Neutral third
A neutral third is an interval that is near 350 cents in size, distinct from the minor third of roughly 300 cents and the major third of roughly 400 cents. A rough tuning range for the neutral third is 330 to 370 cents, as in Schulter's theory of interval regions.
The neutral third range is generally divided at roughly 350 cents into artoneutral (flat) and tendoneutral (sharp) thirds. An artoneutral third has a tendoneutral third as a fifth complement, and so neutral thirds will often be provided in pairs.
In just intonation
The 3-limit and 5-limit do not have simple neutral thirds, so we start with the 7-limit:
- The 7-limit artoneutral and tendoneutral thirds are the ratios of 60/49 and 49/40 respectively, and they are slightly flat of and slightly sharp of 351 cents respectively.
- The 11-limit alpharabian artoneutral and tendoneutral thirds are the ratios of 11/9 and 27/22 respectively, and they are about 347 and 355 cents respectively.
- The 13-limit artoneutral and tendoneutral thirds are the ratios of 39/32 and 16/13 respectively, and they are about 342 and 359 cents respectively.
- The 17-limit supraminor and submajor thirds are the ratios of 17/14 and 21/17 respectively, and they are about 336 and 366 cents respectively.
In EDOs
The following table lists the best tuning of 39/32 and 16/13 in various significant EDOs. For applicable EDOs, it also lists one half of the EDO's perfect fifth, approximating 1\2edf, which, while not a just interval, is the "canonical" neutral third tuning, as stacking two of them gives 3/2.
| EDO | 1\2edf | 39/32 | 16/13 |
|---|---|---|---|
| 7 | 343c | ||
| 17 | 353c | ||
| 24 | 350c | ||
| 25 | - | 336c | |
| 26 | - | * | 369c |
| 27 | 356c | ||
| 29 | - | 331c | * |
| 31 | 348c | ||
| 34 | 353c | ||
| 41 | 351c | ||
| 53 | - | 340c | 362c |
In regular temperaments
Temperaments generated by neutral thirds often involve tempering a pair of neutral thirds together. As such, each pair of neutral thirds has a corresponding temperament, which equates both neutral thirds to half of a perfect fifth:
| Pair of neutral thirds | Temperament |
|---|---|
| 60/49, 49/40 | Breedsmic |
| 11/9, 27/22 | Rastmic |
| 39/32, 16/13 | Temperament of 512/507 |
| 17/14, 21/17 | Temperament of 294/289 |