Neutral second
A neutral second, as a concrete interval region, is typically near 150 cents in size, distinct from semitones of roughly 100 cents and major seconds of roughly 200 cents. A rough tuning range of the neutral second is 130 to 170 cents, and they characteristically are half of some kind of minor third. Neutral seconds are not found in diatonic, but they do exist in a diatonic functional context as "neutral" diatonic intervals.
They are one of the most distinctive-sounding yet versatile xenharmonic intervals, which makes them highly valuable.
Function
The large step of armotonic is always some sort of neutral second, which functions similar to both a whole tone and semitone.
A step of 13ed3 is a middle neutral second. It is notable for having an unusually good approximation of LCJI for a system generated by neutral seconds.
Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds.
Porcupine neutral seconds are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (~10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.
In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
In just intonation
By prime limit
The 3-limit does not have a simple neutral second, so we start with the 5-limit:
- The 5-limit acute minor second or large limma is a ratio of 27/25, and is about 133¢.
- The 7-limit neutral second is a ratio of 35/32, and is about 155¢.
- There is also a 7-limit swetismic neutral second, which is a ratio of 49/45, and is about 147¢.
- The 11-limit neutral/submajor seconds are the ratios of 12/11 and 11/10, which are about 151¢ and 165¢, respectively; 11/10 in particular can also be analyzed as a major second. Despite that, it is also here for completeness.
- The 13-limit neutral/supraminor seconds are the ratios of 14/13 and 13/12, which are about 128¢ and 139¢, respectively; 14/13 in particular can also be analyzed as a semitone. Despite that, it is also here for completeness.
In mos scales
Intervals between 120 and 171¢ generate the following mos scales:
These tables start from the last monolarge mos generated by the interval range.
Scales with more than 12 notes are not included.
| Range | Mos | |
|---|---|---|
| 120–133¢ | 1L 8s | 9L 1s |
| 133–150¢ | 1L 7s | 8L 1s |
| 150–171¢ | 1L 6s | 7L 1s |
Categorization
Proposal: Ground's Neutral Second Categorization System
As neutral intervals, neutral seconds can be named based on interval splitting.
| ¢ | Definition | Name (accepted names are bold) | Edo | ¢ | Error ¢ |
|---|---|---|---|---|---|
| 119.443 | 15/14 | Septimal Major Semitone | 1\10 | 120.000 | 0.557 |
| 121.243 | 7/6 / 4√(7/5) | Quadranseptimal Supraminor Second | 1\10 | 120.000 | -1.243 |
| 124.511 | 4√(4/3) | Quadranpyth Supraminor Second | 3\29 | 124.138 | -0.373 |
| 128.298 | 14/13 | Tridecimal Supraminor Second | 3\28 | 128.571 | 0.273 |
| 133.435 | √(7/6) | Semiseptal Neutral Second | 1\9 | 133.333 | -0.102 |
| 138.573 | 13/12 | Tridecimal Neutral Second | 3\26 | 138.462 | -0.111 |
| 140.391 | 5√(3/2) | Quintanpyth Neutral Second | 2\17 | 141.176 | 0.785 |
| 145.628 | 4√(7/5) | Quadranseptimal Neutral Second | 4\33 | 145.455 | -0.174 |
| 150.637 | 12/11 | Undecimal Neutral Second | 1\8 | 150.000 | -0.637 |
| 157.821 | √(6/5) | Semipental Neutral Second | 5\38 | 157.895 | 0.074 |
| 165.004 | 11/10 | Undecimal Submajor Second | 4\29 | 165.517 | 0.513 |
| 166.015 | 3√(4/3) | Trienpyth Submajor Second | 4\29 | 165.517 | -0.498 |
| 170.013 | 6/5 / 4√(7/5) | Quadranseptimal Submajor Second | 1\7 | 171.429 | 1.415 |
| 175.489 | 4√(3/2) | Quadranpyth Major Second | 6\41 | 175.610 | 0.121 |
