User:VectorGraphics/Diatonic major third
| MOS | 5L 2s |
| Other names | Major 2-diastep |
| Generator span | +4 generators |
| Tuning range | 343c - 480c |
| Basic tuning | 400c |
| Chromatically adjacent interval | Diatonic minor third |
| Adjacent tunings | Antidiatonic minor third, oneirotonic minor fifth |
| Parent interval | Pentic diminished third |
| Daughter intervals | M-chromatic minor fifth, P-chromatic major fifth |
| Associated just intervals | 5/4, 81/64 |
In the diatonic scale, the major third is the major variant of the 2-diastep, or third. It is generated by stacking 4 diatonic perfect fifths and octave-reducing. It can be stacked with a diatonic minor third to form a perfect fifth, and as such is often involved in chord structures in diatonic harmony.
Name
In TAMNAMS, this interval is called the major 2-diastep. However, because the diatonic scale is the standard scale of Western theory, it is more commonly called a major third, such as in standard Western notation, chain-of-fifths notation and quasi-diatonic MOS notation.
Tunings
Being an abstract MOS degree, and not a specific interval, the diatonic major third doesn't have a fixed tuning, but instead has a range of ways it can be tuned, based on the tuning of the generator used in making the scale.
The tuning range of the diatonic major third ranges from 342.8 cents to 480 cents. Sharp of this, it becomes an oneirotonic minor fifth, and flat of this, it becomes an antidiatonic minor third.
The diatonic major third is itself a type of pentic diminished third, and contains the categories of m-chromatic minor fifth and p-chromatic major fifth, corresponding to the flat-of-basic and sharp-of-basic tunings of the major third respectively.
| Tuning | Step ratio | Edo | Cents |
|---|---|---|---|
| Equalized | 1:1 | 7 | 343c |
| Supersoft | 4:3 | 26 | 369c |
| Soft | 3:2 | 19 | 379c |
| Semisoft | 5:3 | 31 | 387c |
| Basic | 2:1 | 12 | 400c |
| Semihard | 5:2 | 29 | 414c |
| Hard | 3:1 | 17 | 424c |
| Superhard | 4:1 | 22 | 436c |
| Collapsed | 1:0 | 5 | 480c |
In regular temperaments
If the diatonic perfect fifth is treated as 3/2, approximating various intervals with the diatonic major third leads to the following temperaments:
| Just interval | Cents | Temperament | Tempered comma | Generator (eigenmonzo tuning) |
|---|---|---|---|---|
| 27/22 | 355c | Io | 33/32 | Perfect fifth ≈ 689c |
| 16/13 | 359c | Superflat | 1053/1024 | Perfect fifth ≈ 690c |
| 21/17 | 366c | Temperament of 459/448 | 459/448 | Perfect fifth ≈ 692c |
| 5/4 | 386c | Meantone | 81/80 | Perfect fifth ≈ 697c |
| 81/64 | 408c | Pythagorean | 1/1 | Perfect fifth ≈ 702c |
| 14/11 | 418c | Parapyth/Pentacircle | 896/891 | Perfect fifth ≈ 705c |
| 9/7 | 435c | Archy/Superpyth | 64/63 | Perfect fifth ≈ 709c |
| 13/10 | 454c | Oceanfront/Temperament of 416/405 | 416/405 | Perfect fifth ≈ 714c |