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.

For example, supermajor thirds may be found between about 429-446 cents, and subminor thirds may be found between about 256-273 cents.

Common supermajor and subminor intervals may be found as simple 7-limit intervals, and include:

• 8/7 (231c), supermajor second
• 7/6 (267c), subminor third
• 9/7 (435c), supermajor third
• 14/9 (765c), subminor sixth
• 12/7 (933c), supermajor sixth
• 7/4 (969c), subminor seventh

Supermajor and subminor intervals are found in diatonic scales where the fifth is tuned significantly sharper than just - depending on the desired interval category, between 709 and 715 cents. For a given neutral interval k in cents, the supermajor quality ranges from around k+78 to k+95, and the pental minor quality ranges from around k-95 to k-78.

Optionally, the category of supermajor or subminor may be split into two smaller categories. Tuning ranges have been provided in terms of thirds:

• Supermajor and subminor, for thirds, may more precisely refer to the ranges between about 429-438 and 264-273, respectively. These are the ranges more closely focused around septimal intervals. Supermajor seconds, under this definition, range from about 225 to 234 cents. For a given neutral interval k in cents, the supermajor version in this sense is found at around k+84, and the subminor version is found at around k-84.
• Sensamajor and sensaminor, for thirds, refer to the ranges between about 438-446 and 256-264 cents, respectively. These are more extreme than the septimal ranges. Sensamajor seconds, under this definiiton, range from about 234 to 242 cents, containing the 5edo second of 240 cents. For a given neutral interval k in cents, the sensamajor version is found at around k+90, and the sensaminor version is found at around k-90.

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 <715 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

See also

• Neutral (interval quality) - halfway between major and minor
• Submajor and supraminor - roughly 15 to 24 cents sharp or flat of neutral
• Pental major and minor - roughly 24 to 43 cents sharp or flat of neutral
• Novamajor and novaminor - roughly 43 to 60 cents sharp or flat of neutral
• Neogothic major and minor - roughly 60 to 78 cents sharp or flat of neutral
• Ultramajor and inframinor - more extreme than 95 cents sharp or flat of neutral