User:Akselai/On the infinite division of the octave: Difference between revisions

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Equal divisions of the octave ([[edo]]s) are, historically, a trick to deal with the (countably) infinite pitches in [[just intonation]] (JI), arguably the basis of almost all music and hearing. It reduces the infinite to the finite (after octave equivalence), and multiplication to addition. In light of the '''additive structure''' of edos, people have constructed larger and larger edos to approximate just intonation more and more accurately. A relatively famous example is [[11358058edo]].

A natural extension of this is called ∞edo, the infinite division of the octave. We already know from the definition of edos, that for all integers ''n, k''>1, that ''n''edo is a ''subset'' of (''kn'')edo, and is in fact a ''subgroup''. So we also '''suppose that ∞edo contains ~~all~~ finite edos'''. I put this in bold because this is a key assumption in our investigation of ∞edo.

A natural extension of this is called ∞edo, the infinite division of the octave. We already know from the definition of edos, that for all integers ''n, k''>1, that ''n''edo is a ''subset'' of (''kn'')edo, and is in fact a ''subgroup''. So we also '''suppose that ∞edo contains finite edos'''. I put this in bold because this is a key assumption in our investigation of ∞edo.

This construction is evidently problematic. The first and most obvious problem is that the step sizes are not well defined. What does it mean by 1 step of ∞edo? We have, by definition, 1 step of ''n''edo equal to

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== Akselai's construction of ∞edo ==

Remember our key assumption: '''we suppose that ∞edo contains ~~all~~ finite edos''', in a natural way compatible with the embedding of a smaller edo into a larger edo. Every edo has a mapping (called a [[val]]) to a subgroup of JI, specified with a (co)vector with finitely many coordinates. For example 12edo has the val 2.3.5 <12 19 28] because it maps 12 steps to the harmonic 2, 19 steps to 3, and 28 steps to 5.

Remember our key assumption: '''we suppose that ∞edo contains finite edos''', in a natural way compatible with the embedding of a smaller edo into a larger edo. Every edo has a mapping (called a [[val]]) to a subgroup of JI, specified with a (co)vector with finitely many coordinates. For example 12edo has the val 2.3.5 <12 19 28] because it maps 12 steps to the harmonic 2, 19 steps to 3, and 28 steps to 5.

We also recall the concept of an edo extension. The better known construction is in the case of 12edo to 24edo, where the mapping of the 11th harmonic is adjoined to our tone system. In this system, we give the val 2.3.5 <24 38 56] to 24edo, since the amount of scale steps is doubled, and also the val 2.3.5.11 <24 38 56 83] to accomodate the 11th harmonic.

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 Equal divisions of the octave ([[edo]]s) are, historically, a trick to deal with the (countably) infinite pitches in [[just intonation]] (JI), arguably the basis of almost all music and hearing. It reduces the infinite to the finite (after octave equivalence), and multiplication to addition. In light of the '''additive structure''' of edos, people have constructed larger and larger edos to approximate just intonation more and more accurately. A relatively famous example is [[11358058edo]].
-A natural extension of this is called ∞edo, the infinite division of the octave. We already know from the definition of edos, that for all integers ''n, k''>1, that ''n''edo is a ''subset'' of (''kn'')edo, and is in fact a ''subgroup''. So we also '''suppose that ∞edo contains all finite edos'''. I put this in bold because this is a key assumption in our investigation of ∞edo.
+A natural extension of this is called ∞edo, the infinite division of the octave. We already know from the definition of edos, that for all integers ''n, k''>1, that ''n''edo is a ''subset'' of (''kn'')edo, and is in fact a ''subgroup''. So we also '''suppose that ∞edo contains finite edos'''. I put this in bold because this is a key assumption in our investigation of ∞edo.
 This construction is evidently problematic. The first and most obvious problem is that the step sizes are not well defined. What does it mean by 1 step of ∞edo? We have, by definition, 1 step of ''n''edo equal to
@@ Line 25: / Line 25: @@
 == Akselai's construction of ∞edo ==
-Remember our key assumption: '''we suppose that ∞edo contains all finite edos''', in a natural way compatible with the embedding of a smaller edo into a larger edo. Every edo has a mapping (called a [[val]]) to a subgroup of JI, specified with a (co)vector with finitely many coordinates. For example 12edo has the val 2.3.5 <12 19 28] because it maps 12 steps to the harmonic 2, 19 steps to 3, and 28 steps to 5.
+Remember our key assumption: '''we suppose that ∞edo contains finite edos''', in a natural way compatible with the embedding of a smaller edo into a larger edo. Every edo has a mapping (called a [[val]]) to a subgroup of JI, specified with a (co)vector with finitely many coordinates. For example 12edo has the val 2.3.5 <12 19 28] because it maps 12 steps to the harmonic 2, 19 steps to 3, and 28 steps to 5.
 We also recall the concept of an edo extension. The better known construction is in the case of 12edo to 24edo, where the mapping of the 11th harmonic is adjoined to our tone system. In this system, we give the val 2.3.5 <24 38 56] to 24edo, since the amount of scale steps is doubled, and also the val 2.3.5.11 <24 38 56 83] to accomodate the 11th harmonic.