Major second

A major second (M2) is an interval that spans one scale step in the diatonic scale with the major (wider) quality. It is generated by stacking 2 fifths octave reduced, and depending on the specific tuning, it ranges from 171 to 240 ¢ (1\7 to 1\5). It can be considered the large step of the diatonic scale.

In just intonation, an interval may be classified as a major second if it is reasonably mapped to 1\7 and 4\24 (precisely one step of the diatonic scale and two steps of the chromatic scale). The use of 24edo's 4\24 as the mapping criteria here rather than 12edo's 2\12 better captures the characteristics of many intervals in the 11- and 13-limit.

As a concrete interval region, it is typically near 200 ¢ in size, distinct from the semitone of roughly 100 ¢ and the neutral second of roughly 150 ¢. A rough tuning range for the major second is about 180 to 240 ¢ according to Margo Schulter's theory of interval regions.

This article covers intervals between 160 and 260 ¢. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they're looking for easily.

In just intonation

By prime limit

The Pythagorean (3-limit) major second is 9/8, which is 204 ¢ in size and corresponds to the mos-based interval category of the diatonic major second. It is generated by stacking two just perfect fifths of 3/2. There is also a Pythagorean diminished third of 65536/59049, which is about 180 cents in size. While called a "third", it is within the range of major seconds.

Other major seconds exist in higher limits, however, for example:

• The 5-limit ptolemaic major second is a ratio of 10/9, however in 5-limit harmony it is used alongside 9/8. It is about 182 ¢.
• The 7-limit (septimal) supermajor second is a ratio of 8/7, and is about 231 ¢.
• The 11-limit (undecimal) submajor second is a ratio of 11/10, and is about 165 ¢, though it can also be analyzed as a neutral second. Despite that, it is also here for completeness.
• The 13-limit (tridecimal) ultramajor second is a ratio of 15/13, and is about 248 ¢, though it can also be analyzed as an inframinor third. Despite that, it is also here for completeness.

By delta

See Delta-N ratio. Ratios that are marginal within the interval category and ambiguous with an adjoining one are marked with an asterisk.

Delta-1		Delta-2		Delta-3
8/7	231 ¢	15/13*	248 ¢	22/19*	253 ¢
9/8	204 ¢	17/15	217 ¢	23/20*	242 ¢
10/9	182 ¢	19/17	193 ¢	25/22	221 ¢
11/10*	165 ¢	21/19	173 ¢	26/23	212 ¢
				28/25	196 ¢
				29/26	189 ¢
				31/28	176 ¢
				32/29*	170 ¢

In edos

The following table lists the best tuning of 10/9, 9/8, and 8/7, as well as other major seconds if present, in various significant edos.

Edo	10/9	9/8	8/7	Other major seconds
5	240 ¢
7	171 ¢
12	200 ¢
15	160 ¢	240 ¢
16	*	225 ¢
17	212 ¢
19	189 ¢		253 ¢
22	164 ¢	218 ¢
24	200 ¢		250 ¢
25	192 ¢		240 ¢
26	185 ¢		231c
27	178 ¢	222 ¢
29	166 ¢	207 ¢	248 ¢
31	194 ¢		232 ¢
34	176 ¢	212 ¢	247 ¢
41	176 ¢	205 ¢	234 ¢
53	181 ¢	204 ¢	226 ¢	249 ¢ ≈ 15/13

In mos scales

Being a small interval, major seconds generate a number of monosmall and monolarge mos.

These tables start from the last monolarge mos generated by the interval range.

Scales with more than 12 notes are not included.

Range	Mos
150–171 ¢	1L 6s	7L 1s
171–200 ¢	1L 5s	6L 1s
200–218 ¢	1L 4s	5L 1s	6L 5s
218–240 ¢			5L 6s
240–267 ¢	1L 3s	4L 1s	5L 4s

Temperament interpretations

The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as generators for a variety of regular temperaments.

• The generator of the 7L 1s scale can be interpreted as a 10/9 major second, that is equated to 11/10 and 12/11 neutral seconds by porcupine, so that three generators reach 4/3. Its tuning range is therefore somewhat ambiguous between major and neutral second.
• The generator of the 6L 1s and 6L 7s scales can be interpreted in terms of 2.5.7 didacus, whose generator represents 28/25 and which splits the septimal tritone 7/5 in three, with one step making the generator 28/25 and two making 5/4. This generator can also stand in for 10/9 and 9/8 in the 2.9.5.7 subgroup, if it is treated as an index-2 restriction of septimal meantone.
• The generator of the 5L 6s scale can be interpreted as 8/7 in 2.3.7 slendric, where three of them are equated to 3/2.
• The generator of the 5L 4s scale can be interpreted in terms of 2.3.7 semaphore, where 8/7 is equated to the subminor third 7/6 so that two generators reach 4/3, or more accurately as 2.3.13/5 barbados if 8/7 is eschewed in favor of 15/13. Either way, it is tuned as an interseptimal ambiguous between a major second and minor third.

Todo: complete list