Major second

A major second (M2) in the diatonic scale is an interval that spans one scale step 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 cents 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 cents.
• The 7-limit (septimal) supermajor second is a ratio of 8/7, and is about 231 cents.
• The 11-limit (undecimal) submajor second is a ratio of 11/10, and is about 165 cents.
• The 13-limit (tridecimal) ultramajor second is a ratio of 15/13, and is about 248 cents, but it might be better analyzed as an inframinor third. Despite that, it is also here for completeness.

By delta

See Delta-N ratio.

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

In regular temperaments

The three simplest major second ratios are 10/9, 9/8, and 8/7. The following notable temperaments are generated by them:

Temperaments generated by 8/7

• Slendric, where a stack of three 8/7s is equated to 3/2

Todo: complete list