Supermajor and subminor

The interval qualities "supermajor" and "subminor" essentially refer to intervals sharper than major and flatter than minor, respectively; a supermajor interval is sharper than the corresponding ~12edo interval by approximately a sixth-tone or diesis, and a subminor interval is flat of the corresponding ~12edo interval by approximately a sixth-tone or diesis.

In just intonation

In some notations and interval naming systems for just intonation, "supermajor" and "subminor" indicate sharping or flatting by a specific predefined comma, such as 64/63 (to reach septimal intervals), 81/80 (to reach acute and grave intervals), or 2048/2025 (to reach 5-limit supermajor and subminor intervals).

In other notations

In, say, 41edo or 53edo (or other similar systems), "upmajor" corresponds to "supermajor", and "downminor" to "subminor". Here is a rough list of EDOs where this is true in regards to thirds (i.e. the (anti)diatonic major third is >370c and <415c and the upmajor third is >425c and <460c). The restriction on normal major thirds is placed to ensure that the chosen diatonic major thirds are not already within the supermajor range.

EDO	Major	Upmajor/supermajor
16	375	450
19	379	442
24	400	450
25b	384	432
29	414	455
31	387	426
32	413	450
36	400	433
41	410	439
48	400	425
53	408	430
58	414	434
70	411	429
87	414	427

Similarly, as mentioned, diatonic thirds can be supermajor, and thus other diatonic intervals supermajor or subminor:

With our previously established supermajor range, this corresponds to a diatonic fifth of >706.25 cents and <415 cents; here are all EDOs which have that as a patent val fifth, excluding contorted EDOs (i.e. those which have the same fifth as a smaller EDO).

EDO	Major
22	436
27	444
32	450
37	454
39	431
42	457
49	441
56	429
59	447
61	433
71	439
73	427
83	434
90	426
95	430
107	426
124	426