Functional harmony in rank-2 temperaments
Using functonal harmony in rank-2 temperaments can be challenging. Diatonic functional harmony is virtually based on one temperament over the centuries: meantone, but what works in meantone may or may not work in other rank-2 temperaments. This guide discusses some techniques that may be helpful for transferring functional harmony to those temperaments.
Functions based on septimal harmony
In septimal meantone, the interval class of 5 is reached by a relatively low number of generator steps whereas the interval class of 7 is way more complex, so classical intervals are very accessible in the diatonic scale whereas septimal intervals are only available in the chromatic or even enharmonic scale (except for ~7/5, which is mapped to the tritone). This greatly shapes the way we use harmony in it: the classical chords are the basis of harmony, and septimal chords are only used sparingly for additional spices. For other temperaments generated by a fifth, the condition tends to differ. In superpyth, the interval class of 7 is only -2 generator steps, while the interval class of 5 is +9 steps away, so the temperament inherently favors prime 7 over 5. If we continue to use it the old way, the harmonic resource we get from a finite scale will be sparse. To address that, we can swap the functional roles of primes 5 and 7 according to their temperamental complexities (number of generator steps), and this means using septimal chords as the basic form of harmony, only inserting classical chords occasionally for additional spices.
Example: superpyth
Intuitively, one can think of the basic form of septimal harmony as follows:
- 1–7/6–3/2 (0–(−3)–1)
- 1–9/7–3/2 (0–4–1)
As Flora Canou's essay Analysis on the 13-Limit Just Intonation Space: Episode I has identified, the classical triads 1–5/4–3/2 and 1–6/5–3/2 are rooted and uprooted, respectively, but these septimal triads are not. Another possible complaint is that the chroma between the "major" and "minor" triads is too large. One solution is to insert another tone to the chords, resulting in
- 1–7/6–21/16–3/2 (0–(−3)–(−1)–1)
- 1–8/7–9/7–3/2 (0–2–4–1)
Particular to superpyth, note that 1–7/6–21/16–3/2 can also be expressed as 1–7/6–4/3–3/2 and that 1–8/7–9/7–3/2 can also be expressed as 1–9/8–9/7–3/2.
Another solution, proposed in the essay, is to adopt semiquartal harmony, a chord construction scheme analogous to tertian harmony but for septimal chords:
- 1–8/7–4/3 (0–2–(−1))
- 1–7/6–4/3 (0–(−3)–(−1)))
Flora specifically proposes this voicing:
- 1–7/4–3 (0–(−2)–1)
- 1–12/7–3 (0–3–1))
which is suppoed to sound better with very wide voicing variations.
Superpyth[7], the diatonic scale of superpyth, contains 3 copies of the tetrads and 4 copies of the rooted/uprooted triads, and with Superpyth[12], the chromatic scale, classical chords are available here and there.
Functions based on generators
In septimal meantone, the most consonant interval besides the octave, the perfect fifth (3/2), is a generator, meaning it is the most abundant interval in any mos scale. This is often not true for other temperaments. Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.
Example: magic
In terms of generator steps of magic, the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. We also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.
Example: hanson
In terms of generator steps of hanson, the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.
Combination of techniques
We can combine the two techniques discussed above, which means we can:
- Treat septimal chords as the basic form of harmony, and
- Define the functions on the generator.
Example: orwell
Orwell is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and tetrads alike.
Example: sensi
Sensi is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.
Other role-swapping techniques
All the techniques discussed above may be collectively called role swapping. As the name suggests, they all involve swapping the functional roles of certain primes according to their temperamental complexities. To better illustrate this, we abstract the following hierarchy of functional roles for intervals of 3, 5, and 7 from meantone:
- Primary role
- The spine on which the functions are defined.
- Secondary role
- The main flavors for harmony.
- Tertiary role
- The additional spices for harmony.
Note that "intervals of 5" may be 5, 5/3, sometimes 9/5 or even 15, all depending on the temperament; same with intervals of 7. Other primes may be swapped in too. For instance, intervals of 13 have a great tertiary role in hanson. The following table shows a more typical role swapping result in orwell.
| Temperament | Intervals of 3 | Intervals of 5 | Intervals of 7 |
|---|---|---|---|
| Meantone | Primary | Secondary | Tertiary |
| Orwell | Secondary | Tertiary | Primary |
We can, as a bonus, identify the "nullary" role for prime 2:
- Nullary role
- The interval of equivalence.
Adding that to the table, it might give us a clue about using temperaments in non-octave settings.
Breaking out of mos
In contrast to role swapping, there is a one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain.
This approach entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always ~3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about ratios and comma pumps. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.
Example: würschmidt
Würschmidt is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.
Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional lattice up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.