Major third
A major third is an interval that is near 400 cents in size, distinct from the minor third of roughly 300 cents. A rough tuning range for the major third is about 360 to 460 cents, though this is extremely wide; some might prefer to restrict it to around 370-440 cents as in Schulter's theory of interval regions. Flat of major thirds (but sharp of minor thirds) are neutral thirds.
Terminology
"Major third" refers both to the ~370-450 cent range as a whole, and to a specific subdivision within it (about ~370-415 cents); major thirds sharp of this are often called "supermajor thirds".
"Major third" may also refer to the diatonic major third, which is an interval generated by stacking 4 tempered fifths and is not the subject of this article.
In just intonation
3-limit intervals in the range of major thirds include the Pythagorean major third of 81/64, about 408 cents in size, which corresponds to the MOS-based interval category of the diatonic major third and is generated by stacking four just perfect fifths of 3/2, and the Pythagorean diminished fourth of 8192/6561, which is flat of 81/64 by one Pythagorean comma, and is about 384 cents in size.
Much simpler major thirds exist in higher limits, however, for example:
- The 5-limit classical major third is a ratio of 5/4, and is about 386 cents.
- The 7-limit supermajor third is a ratio of 9/7, and is about 435 cents.
- The 11-limit neogothic major third is a ratio of 14/11, and is about 418 cents.
- The 13-limit ultramajor third is a ratio of 13/10, and is about 454 cents.
- There is also a 13-limit submajor third, which is a ratio of 26/21, and is about 370 cents.
- The 17-limit submajor third is a ratio of 21/17, and is about 366 cents.
In tempered scales
The following table lists EDOs between 5 and 26 that have intervals in the range of 370 to 415 cents ("major thirds"), as well as the tunings in 27edo, 29edo, 31edo, 34edo, 41edo, and 53edo. All EDOs larger than 26 have an interval in this range.
Due to the width of the range, 29edo, 41edo, and 53edo have two intervals within it.
| EDO | Major third |
|---|---|
| 29 (flat third) | 372c |
| 16 | 375c |
| 19 | 379c |
| 41 (flat third) | 381c |
| 22 | 382c |
| 25 | 384c |
| 53 (flat third) | 385c |
| 31 | 387c |
| 34 | 388c |
| 6, 9, 12, 15, 18, 21, 24, 27 | 400c |
| 53 (sharp third) | 408c |
| 41 (sharp third) | 410c |
| 29 (sharp third) | 414c |
The following table lists EDOs between 5 and 34 that have intervals in the range of 415 to 450 cents ("supermajor thirds"), as well as the tunings in 41edo and 53edo. All EDOs larger than 34 have an interval in this range.
| EDO | Supermajor third |
|---|---|
| 26 | 415c |
| 23 | 417c |
| 20 | 420c |
| 17, 34 | 424c |
| 31 | 426c |
| 14, 28 | 429c |
| 53 | 430c |
| 25 | 432c |
| 11, 22, 33 | 436c |
| 41 | 439c |
| 30 | 440c |
| 19 | 442c |
| 27 | 444c |
| 8, 16, 24, 32 | 450c |
The following table lists the best tuning of 5/4 and 9/7* in several significant EDOs.
*Note that 9/7 may not always be the actual best note in the EDO, this is because of the way EDO tunings try to match how intervals in just intonation are stacked to form new intervals. See Val for more information. Essentially, this is what 9/7 should be based on the best tuning of 7 itself.
| EDO | 5/4 | 9/7 |
|---|---|---|
| 9 | 400c | |
| 12 | 400c | |
| 13 | 369c | 461c |
| 16 | 375c | 450c |
| 17 | 353c | 424c |
| 19 | 379c | 442c |
| 22 | 382c | 436c |
| 24 | 400c | 450c |
| 25 | 384c | 432c |
| 26 | 369c | 415c |
| 27 | 400c | 444c |
| 29 | 372c | 455c |
| 31 | 388c | 426c |
| 34 | 388c | 459c |
| 41 | 381c | 439c |
| 53 | 385c | 430c |
In regular temperaments
In regular temperaments, major-third-sized intervals may be used in various ways, such as generators or generated intervals. The following lists go over the uses of 5/4 and 9/7 in temperaments.
This list is incomplete.
Temperaments that use 5/4 as a generator
- Magic, which generates 3/2 by stacking five 5/4s (octave-reduced).
- Father, a very inaccurate temperament which equates 4/3 and 5/4 as a single "fourth-third" interval
- Dicot, a somewhat inaccurate temperament which equates 5/4 and 6/5, tempering out the chromatic semitone that usually separates them
Temperaments that use 9/7 as a generator
- TBD
Temperaments that generate 5/4
- Meantone, a familiar temperament which flattens 3/2 so that four of them (octave-reduced) stack to 5/4
- Mavila, a temperament which sharpens 4/3 so that three of them (octave-reduced) stack to 5/4
Temperaments that generate 9/7
- Archy, a temperament which sharpens 3/2 so that four of them (octave-reduced) stack to 9/7