User:Eufalesio/Harmonic tiers

This is mainly a description of my mental categories of primes and odds and how they make sense in my brain.

Preamble

Harmonics are an extremely important basis to judge the efficacy of a tuning system, and though there are infinite harmonics, some are more important than others. How would one determine such rating? And which harmonics merit a rating?

One way is to determine some sort of harmonic uniqueness. Since harmonics are essentially integer frequency multiples, primes fit this category perfectly, as any composite harmonic can be uniquely built by a combination of prime harmonics. This is an extension of the Fundamental Theorem of Arithmetic, applied to music. I will

Prime ratings

Tier I

There are two members here: Primes 2 and 3, octave reducing to... the octave... and the fifth. Technically speaking, octaves would belong to Tier 0, but that's a trivial category.

This tier is built from primes that are extremely important. A tuning system will very likely have these primes and tune them extremely precise, if not exact; for example, all edos by definition tune the octave perfectly, and edos that tune the fifth accurately are also very important, or use stack them to build all other intervals, as is the case with many rank-2 temperaments and historical temperaments. These are the only primes can stack easily with everything else.

• Prime 2: So important that you only care about it when you work without it. Some people, like Aura, treat it differently as the rest of primes, in that it adds luminosity to an interval class, instead of being treated as its own interval class. I treat it as its own interval class, distinct from the unison (the fundamental). Melodically, it adds a lot of punch to melodies, but it should only be used once.
• Prime 3: Extremely important. It's the first prime that adds sonic variety of any kind, and with octaves, it builds an infinite universe called the 3-limit. Melodically, it makes for an incredible leap to drive momentum, and when stacked and octave reduced, it makes for incredible "low complexity" steps, such as 256/243 and 9/8.

Tier II

There are two members here: Primes 5 and 7, octave reducing to the just major third and the septimal harmonic seventh.

This tier is built from primes that are quite important. A tuning system will probably make an effort to tune this accurately, but it's not a necessity. However, these primes add flavor to music. At least, that's what I think.

• Prime 5: Important mostly due to tradition, as it sits right beneath the unison and the fifth and makes for some sweet sweet 4:5:6 chords. It leaves a good taste on the mouth when using this interval to finish a cadence. Melodically, it is perfect for small leaps forward and it combines perfectly with 3.
• Prime 7: Often overlooked in tradition, it's an interesting prime. A bit alien, due to its notable absence in 12edo. Because its octave complement 8/7 is close to the unison, some people consider it a weird mix between consonance and dissonance. I personally consider it a consonance. Less than 5, but consonant nonetheless, specially in the chord 6:7:9. Melodically, it is a bit jarring, as it tends to create tiny intervals in scales with plenty of 7, namely 28/27 or 49/48.

Tier III

There are two primes here: 11 and 13, octave reducing to the undecimal harmonic fourth, and the tridecimal harmonic sixth.

This tier is built from truly xenharmonic primes. Western music theory considers these harmonics truly alien, but I've grown so accustomed to these primes that they now feel natural. These primes are less consonant, but still can be treated as consonances.

Melodically, they are both the same. There are pairs of 11- and 13- limit intervals that sound alike, like 39/32~11/9, 27/22~16/13. The comma that equates both pairs is 352/351, so it can be safely fudged or tempered. They work extremely well melodically, giving a nice xenharmonic punch to any music rooted in the 5- or 7- limit.

• Prime 11: It's the first xenharmonic prime. Good flavor, but in my opinion, it's harder to use than 13. Since it can be considered a type of "fourth", "augmented fourth" or "half-augmented fourth" (depending on how precise or pedantic you want to be), I see it as a deviation from a very consonant interval: the fourth 4/3.
• Prime 13: The second xenharmonic prime. Technically less consonant than 11, but it tends to be easier to work with. Part of this is the fact that it is extremely well approximated by 10n edos, and by Pythagorean tuning, placing it as the triple augmented fourth. It is also extremely easy to reach through the 7-limit too, connecting a chain of 5*7 to 13 through the schismina. The fact that it is a kind of sixth means that its interval class is not perfect, so it lends itself to readily accept xenharmonic (neutral) values. Might be just my bias, but I just really like it. And it also appears in the 12:13:14:15:16, which I really like too.

Tier IV

There are four primes here: 17, 19, 23, 29, 31, octave reducing to the harmonic septendecimal semitone, the harmonic decimononal third, the vigesimotertial harmonic tritone, the vigesimononal harmonic seventh, and the trigesimoprimal harmonic suboctave. In my own mental category, I'd make two subtiers: Tier IVa and Tier IVb, for reasons that will become clear.

These tiers contain primes that are not very consonant, some of them being in fact quite dissonant, at least in my opinion.

Tier IVa

It includes primes 17 and 19. They don't sound xenharmonic at all, as they are very well aproximated by 12edo. They

• Prime 17: It's dissonant. It's good for making locked-in harmonic chords, but that's it. I don't like it harmonically. Melodically, works as your typical diatonic semitone, but I often just turn to 16/15 because it's less complex.
• Prime 19: Despite being a very weak consonance, it is extremely close to a Pythagorean major third, which automatically makes it very easy to use. It's great to build otonal minor chords and get somber feelings whilst being locked on the fundamental, and specially for ending cadences, for which 6/5 might make for a worse minor third. It combines marvelously with 3. Melodically, it works the same as 5... so, great!

Tier IVb

It includes primes 23, 29, and 31, which sound quite xenharmonic.

• Prime 23: It's kind of a weird prime in my opinion. Since it sounds like a very sharp augmented fourth, but shares a similar consonance level to 19, so it should be in the same tier. Its complexity makes it hard to use. Melodically, I'd treat it like an augmented fourth. There are better options available, in my opinion.
• Prime 29: Despite being one of the less consonant primes, it makes for a nice supraminor flavor. It's very well approximated by 7n edos, so it also has that going for it. Melodically, I'd say it goes best when used in the tetrachord 24:26:29:32, serving as a more pungent, xenharmonic version of 15:16:18:20. The interval 29/24 is great for momentary supraminor flavors hidden in otonal passages.
• Prime 31: It's arguably the most dissonant of all the primes seen so far. Its octave complement, 32/31 is also incredibly close to 33/32, so by fudging or tempering 1024/1023, one can get a good approximation of 31 by reducing it to the 11-limit, which at least makes it somewhat approachable, but its complexity makes it a very difficult prime to digest and work with. Melodically, it's the same as 64/33, so it is incredibly jarring. Its octave complement is no better either,

Tier V and beyond

There would theoretically be infinite tiers, each having exponentially more primes than the one before, but I really don't care for primes beyond 29. They become far too complex for my ears. The only way they can be reliably and practically explored for most cases is by using the serendipitous 311edo, I'd say.

Odd ratings

TBA