User:Squib/Unnamed music theory
I decided to formalize the ideas that I have been exploring into a music theory, inspired by Shastavistic music theory, partly as a response to my complaints about how xen is done here and in general, and partly because it was naturally happening anyway. And it sounded fun.
Terms
Most of these terms are widely-used but I might be using them slightly differently than most people.
Interval: The frequency ratio between two notes. Can be a measurement of specific notes or just a ratio.
Harmonic: An interval which is an integer ratio (including 1/1). Also, a note which is a harmonic interval above another note.
Harmonic Series: The set of notes which are harmonics of the fundamental, including the fundamental itself.
Overtone: A tone produced with the fundamental, contributing to the timbre of a single note.
Theory
This theory organizes pitches as points on a 2D lattice defined by primes 2 and 3, then divides that lattice into a fixed number of pitch classes ("colors") which can be used to approximate other primes. The result of this is a clearly specified tuning and layout that uses colors to intuitively show harmonic relationships and progression.
Derivation
The simplest tuning is Pythagorean, which is 2.3 JI. Although this is generally thought of as notes created by stacking 2/1 and 3/1, here 3/2 and 4/3 are combined instead. This is much more natural.
To use prime 5, which cannot be created from fifths and fourths (I'll get to meantone later), multiple sets of Pythagorean are used, offset by 5/4. However, this is difficult to use practically since most keyboards are not three-dimensional. So instead, sets of pyth are added interlaced with the original, such that the layout is still two-dimensional.
Although this is useful and allows for true 5-limit JI, it removes the important property of isomorphism, meaning that intervals available in some places may not be available in others. So instead, the sets are tuned to be equally spaced, keeping the major third approximately in tune while making the tuning equal again.
Then, since the tuning is equal, instead of adding new sets of pyth and retuning them to be equal, we equivalently divide the existing set. Thus, all pitches in a tuning are created by multiplying together some (perhaps fractional) number of 2/1 and 3/1. This divides 3-limit JI analogously to how edos devide 2-limit JI.
This avoids the unnecessary clutter and math associated with RTT by specifying which notes are available in a tuning instead of specifying equivalences. For example, without already knowing that 243/242 is the difference between 3/2 and a stack of two 11/9s, it's very difficult to tell that tempering it out spits the fifth in half. It's much more natural to simply say that the fifth is split in half, and then that hemififth can be easily used as 11/9 without having to factor 243 first.
section break?
With edos, simply specifying the number of divisions is enough, since there is only 1 way to arrange any number of copies of 2-limit JI isomorphically. But here, there are multiple ways. The number of tunings that use N copies of pyth is equal to the sum of the unique factors of N.
(I don't know how much more I can explain without the diagrams.)
Overview
For now, this is just a list of the things I want to explain here.
- intro
- what this is & why i made it
- how much i'm taking credit for
- no, i don't expect this to become widely-used. although that would be cool.
- philosophy
- creating the theory based on experience and what actually works/is helpful
- intentionally keeping it simple; this makes it more useful even if it's not what i would naturally gravitate to otherwise
- as a result of this: models emerge!
- models are abstractions, describing the important attributes of a complex phenomenon in a simple way.
- the foundation
- what are notes, anyway?
- harmonic series as a model of complex timbres
- overtones as a model of consonance, harmony, and progression
- rational numbers as a model of intervals
- chords & the harmonic series
- why chords are a useful but incomplete model of harmony
- why harmonic series segments are a useful but incomplete model of chords
- so what do we do? keep reading to find out :)
- 2d tunings & layouts
- limited space & the tradeoff between complexity and range
- 3d tunings in a 2d space
Explanation
The simplest interval is the unison, 1/1. This is the interval between a note and itself. Stacking it multiple times doesn't do anything, which isn't very interesting, so let's consider the next simplest interval: the octave, 2/1. Although notes an octave apart are not considered equivalent, they do have a special relationship: the higher note is a harmonic of the lower. 3/1 is also a harmonic interval. Stacking these gives us intervals like 4/1, 6/1, and 8/1, which are also harmonics. Since only harmonics can be created, this does not allow for much progression, since every note only contains overtones from the root note. So we will now allow intervals to be subtracted from each other, creating nonharmonic intervals like 3/2 which have new overtones. Going up by harmonics removes some overtones and strengthens the remaining ones; going down by harmonics weakens existing overtones and adds new ones in the gaps. Combining these gives us nonharmonic intervals, and stacking them adds more and more new overtones.
Every culture's music is influenced by the tools available to them. The primary tool available to me is a Launchpad with a 9x9 grid of LED buttons (minus the top right corner). So, the most immediately obvious set of pitches to use is a lattice where moving one button to the right is by 2/1 and one button up is 3/1. This theoretically allows these intervals to be combined in any combination.
There's a problem. 9/8, the whole tone, is as easy to reach as 108/1! Not only does this make useful intervals like 9/8 more difficult to use, but since keyboard space is limited, this adds extremely high and low notes that render much of the space unusable. So instead, notes can be represented as any combination of fourths (4/3) and fifths (3/2), the two simplest non-harmonic intervals. This leaves 2/1 and 3/1 easily accessible while moving very large intervals further away, as well as moving more complex smaller intervals closer together. It's also much more natural to think of 9/8 as the difference between a fourth and a fifth than it is to think of it as the difference between three 2/1s and two 3/1s, for example.
Thus, the fourths-fifths layout is the most basic layout of this music theory, and it works very well. Except for absolute pitch, every place on the keyboard is the same, a property called isomorphism. This allows any interval or chord to be placed anywhere on the keyboard.
But what about ratios involving 5? 4 can be created from 2×2, and 6 can be created from 2×3, but 5 is prime. It could be represented using a third dimension, but that would be difficult to visualize and next to impossible to play, since most keyboards are at most 2-dimensional, including the Launchpad. So how can we get more notes?
One way is to use the notes we already have. Two 9/8s make a very rough approximation of a major third, but two 4/3s minus three 9/8s is a much closer one and is nearly indistinguishable from 5/4. It's awkward to use, but we can adjust the layout to change that.
Just like when the layout was changed to fourths and fifths, bringing more complex intervals closer pushes larger intervals further away, but this time the range is reduced to only a couple octaves and the intervals far to the side are very complex and not very useful. We can make convenient ether very large or very complex intervals, but not both. The layout shown is in my opinion the best trade-off; it's possible to make 5/4 and especially 10/9 more convenient, but then it can barely fit a single 3/1! So is there a less complex way to approximate 5?
The other way to approximate 5/4 is to indeed add more notes between the existing ones. The tuning we already have is called Pythagorean tuning because it uses only ratios of 2 and 3. We can take two sets of Pythagorean and put them together. There are an infinite number of ways to do this, but only 3 preserve isomorphism. In fact, the number of equal tunings with N sets of pythagorean is exactly equal to the sum of the unique factors of N. Don't ask me why, although I suspect figuring it out will help me name them.
Explanation (that I don't entirely understand) from a mathy discord server I'm in: "These arrangements of n integer lattices must be lattices themselves, so they have a basis. As a basis, we can take {(a,0), (b,c)}, where a is the least positive value such that (a,0) is in the lattice, c is the least positive value such that (x,c) is in the lattice for some x, and b is the least nonnegative value such that (b,c) is in the lattice. a has to be 1/k for integer k because (0,1) is in the lattice, and since the lattice has n copies of the integer lattice, k must be a divisor of n. This means there are n/k integer lattices with points on the x-axis, so c must be k/n. In order for the lattice to close, b must be i/n for some integer i, and since b is minimal, 0 ≤ i/n < 1/k, so 0 ≤ i < n/k and there are n/k possible values of i. Since all divisors of n are of the form n/k for some k | n, the number of lattices is the sum of the divisors of n."
Random
In two dimensions, the number of tunings with N colors is the sum of the unique factors of N. So there are 6+3+2+1=12 6-color tunings in two dimensions. The amount of *three*-dimensional tunings with N colors is the sum of the factors of N times the sum of *their* factors. So the number of 3D 6-color tunings is 6(6+3+2+1)+3(3+1)+2(2+1)+1(1) = 91.