Major third (diatonic interval category): Difference between revisions
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Being an abstract mos degree, and not a specific interval, the diatonic major third does not 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. | Being an abstract mos degree, and not a specific interval, the diatonic major third does not 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 to 480{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic major third can be found by {{ | The tuning range of the diatonic major third ranges from 342.8 to 480{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic major third can be found by {{nowrap| (''n'' + 2400)/4 }}. For example, the third 384{{c}} gives us {{nowrap| (384 + 2400)/4 {{=}} 2784/4 {{=}} 696{{c}} }}, corresponding to 50edo. | ||
Several example tunings are provided below: | Several example tunings are provided below: | ||
{| class="wikitable" | {| class="wikitable center-all left-1" | ||
|+ style="font-size: 105%;" | Tunings of the major third | |+ style="font-size: 105%;" | Tunings of the major third | ||
|- | |- | ||
| Line 84: | Line 84: | ||
If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major third leads to the following temperaments: | If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major third leads to the following temperaments: | ||
{| class="wikitable" | {| class="wikitable center-2 center-5" | ||
|- | |- | ||
! Just interval | ! Just<br>interval | ||
! Cents | ! Cents | ||
! Temperament | ! Temperament | ||
! | ! Vanishing<br>comma | ||
! Generator (eigenmonzo tuning) | ! Generator<br>(eigenmonzo tuning) | ||
|- | |- | ||
| [[27/22]] | | [[27/22]] | ||
| Line 96: | Line 96: | ||
| [[Io]] | | [[Io]] | ||
| [[33/32]] | | [[33/32]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 689{{c}} }} | ||
|- | |- | ||
| [[16/13]] | | [[16/13]] | ||
| Line 102: | Line 102: | ||
| [[Superflat]] | | [[Superflat]] | ||
| [[1053/1024]] | | [[1053/1024]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 690{{c}} }} | ||
|- | |- | ||
| [[21/17]] | | [[21/17]] | ||
| Line 108: | Line 108: | ||
| Temperament of 459/448 | | Temperament of 459/448 | ||
| 459/448 | | 459/448 | ||
| {{nowrap| | | {{nowrap| P5 ≈ 692{{c}} }} | ||
|- | |- | ||
| [[5/4]] | | [[5/4]] | ||
| Line 114: | Line 114: | ||
| [[Meantone]] | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 697{{c}} }} | ||
|- | |- | ||
| [[81/64]] | | [[81/64]] | ||
| 408{{c}} | | 408{{c}} | ||
| [[Pythagorean tuning | | [[Pythagorean tuning]] | ||
| [[1/1]] | | [[1/1]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 702{{c}} }} | ||
|- | |- | ||
| [[14/11]] | | [[14/11]] | ||
| 418{{c}} | | 418{{c}} | ||
| [[ | | [[Pepperoni]] | ||
| [[896/891]] | | [[896/891]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 705{{c}} }} | ||
|- | |- | ||
| [[9/7]] | | [[9/7]] | ||
| Line 132: | Line 132: | ||
| [[Superpyth|Archy/superpyth]] | | [[Superpyth|Archy/superpyth]] | ||
| [[64/63]] | | [[64/63]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 709{{c}} }} | ||
|- | |- | ||
| [[13/10]] | | [[13/10]] | ||
| 454{{c}} | | 454{{c}} | ||
| [[Oceanfront]]/ | | [[Oceanfront]] / temperament of 416/405 | ||
| [[416/405]] | | [[416/405]] | ||
| {{nowrap| | | {{nowrap| P5 ≈ 714{{c}} }} | ||
|} | |} | ||
Revision as of 12:46, 17 April 2025
| MOS | 5L 2s |
| Other names | Major 2-diastep |
| Generator span | +4 generators |
| Tuning range | 343–480 ¢ |
| Basic tuning | 400 ¢ |
| Function on root | Mediant |
| Interval regions | Neutral third, major third, naiadic |
| Associated just intervals | 5/4, 81/64 |
| Octave complement | Minor sixth |
A major third (M3) is an interval that spans two scale steps in the diatonic scale with the major (wider) quality. It is generated by stacking 4 fifths octave reduced, and depending on the specific tuning, it ranges from 343 to 480 ¢ (2\7 to 2\5). In just intonation, an interval may be classified as a major third if it is reasonably mapped to two steps of the diatonic scale and four steps of the chromatic scale, or formally 2\7 and 8\24. The use of 24edo's 8\24 as the mapping criteria here rather than 12edo's 4\12 better captures the characteristics of many intervals in the 11- and 13-limit.
The major third can be stacked with a minor third to form a perfect fifth, and as such is often involved in chord structures in diatonic harmony.
In TAMNAMS, this interval is called the major 2-diastep.
Scale info
The diatonic scale contains three major thirds. In the Ionian mode, major thirds are found on the first, fourth, and fifth degrees of the scale; the other four degrees have minor thirds. This roughly equal distribution leads to diatonic tonality being largely based on the distinction between major and minor thirds and triads.
Tunings
Being an abstract mos degree, and not a specific interval, the diatonic major third does not 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 to 480 ¢. The generator for a given tuning in cents, n, for the diatonic major third can be found by (n + 2400)/4. For example, the third 384 ¢ gives us (384 + 2400)/4 = 2784/4 = 696 ¢, corresponding to 50edo.
Several example tunings are provided below:
| Tuning | Step ratio | Edo | Cents |
|---|---|---|---|
| Equalized | 1:1 | 7 | 343 ¢ |
| Supersoft | 4:3 | 26 | 369 ¢ |
| Soft | 3:2 | 19 | 379 ¢ |
| Semisoft | 5:3 | 31 | 387 ¢ |
| Basic | 2:1 | 12 | 400 ¢ |
| Semihard | 5:2 | 29 | 414 ¢ |
| Hard | 3:1 | 17 | 424 ¢ |
| Superhard | 4:1 | 22 | 436 ¢ |
| Collapsed | 1:0 | 5 | 480 ¢ |
In regular temperaments
P5 = 3/2
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 | Vanishing comma |
Generator (eigenmonzo tuning) |
|---|---|---|---|---|
| 27/22 | 355 ¢ | Io | 33/32 | P5 ≈ 689 ¢ |
| 16/13 | 359 ¢ | Superflat | 1053/1024 | P5 ≈ 690 ¢ |
| 21/17 | 366 ¢ | Temperament of 459/448 | 459/448 | P5 ≈ 692 ¢ |
| 5/4 | 386 ¢ | Meantone | 81/80 | P5 ≈ 697 ¢ |
| 81/64 | 408 ¢ | Pythagorean tuning | 1/1 | P5 ≈ 702 ¢ |
| 14/11 | 418 ¢ | Pepperoni | 896/891 | P5 ≈ 705 ¢ |
| 9/7 | 435 ¢ | Archy/superpyth | 64/63 | P5 ≈ 709 ¢ |
| 13/10 | 454 ¢ | Oceanfront / temperament of 416/405 | 416/405 | P5 ≈ 714 ¢ |
In just notation systems
Due to the way the primes 7 and 11 are notated, in many systems of notation for just intonation, the interval 14/11 is not considered to be a major third, but instead belongs to the enharmonic category of diminished fourth.
See also
- Major third (disambiguation page)