Neutral second
Neutral seconds are intervals with a size in between a whole tone and a semitone. 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.
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 (T10/9), 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 T10/9 is a slightly larger 168¢ in exchange for making the T81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
Categorization
Neutral seconds are generally considered to be between 1\9 and 1\7, being half of some sort of minor third. If semifourths are included, they can be as narrow as 1\10.
Proposal: Ground's Neutral Second Categorization System
Import splitting table here.
| ¢ | Definition | Name (accepted names are bold) | Edo | ¢ | Error ¢ |
|---|---|---|---|---|---|
| 109.050 | 7/6 / √(6/5) | Sepsemipental Semitone | 1\11 | 109.091 | 0.041 |
| 111.731 | 16/15 | Pental Semitone | 3\32 | 112.500 | 0.769 |
| 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 |
| 182.206 | 6/5 / √(7/6) | Pensemiseptal Major Second | 5\33 | 181.818 | -0.388 |
| 182.404 | 10/9 | Minor Whole Tone | 5\33 | 181.818 | -0.586 |
