User:Squib/Unnamed music theory: Difference between revisions

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~~{{todo|inline=1|fix formatting from page transfer}}~~

I decided to formalize the ideas that I have been exploring into a music theory, inspired by [https://youtu.be/q8A7utFpS9E 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.

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Overtone: A tone produced with the fundamental, contributing to the timbre of a single note.

===~~Explanation~~===

==Theory==

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.

~~[[File:Octaves-tritaves layout.png|300px|thumb|none|as powers of 2 and 3]]~~

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.

~~[[File:Fourths-fifths layout with schismic major thirds.png|300px|thumb|none|"squished" to be powers of 4/3 and 3/2]]~~

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.

~~[[File:Whole tone - pyth minor third layout.png|x300px|thumb|none|"squished" again]]~~

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?

==Derivation==

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."

~~[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n~~=~~1 to n~~=~~6 in no particular order, and example intervals in the top right]]~~

===Overview===

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** 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.

[[File:Octaves-tritaves layout.png|300px|thumb|none|as powers of 2 and 3]]

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.

[[File:Fourths-fifths layout with schismic major thirds.png|300px|thumb|none|"squished" to be powers of 4/3 and 3/2]]

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.

[[File:Whole tone - pyth minor third layout.png|x300px|thumb|none|"squished" again]]

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."

[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]

@@ Line 1: / Line 1: @@
-{{todo|inline=1|fix formatting from page transfer}}
 I decided to formalize the ideas that I have been exploring into a music theory, inspired by [https://youtu.be/q8A7utFpS9E 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.
@@ Line 16: / Line 14: @@
 Overtone: A tone produced with the fundamental, contributing to the timbre of a single note.
-===Explanation===
+==Theory==
-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.
-[[File:Octaves-tritaves layout.png|300px|thumb|none|as powers of 2 and 3]]
-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.
-[[File:Fourths-fifths layout with schismic major thirds.png|300px|thumb|none|"squished" to be powers of 4/3 and 3/2]]
-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.
-[[File:Whole tone - pyth minor third layout.png|x300px|thumb|none|"squished" again]]
-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?
+==Derivation==
-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."
-[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]
 ===Overview===
@@ Line 67: / Line 44: @@
 ** 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.
+[[File:Octaves-tritaves layout.png|300px|thumb|none|as powers of 2 and 3]]
+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.
+[[File:Fourths-fifths layout with schismic major thirds.png|300px|thumb|none|"squished" to be powers of 4/3 and 3/2]]
+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.
+[[File:Whole tone - pyth minor third layout.png|x300px|thumb|none|"squished" again]]
+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."
+[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]

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