User:Eufalesio/Harmonic tiers

This is mainly a description of my mental categories of primes 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 integer frequency multiples, primes fit this category perfectly, as any composite harmonic can be uniquely built by a combination of prime harmonics. Basically, the Fundamental Theorem of Arithmetic applied to music. Here is how I would rate primes. You might not agree, but you're also not me.

P-tier rating

This rating will deal predominantly with primes as a whole, not just the odds that appear - sorted by tiers. The tier is dependent on the denominator the prime octave-reduces to, starting with the octave at Tier 0... but that's a trivial category so I put it in Tier I.

P-Tier I

There are two members here: Primes 2 and 3, the octave and the octave-reduced just fifth.

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. These are very stackable primes.

• 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). I treat it as a "hyperconsonance" of sorts, which is distinct from a consonance in that it can be stacked to thicken. 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 of 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. Important to remark is 3/1, which I also consider a hyperconsonance and very distinct from 3/2. Less than 2/1 obviously, but nothing to scoff at. Stacking tritaves is a powerful move, that I deem suitable for a hollower and more peculiar kind of thickening.

P-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 them accurately, but it's not a necessity. However, these primes add flavor to music. At least, that's what I think. Despite being kinda bad for building scales (horrible in case of 5), they combine extremely well with each other.

• 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 my mouth when using 5/4 to finish a cadence. Melodically, it is perfect for small leaps forward and it combines perfectly with 3. It allows many new flavors of tones and semitones, like 16/15, 25/24, 27/25, 135/128, 10/9... etc. Technically these already exist in the 3-limit, if you fudge the schisma.
• Prime 7: Often overlooked in tradition, it's an interesting prime. A bit xenharmonic, due to its notable innacuracy in 12edo (which for some purposes... it may be just enough). 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, but it makes a killer augmented sixth chord. 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 based on 7, namely 28/27 or 49/48. It can still be well employed with less jarring limmoid steps like 21/20. NOT using it for diasem, near-just diasem kinda sucks melodically.

P-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 basically 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 truly 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, which is part of the reason why I don't like it as much. However, it adds a great otonal flavor from time to time. This prime is also seemingly connected to prime 7 in a number of ways, but the most natural are either orgone or gary; gary being a better alternative in my opinion.
• Prime 13: The second xenharmonic prime. Technically less consonant than 11, but I find it easier to work with. It is extremely well approximated by 10n edos and extremely easy to reach through the 7-limit too, connecting a chain of 5*7 to 13 through the minisma. 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 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. This prime is also seemingly connected to prime 5 in a number of ways, most notably through marveltwin or will.

P-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.

P-Tier IVa

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

• Prime 17: It's dissonant. It's good for lowering the complexity of wide otonal chords, but that's it. I don't like it harmonically that much on its own. Melodically, works as a 12edo-ish semitone, but I often just turn to 16/15 because I like it more.
• Prime 19: Despite being a very weak consonance, it is extremely close to a Pythagorean minor 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 great with 3 in the 16:19:24 chord. Melodically, it works the same as 5... so, great! I only really use it for a couple intervals: 19/16, 19/12.

P-Tier IVb

From this tier onward the likelyhood of me using these is minimal. 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, it should be in the same tier. But I don't like it that much on its own. The only time I'd use it is in a super wide otonal chord and for that it excels. Melodically, I'd treat it like a sharper 10/7, that is... a slightly sharper tritone.
• Prime 29: Despite being one of the less consonant primes, it makes for a nice supraminor flavor. Apart from that, since it is a kind of seventh, it accepts xenharmonic values more easily. 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 (as the interval 29/24). Least hard to use of the bunch.
• 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 integer 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, being a quartertone.

P-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 19. I'm a fairly classical guy and I want to limit myself to a set of primes I can control.