Third
For intervals with a denominator of 3, see Perfect fourth and 5/3.
A third is an interval that spans, or could reasonably span, two steps of the 7-form. The most significant harmonic feature of thirds is that two thirds of opposite qualities stack to a perfect fifth and form a tertian triad (and therefore, qualities of thirds can be defined with respect to the size of the perfect fifth). Therefore, xenharmonic thirds are a way to extend familiar Western triadic concepts to new tunings. Basic just intonation thirds include 9/7, 5/4, 6/5, and 7/6. Thirds are generally split into major, neutral, and minor.
In tertian harmony, the quality of the third in a triad often determines the character of said triad, giving the third a central role in diatonic harmony. It is for this reason, and the reason that a set of diatonic intervals sharing a quality may be constructed trivially from a third of said quality and a perfect fifth (although this applies to all diatonic ordinals), that the structure and tuning of the thirds in a given tuning system, together with the tuning of the fifth, are seen as somewhat representative of the characteristics of the tuning system as a whole.
Name
The term third comes from conventional music theory, wherein the degrees of the diatonic scale are numbered from 1 and their corresponding intervals are given the numbers' ordinals. Thus, a third encompasses three staff positions. This is somewhat counterintuitive, as thirds span two heptatonic scale steps.
Several other coincidental ways in which "third" describes the interval region have been noted in the xenharmonic community:
- A major third is approximately 1\3 ("one third" of an octave, logarithmically speaking).
- A minor third is the third interval (zero-indexed) of the 12-form.
- Consequently, the cent values of many thirds begin with the digit 3.
- A third is the third interval (zero-indexed) of the 10-form.
In heptatonic scales
The concept of a third may be defined in terms of a heptatonic moment-of-symmetry scale, which always has two varieties of thirds:
- Diatonic has three major thirds ranging from 343 to 480 cents (basic tuning: 400 cents), and four minor thirds ranging from 240 to 343 cents (basic tuning: 300 cents). These are the diatonic major third and diatonic minor third, and are often tuned specifically to approximate certain qualities of third in various regular temperaments, and in their just tunings of 81/64 = ~408c and 32/27 = ~296c respectively are usually the intervals meant when an unspecified "major" or "minor" is present in diatonic theory/notation or most (pyth-spine) JI notation systems.
- In rank-3 diatonic scales under the scale quality scheme, the tuning of thirds is linked to that of sixths, sevenths, and optionally seconds, but independent of that of fifths.
- Antidiatonic has four major thirds ranging from 343 to 600 cents (basic tuning: 400 cents), and three minor thirds ranging from 0 to 343 cents (basic tuning: 267 cents).
- Onyx has two major thirds ranging from 343 to 1200 cents (basic tuning: 450 cents), and five minor thirds ranging from 0 to 343 cents (basic tuning: 300 cents).
- Archaeotonic has five major thirds ranging from 343 to 400 cents (basic tuning: 369 cents), and two minor thirds ranging from 200 to 343 cents (basic tuning: 277 cents).
- Mosh has six perfect thirds ranging from 343 to 400 cents (basic tuning: 360 cents), and one diminished third ranging from 0 to 343 cents (basic tuning: 240 cents).
- Smitonic has six perfect thirds ranging from 300 to 343 cents (basic tuning: 327 cents), and one augmented third ranging from 343 to 600 cents (basic tuning: 436 cents).
As an interval region
Thirds as an interval region generally range from about 260 to 440 cents, with major thirds in the larger portion of the range and minor thirds in the smaller portion, and interordinal intervals from 240-260 and 440-460 cents can also function as thirds.
Major thirds
Major thirds are the larger variety of third, at approximately 400 cents in size. 5/4 and 9/7 are examples of major thirds.
Minor thirds
Minor thirds serve as the fifth-complements of major thirds: for every major third, there is a unique corresponding minor third which stacks together with it to form a perfect fifth (given a particular tuning of said fifth). Minor thirds are approximately 300 cents in size.
Neutral thirds
Neutral thirds are "in-between" major and minor thirds. Two neutral thirds stack to a perfect fifth; pairs of neutral thirds are at most distinct from one another by a comma-sized interval, which may be tempered out to equate the thirds to the perfect semififth, so that two equal neutral thirds can produce a 3/2 fifth. In a matching pair of unequal neutral thirds, the larger is called tendoneutral, and the smaller is called artoneutral; this is not to be confused with the separate "arto" and "tendo" interval qualities. Neutral triads are often described as being ambiguous or "neutral" in quality between those of major and minor; in the absence of major and minor triads, they may function as both simultaneously (as in 7edo).
Thirds as scale generators
- Thirds flat of 267 cents generate semiquartal. This is associated with semaphore temperament, although semaphore generators tend to be tempered closer to seconds than thirds.
- Thirds between 267 and 300 cents generate gramitonic; this is the characteristic scale of orwell temperament (generator around 272 cents).
- Thirds between 300 and 343 cents generate smitonic; this is the characteristic scale of amity (~336c), kleismic (~317c), and myna (~310c) temperaments, as well as many keemic temperaments with generators around 323 cents.
- Thirds between 343 and 400 cents generate mosh; this is the characteristic scale of rastmic (and thus mohajira) (~348c) alongside magic (~380c) and wurschmidt (~387c).
- Thirds sharp of 400 cents generate checkertonic; this is the characteristic scale of squares (~425c) and sensi (~445c) temperament.
In just intonation
TODO: explain comma inflections and Pythagorean thirds
In Pythagorean tuning, the perfect hemififth is sqrt(3/2), or about 351 cents. This is the "perfect" neutral third, and stacks twice to reach the perfect fifth. It is not a just interval, but is treated equivalently to one in systems such as Neutral FJS.
5-limit
The most common major third in 5-limit just intonation is 5/4, the classical major third (in JI notation systems, the 5-flat major third), which is tuned to approximately 386 cents in pure tuning. In a triad bounded by a fifth, 5/4 produces a 4:5:6 otonality, which functions as a major chord and explains the stability and consonance of major chords in general. 5/4 is also the octave-reduced 5th harmonic, giving it significance in the structure of the 5-limit. Building a scale by stacking 4:5:6 triads produces the Zarlino diatonic major scale. The 4:5:6 triad is well-represented in 15edo (which has a stretched triad), 19edo, 22edo, 31edo, 34edo, 41edo, and 53edo.
The minor third corresponding to 5/4 is 6/5. In JI notation systems, 6/5 is the 5-sharp minor third. The two share the property of being superparticular, which is unique to them out of any pair of thirds, and produces a contrast between otonalities and utonalities involving them. For instance, the classical minor triad 10:12:15 is significantly more complex when expressed as an enumeration, and may be written as a utonality /(4:5:6). Building a scale by stacking 10:12:15 triads produces the Zarlino minor scale. The difficulty of tuning 6/5 is somewhat in between that of 5/4 and 9/7, as it is in between them in terms of prime factorization complexity and does not have any repeated prime factors. For instance, while 22edo accurately tunes 4:5:6, its approximation of 6/5 (and thus especially 10:12:15) is insufficient for many people. 27edo has a similar tuning tendency for the fifth, and tunes 6/5 accurately at the cost of 5/4. Edos with accurate fifths, however, such as 12, 34, 41, and 53, are able to tune both thirds to comparable accuracy.
7-limit
The characteristic major third of the 2.3.7 subgroup is 9/7, which is a septimal supermajor third (in JI notation systems, the 7-sharp major third), about 435 cents in size. Conversely to the classical major third, this is a very unstable interval when used in a triad (14:18:21), combining the melodic quality usually associated with major with the harmonic quality characteristic of minor. 9/7 contains a factor of 9, therefore it is very sensitive to tuning differences in the 3rd harmonic. Therefore, it is difficult to tune correctly in edos and regular temperaments without severely detuning the 7th harmonic. The 14:18:21 triad is well-tuned in 14edo (which has a compressed triad), 22edo, 27edo, 31edo, 36edo, 39edo, 41edo, 46edo, and 53edo; out of these, the only edos that derive an accurate 7th harmonic from their approximations are 31edo, 36edo, 41edo, 46edo, and 53edo. Notably, the first four of these support slendric, making it the first 2.3.7 temperament that can tune both 7/4 and 9/7 well.
The minor third corresponding to 9/7 is 7/6. 7/6 shares a number of properties with 6/5 in terms of its ratio, being superparticular (and thus appearing between adjacent notes in the harmonic series, most notably in the tetradic form of the classical major chord, 4:5:6:7) and having a similar complexity in terms of its prime factorization, substituting 5 for 7. (In JI notation systems, 7/6 is the 7-flat minor third.) As such, many 2.3.7 edos that mistune 9/7 have a more accurate 7/6, such as 32edo. However, 7/6 is more analogous to 5/4 in its function, both in terms of being the first interval in the more stable and consonant of the two septal tertian triads (6:7:9), and being the "major" version of its own class of intervals called chthonics (loosely centered around interordinal inframinor thirds), which it shares with 8/7.
11-limit
Neutral thirds
The 11-limit neutral thirds are 11/9 (~347c) and 27/22 (~355c). 11/9 may be obtained by taking the mediant of 5/4 and 6/5, and 27/22 is its fifth complement. This is the most common pair of neutral thirds, and is generally associated with Middle-Eastern music theory (for instance, 11/9 is found in the equable diatonic tetrachord 9:10:11:12). The two are so close to one another that they are often tempered together without much damage to either, such as in 41edo. However, they are very unstable in terms of tuning due to the powers of 3 in their prime factorizations; porcupine (a temperament of reasonable accuracy overall) detunes 11/9 all the way to equate it with 6/5 (due to equalizing the aforementioned tetrachord), and even 53edo must treat it as supraminor. Therefore, the temperaments that tune 11/9 and 27/22 the best are often those that equate them to one another. These include 7edo, 17edo, 24edo, 27edo (with the second-best 11/8), 31edo, 34edo, and 41edo. However, of these, only 41edo, 17edo, and 24edo, and possibly 31edo, have reasonable tunings for 11/8.
11/9 may be notated as the 11-(semi)flat neutral third or the 11-sharp minor third in JI notation systems, depending on the uninflected interval qualities available.
Major and minor thirds
The simplest pair of 11-limit major and minor thirds are:
- 14/11 (~418c), an 11-limit interval that is the size of a major third but gets represented as an imperfect fourth in many systems of JI notation (as the 11-flat 7-flat fourth), and which is between 5/4 and 9/7 in size (serving as a neogothic major third),
- 33/28 (~284c), the minor counterpart of 14/11, functioning as a neogothic minor third
13-limit
Neutral thirds
The 13-limit neutral thirds are 16/13 (~359c) and 39/32 (~343c). Unlike the 11-limit ones, these are very stable tuning-wise, as 16/13 is the octave complement of the 13th harmonic, so it is always mapped to its closest direct approximation (assuming pure octaves). These, therefore, show up as the most prominent neutral thirds generated by temperaments that detune 11/9, such as in porcupine. However, unless the fifth is sharpened considerably so that halving it approaches 16/13 (as in 10edo and 37edo), it is, unlike with 11/9 and 27/22, often best to leave the 13-limit neutral thirds separate from one another, such as in 36edo.
In JI notation systems, 16/13 may be notated as the 13-flat major third (in FJS), the 13-sharp minor third (in HEJI), or the 13-sharp neutral third depending on the choice of formal comma and the uninflected interval qualities available.
Arto and tendo thirds
The tridecimal ultramajor (or tendo) third, 13/10 (~453c), a 13-limit interordinal interval that can function as a very sharp supermajor third, or "ultramajor third"; 13/10 is more stable in a triad than its corresponding minor third
15/13 (~247c) is the tridecimal inframinor (or arto) third, and the minor counterpart of 13/10, which serves as the approximate center of the previously mentioned chthonic category, and as the less stable counterpart of 13/10 in a triad. 15/13 and 13/10 are far enough apart that they can be played simultaneously over a root without clashing; see Arto and tendo theory.
Other 13-limit thirds
26/21 (~370c) is a 13-limit major third that is flatter than 5/4, and can be called a "submajor third".
13/11 (~289c) is a 13-limit interval close to 14/11 which is often tempered together with 14/11 in neogothic harmony; this tunes the fifth sharply enough to generate these thirds as the mosdiatonic major and minor thirds.
17-limit
17/14 (~336c) is a 17-limit minor third that is sharper than 6/5, and can be called a "supraminor third"
19-limit
19/16 (~297c) is the octave-reduced 19th harmonic. Its triad, 16:19:24, is considered by some to be a major factor in the perceived rootedness of the 12edo minor triad, given how well 12edo approximates it compared to the simpler but less rooted 6/5. 24/19 is its major counterpart.
There are also similarly tuned thirds of 19/15 (~409c) and 45/38 (~293c).
