Archy
Archy (22 & 27) is the temperament that tempers out the archytas comma, 64/63, equating septal intervals with nearby diatonic ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened fourths of about 490 cents stack to a sharply tuned 7th harmonic of about 980 cents. Equivalently, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.
Archy is usually tuned such that the subminor (7/6) third is close to accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale. The tuning that justly tunes the harmonic seventh places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 cents.
As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard diatonic notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an equipentatonic scale, so perhaps diamond-MOS or KISS notation with pentic would be better suited for it.
Structural theory
Extensions
The following are extensions to prime 5 (i.e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy).
Archy can be defined as 5 & 7, which is the meantone extension dominant in the full 7-limit. Other extensions follow:
5/4 as limma-flat major third (22 & 27), often called "Superpyth"
The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the septimal quartertone (36/35, the interval between 5/4 and 9/7) and the syntonic comma. It can be seen as the 22 & 27 temperament. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred. It is a 5-cluster temperament, as indicated by the edo join (27 - 22 = 5).
5/4 as doubly limma-flat major third (5 & 37)
This is an alternative extension, best tuned at least as sharp as 32edo. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.
This range is often interpreted as Oceanfront or Ultrapyth, equating the diatonic major third (already interpreted as 9/7) to 13/10. A recommendable tuning for this temperament is 37edo.
Machine temperament
Machine, 11 & 17, is a weak restriction to 2.9.7.11 of Supra, the [17 & 22] subrange of Archy. It is generated by 9/8~8/7, three of which make a 16/11. It's straddle-3 in that the ~14/9 is the >3 and the ~16/11 is the <3.
- 11edo nearly divides 9/7 in half
- 17edo provides a near-isodifferential tuning of ~8:11:14, actually much closer to 13:18:23
- 28edo (generator 5\28, 214.3c) provides a near-isodifferential ~7:9:11, which is actually much closer to 32:41:50.
Machine generates machinoid (5L1s) and 6L5s.
Compositional theory
Chords
In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the supermajor triad and subminor triad respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The triad [0 4/3 7/4~14/9] is an important essentially tempered chord, although HKM finds that its other closed-voice inversions do not sound as if they contain septimal intervals unless the fifth is tuned as sharp as that of 37edo.
Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad.
Full 7-limit harmony
This section assumes a reasonably accurate extension to prime 5 is used. The precise extension does not particularly matter, but an up / down symbol represents the difference between 9/7 and 5/4.
The primary characteristc of archy in a full 7-limit context is that a zarlino dominant chord (found in the zarlino tuning of Mixolydian) is a 4:5:6:7 harmonic seventh chord.
Leading tones
The semitones found in archy's MOS diatonic are too narrow to use as leading tones. The nearminor seconds provided by a 5-limit extension may be seen as too wide (usually exceeding the "optimal" size of a leading tone presented by George Secor at 70 cents, depending on the tuning). However, they align with Aura's system of functional harmony, which places the 70-cent leading tone at the intersection of two other functional categories at around 110 cents and 50 cents respectively - the collocant and gradient functions. The collocant functions as a conventional leading tone, whereas the gradient functions as a passing tone to either jump past the tonic or resolve to the collocant. In this case, the larger nearminor second represents the collocant, meanwhile the smaller subminor second represents the gradient.
Further functional harmony
There are four distinct "keys" in the 7-limit (nearmajor, supermajor, nearminor, subminor), as compared to two in MOS diatonic alone, where a key is defined as a system of tonal hierarchy based around a certain interval quality or tonic chord (independent of absolute pitch), which will be elaborated on below. Note that relative major or minor depends on whether the key is near- or super/sub, and that, for instance, nearmajor and supermajor use different scales that are not rotations of one another. In specific, using ups and downs notation, C Nearmajor corresponds to vA Nearminor, meanwhile C Supermajor corresponds to A Subminor, and in general nearmajor-nearminor relative correspondences acquire an additional down accidental compared to standard MOSdiatonic correspondences.
The chirality of the nearmajor or nearminor scale in question is ultimately of little relevance (see blackdye; in short, the major second in nearmajor (and the fourth in nearminor) may be either note depending on context), but in general the right-handed version of nearmajor is assumed due to having a non-wolf V chord, and the left-handed version of nearminor is assumed due to having a non-wolf fourth over the tonic.
The heptatonic interval functions remain as they are in 12edo, although with the caveat that the ideal leading tone ends up at the nearminor second rather than the semitone found in MOSdiatonic, which has implications for the subminor and supermajor keys and turns the use of the diatonic scale into a balancing act between the functional utility of MOSdiatonic and the tension of the leading tones in zarlino diatonic. (In particular, it suggests the use of a "harmonic supermajor" by flattening the seventh of supermajor by an edostep.)
Nearmajor key
In nearmajor (the key with the nearmajor tonic chord), the fourth acts as it usually does in MOS major, serving as a tendency tone towards the third. The basic tonal identity for nearmajor is 4:5:6, which extends generally to a nearmajor seventh chord, although a dominant (harmonic in archy temperament) seventh is also possible, and more justified in archy due to naturally extending the harmonic series segment corresponding to 4:5:6.
Nearminor key
Nearminor harmony functions somewhat similarly to how you expect, with the nearminor sixth functioning as a leading tone down to the fifth and the seventh being able to be raised to a nearmajor seventh in order to give a more directed dominant resolution. The whole tone also provides a lead up to the minor third, like in standard diatonic.
Melodic minor scales are somewhat interesting here as well, as there are a couple different reasonable ways to construct them, which would likely depend on the chords being used and the desired melodic contour.
Supermajor key
In supermajor, a lead to the third would be a wolf fourth (11/8), perhaps justifying its inclusion in the scale over the fourth proper, or the functional alternation between the two in different contexts.
The functionality of the seventh grows increasingly complicated in supermajor - while in 12edo, one may only see, for instance, a dominant chord replacing the I chord, in the 7-limit there are four different potential types of seventh, all with justifications. A fifth over the third would be a supermajor seventh (notably serving as the MOSdiatonic maj7, and distinguishing itself from the 12edo maj7 by not leading up to its own root), a tritone (neardim 5; 7/5) over the third would be a nearminor seventh, a lead up to the tonic would be a nearmajor seventh, and finally the MOS diatonic dominant chord utilizes a subminor seventh. Therefore, an alternate version of the supermajor scale usable in certain contexts makes the fourth wolf and the seventh nearmajor.
This also means that the regular perfect fourth isn't as unstable an interval or as functionally dissonant in supermajor - in fact, the third is actually somewhat of a tension compared to it (though the step between them is smaller than the size of a conventional leading tone).
Subminor key
The same kind of justification emerges for harmonic subminor, except that there is little reason to alter the seventh all the way up to a supermajor seventh if the objective is for it to function as a leading tone. In fact, the same logic can be used against a dominant chord with a nearminor seventh in nearmajor - leading inwards to a nearmajor third by equal semitones on either side requires that the initial interval be a neardiminished fifth, and that the chord to be used as a dominant is actually a harmonic 4:5:6:7 on the fifth. (Resolving to a supermajor chord actually wants a dom7 with a nearmajor third and nearminor seventh, if quartertones are not to be used).
In general, archy's functional harmony ends up a lot more context-bound and much less scale-bound than 12edo's, due to the multiple different qualities of intervals and notes doing different things, and the ideal leading tone not matching the standard diatonic structure.
Alternative leading tones
An alternative approach to simplify things is instead to discard Aura's theory of leading in favor of treating the quartertone as the optimal leading tone (as it is the diatonic major seventh), an entirely different paradigm emerges. Supermajor and subminor become definitive, stable diatonic tonality systems, with no awkwardness around leading tones, behaving identically to any MOSdiatonic temperament (albeit with the different, somewhat inverted "moods" presented by the supermajor and subminor intervals). Meanwhile, the nearmajor and nearminor scales acquire new "harmonic" variations, with the final note raised up to a quartertone below the tonic. In effect, supermajor/subminor and nearmajor/nearminor "switch" in regards to some functions. Instead of raising the fourth in supermajor, it is in this system viable to lower it in nearmajor. The best dom7 to resolve to a nearmajor triad on the tonic features a seventh lowered to one step below the subminor seventh, alongside the supermajor third, and can consequently be reanalyzed as a subminor seventh chord on the 9/7 over the tonic. The MOSdiatonic dominant seventh serves to resolve to a MOSdiatonic major triad, as in 12edo.
Consonant vs. tense suspended chords
The wider supermajor second and contrast with the supermajor third actually makes suspended chords somewhat of a point of resolution, rather than a point of tension like in 12edo. It's reasonable to have a suspended chord that doesn't resolve, perhaps making the term "suspended" inaccurate. These suspended chords can function like arto and tendo chords, with a 1-2-4-5 chord structure being plausible, or can be used in modal harmony as a form of "mode-agnostic" anchor point. The sus4 chord in particular is composed of the three octave-reduced perfect consonances, and thus can also be considered the most basic polychordal scale (perhaps a/the "dichordal" scale). However, suspensions that function more like 12edo ones in leading into the MOS diatonic intervals and being more tense can still be found with the nearmajor sus2 and wolf sus4, which lose some of the structural elegance of standard Pythagorean suspensions in favor of a more tense, crowded sound that can easily resolve to even the rather tense supermajor triad.
Scales
Superpyth diatonic
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of porcupine. A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone.
It may be useful, perhaps for extraclassical tonality, to use the full 12-note form of superpyth's scale.
Superpyth pentatonic
Zarlino
Similar to the previous section, this assumes a reasonably accurate extension to prime 5.
List of patent vals
| EDO | Extension to 5 | Generator tuning | Diatonic scale hardness | 7/4 tuning |
|---|---|---|---|---|
| 3 | 800c | N/A | 800c | |
| 8 | 450c | N/A | 900c | |
| 13 | 461.5c | N/A | 923.1c | |
| 5 | [5 & 37], [22 & 27], dominant | 480c | ∞ (collapsed) | 960c |
| 42 | [5 & 37] | 485.7c | 8 | 971.4c |
| 37 | [5 & 37] | 486.5c | 7 | 973c |
| 32 | [5 & 37] | 487.5c | 6 | 975c |
| 59 | 488.1c | 5.5 | 976.3c | |
| 27 | [22 & 27] | 488.9c | 5 | 977.8c |
| 49 | [22 & 27] | 489.8c | 4.5 | 979.6c |
| 22 | [22 & 27] | 490.9c | 4 | 981.8c |
| 17 | [22 & 27] | 494.1c | 3 | 988.2c |
| 12 | dominant | 500c | 2 | 1000c |
| 7 | dominant | 514.3c | 1 (equalized) | 1028.6c |
| 2 | 600c | N/A | 1200c |
