Diaschismic straddle temperaments

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Diaschismic tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for straddling 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.

Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.

Straddle Diaschismic is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:

• Septimal Diaschismic tempers out 126/125, a 2.9.5.7 comma requiring no straddling because it has 9 rather than 3.
• Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1.

These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a. This can be converted to standard RTT to avoid straddling, resulting in 2.9.5.7.11.289 22&30.

Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration a is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

Rank-2 Temperaments

Gens	DP 162.287¢		DS 243.180¢		DH 115.976¢		DW 210.145¢		DO 129.835¢
-14	128	728	195	795	176	776	58	658	582	1182
-13	290	890	439	1039	292	892	268	868	112	712
-12	453	1053	82	682	408	1008	478	1078	242	842
-11	15	615	325	925	524	1124	88	688	372	972
-10	177	777	568	1168	40	640	299	899	502	1102
-9	339	939	211	811	156	756	509	1109	31	631
-8	502	1102	455	1055	272	872	119	719	161	761
-7	64	664	98	698	388	988	329	929	291	891
-6	226	826	341	941	504	1104	539	1139	421	1021
-5	389	989	584	1184	20	620	149	749	551	1151
-4	551	1151	227	827	136	736	359	959	81	681
-3	113	713	470	1070	252	852	570	1170	210	810
-2	275	875	114	714	368	968	180	780	340	940
-1	438	1038	357	957	484	1084	390	990	470	1070
0	0	600	0	600	0	600	0	600	0	600
1	162	762	243	843	116	716	210	810	130	730
2	325	925	486	1086	232	832	420	1020	260	860
3	487	1087	130	730	348	948	30	630	390	990
4	49	649	373	973	464	1064	241	841	519	1119
5	211	811	16	616	580	1180	451	1051	49	649
6	374	974	259	859	96	696	61	661	179	779
7	536	1136	502	1102	212	812	271	871	309	909
8	98	698	145	745	328	928	481	1081	439	1039
9	261	861	389	989	444	1044	91	691	569	1169
10	423	1023	32	632	560	1160	301	901	98	698
11	585	1185	275	875	76	676	512	1112	228	828
12	147	747	518	1118	192	792	122	722	358	958
13	310	910	161	761	308	908	332	932	488	1088
14	472	1072	405	1005	424	1024	542	1142	18	618

Diaschismic Porcupine

Diaschismic Porcupine tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.

The simplest interpretation of the generator is 9/7, which also generates Hedgehog and Stearn. Straddle Diaschismic is equivalent to Stearn in the 2.9.7 subgroup, tempering out 118098/117649 = ((9/7)/(7/6)*(9/7))^2/2. This is an extremely efficient temperament.

Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.

17-limit GTO string: 600 | 2.<3.>3.5.7.11.13.<17.>17 || 0 8 -3 -5 6 -4 -17 8 -3; 162.287¢

One possible erac group resembling the 74edo tuning is 600 | 2.<3.>>3.>5.>7.11.>13.<17.>>17 || 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

Diaschismic Semioceanfront

Diaschismic Semioceanfront has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.

17-limit GTO string: 600 | 2.<3.>3.5.7.11.13.<17.>17 || 0 -2 -7 9 4 -15 1 -2 -7; 243.180¢

Diaschismic Hemiseven

Hemiseven may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.

Subgroup GTO string: 600 | 2.<3.>3.5.7.<13.>13.<17.>17 || 0 6 1 -7 -2 2 -3 6 1; 115.976¢

Diaschismic whole-tone

Diaschismic whole-tone is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

Subgroup GTO string: 600 | 2.<3.>3.5.13.<17.>17.23 || 0 9 -8 -1 4 9 -8 3; 210.145¢

It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

Diaschismic oneirotonic

Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46, which may be called Diaschismic Sensibeta after its 2.9.5.7 commas.

17-limit GTO string: 600 | 2.<3.>3.5.7.11.13.<17.>17 || 0 -4 1 3 -11 -5 -12 -4 1; 129.835¢

It is a straddle variant of Supersharp and Octokaidecal (using >3 as the canonical 3), which temper out 10/9 / (81/80) = 800/729, analogous to Mavila's 9/8 / (16/15) = 135/128.