Major second (interval region)
← Neutral second | Interval region | Minor third → |
9/8 (203.9¢)
10/9 (182.4¢)
A major second (M2), as a concrete interval region, 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 are 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 7L 6s scales can be interpreted as a 10/9 major second in tetracot (which is sometimes equated to 11/10), where four of these seconds reach 3/2.
- 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.
V • T • EInterval regions | |
---|---|
Seconds and thirds | Unison • Comma and diesis • Semitone • Neutral second • Major second • (Interseptimal second-third) • Minor third • Neutral third • Major third |
Fourths and fifths | (Interseptimal third-fourth) • Perfect fourth • Superfourth • Tritone • Subfifth • Perfect fifth • (Interseptimal fifth-sixth) |
Sixths and sevenths | Minor sixth • Neutral sixth • Major sixth • (Interseptimal sixth-seventh) • Minor seventh • Neutral seventh • Major seventh • Octave |