEE: ~3/2 = 704.0090 ¢	CSEE: ~3/2 = 703.9204 ¢	POEE: ~3/2 = 703.9138 ¢
Tenney	CTE: ~3/2 = 704.0164 ¢	CWE: ~3/2 = 703.8520 ¢	POTE: ~3/2 = 703.8121 ¢
Benedetti, Wilson	CBE: ~3/2 = 704.0285 ¢	CSBE: ~3/2 = 703.7782 ¢	POBE: ~3/2 = 703.7642 ¢

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)*	Generator (¢)	Comments
	17/9	698.955
7\12		700.000	12f val, lower bound of 7-, 9- and 11-odd-limit diamond monotone
	3/2	701.955
	15/14	702.778
41\70		702.857	70ef val
	7/5	702.915
	21/20	703.107
	15/11	703.359
	15/13	703.410
34\58		703.448	Lower bound of 13-, 15-, 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	11/10	703.500
	9/7	703.508
	13/10	703.522
	13/11	703.597
	12/7	703.681
	13/9	703.728
	11/9	703.757
	21/13	703.782
	49/48	703.783
	13/12	703.839
61\104		703.846	104c val
	21/11	703.893
	11/6	703.895
	7/4	703.897
	13/8	703.965
	17/14	704.014
	17/13	704.027
	13/7	704.043
	11/8	704.057
	36/35	704.064
	17/11	704.126
	21/16	704.174
	21/17	704.272
27\46		704.348	Upper bound of 11-, 13-, 15-, 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	11/7	704.377
	9/5	704.399
	17/16	704.955
	5/3	705.214
	25/24	705.866
20\34		705.882	34ef val, upper bound of 7- and 9-odd-limit diamond monotone
	17/10	706.214
	5/4	706.843
	17/15	708.343
	15/8	711.731

* Besides the octave