User:Akselai/On the infinite division of the octave

Abstract

A construction of ∞edo by vals is given, such that its structure is compatible with the regular temperament theory of finite edos.

Introduction

Equal divisions of the octave (edos) 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 nedo is a subset of (kn)edo, and is in fact a subgroup. So we also suppose that ∞edo contains finite edos and nothing else. 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 nedo equal to [math]1 \backslash n = 2^{1/n}[/math], so naturally [math]1 \backslash \infty = \lim_{n \rightarrow \infty} 2^{1/n} = 1[/math]. Thus every step of ∞edo is the unison. We are not going anywhere by moving a finite amount of scale steps, and an infinite amount of scale steps (e.g. to get to the octave) is even more absurd since infinity is not a quantity.

Another problem with this is structure. Suppose we divide the octave into countably infinite many steps, whatever that may mean. We can label each interval with a positive integer, according to its appearance in the sequence 1edo, 2edo, 3edo, ... This has the advantage that finite scale steps no longer "pile up infinitesimally near the unison" as we have seen above. But now our labels don't make sense algebraically, i.e. the stacking of the intervals corresponding to 3 and 4 is not the one corresponding with 7.

In fact, by restricting to the countably infinite, there is also a mismatch of cardinality of this construction, if the goal is to (I paraphrase) recreate all harmonics [and intervals] perfectly, since the continuum is uncountable.

So is there even a way to see ∞edo by a formal construction other than by a facetious meme in xenharmonic circles? I say the answer is yes.

The axioms of ∞edo

∞edo is not just any infinite set of intervals in an octave. Since intuition fails for the infinite, it would be nice to see what basic properties ∞edo has to satisfy to deserve the name.

0) ∞edo contains the unison 1/1 and the octave 2/1. (Existence)

1) ∞edo contains an infinite amount of intervals between any two intervals in ∞edo. (Infinity)

2) An interval in ∞edo is an interval in some finite edo. (Inclusion)

3) Intervals in ∞edo are closed under stacking. (Closure)

Axiom 0 ensures that ∞edo is not an empty set of intervals. It also emphasizes that we are dividing the octave and not some other interval.

Axiom 1 means that we are expected to find more and more intervals when we zoom in ∞edo, much like how increasing the number of divisions in a finite edo results in more intervals.

Axiom 2 implies that our set of intervals is at most countably infinite, and will not contain any JI intervals exactly unless if it is a multiple of an octave.

And Axiom 3 implies that ∞edo has the same structure as we would find in a finite edo, that one can never "break out" just by stacking intervals.

Below, we will use the bold letters Existence, Infinity, Inclusion, and Closure in place of Axioms 0, 1, 2, and 3.

With these axioms, we can reconstruct our key assumption.

∞edo exclusively contains finite edos

Suppose we have found some interval s\n in ∞edo, which is s steps out of nedo. This is made possible by Inclusion. By Infinity we can be sure that n > 1, since a finite edo alone does not satisfy Infinity. If s = 1, then applying Closure t times implies that the multiples t\n are also in ∞edo and thus we are done. Since s\n is just 2^s/n, we can assume that gcd(s, n) = 1. Applying Closure under the guise of Bézout's lemma and using the existence of the octave (i.e. n\n) by Existence, we get that 1\n is in ∞edo and thus by repeating the argument above, we are done. [math]\square[/math]

Non-examples

As is dicussed above, an empty set of intervals satisfy Infinity, Inclusion, and Closure, but does not satisfy Existence. Although we haven't got to it yet, a construction of ∞edo below with tritave (3/1) in place of octave (thus yielding ∞edt) also satisfies Infinity, Inclusion, and Closure but not Existence.

Finite edos are precisely the tunings that satisfy Existence, Inclusion, and Closure, but not Infinity.

The set of JI intervals satisfy Existence, Infinity, and Closure, but not Inclusion.

It is quite tricky to come up with a set of intervals that satisfy Existence, Infinity, Inclusion but not Closure. The reader can come up with a solution themself. Here is my solution:

A set of intervals that satisfy Existence, Infinity, Inclusion but not Closure

The union of 1edo and the set of 3ⁿedo intervals with step number equal to 1 modulo 3. Equivalently, the union of 1edo with rings of 3ⁿedo offset by 1\3ⁿ. This will contain the unison, the octave, an infinite number of intervals between any two intervals, and edo intervals exclusively. However, it does not contain 2\3 which is 1\3 + 1\3. [math]\square[/math]

∞edo as a temperament

The axioms above only specify ∞edo as a tuning. To turn ∞edo into a temperament, we have the following:

4) There exists a mapping V from a JI subgroup I to ∞edo such that the regular temperament property holds, i.e. V(α) V(β) = V(αβ) for all α, β ∈ I.

Akselai's construction of ∞edo

Remember our key assumption: we suppose that ∞edo contains finite edos and nothing else, 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 commonly-used 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, represented by 83 steps of 24edo, is adjoined to our tone system. In this system, we expand the val 2.3.5 ⟨24 38 56] to 24edo, since the amount of scale steps is doubled, and give the val 2.3.5.11 ⟨24 38 56 83] to accomodate the 11th harmonic.

A general extension of nedo to (kn)edo multiplies the val by k times before adjoining the approximation of the next JI interval. Here is a tower of edo extensions with length 3, starting from 5-limit 19edo, multiplied by 3 to give the 7-limit 57edo, then multiplied by 5 to give the 11-limit 285edo.

2.3.5 <19 30 44] ⊆ 2.3.5.7 <57 90 132 160] ⊆ 2.3.5.7.11 <285 450 660 800 986]

The extension of an edo is not entirely representative by the behaviour of the larger edo alone, i.e. the val is not always patent. For example, the patent 2.3.5 val in 60edo is <60 95 139], which is not the same thing by multiplying each entry in the 12edo val by 5 (which would be ⟨60 95 140], the mapping of 5 is due to the inflection of the syntonic comma). Thus, the information of the smaller edos are actually important.

By extending this tower to the infinity prime limit, we obtain an "ascending" tower of edo mappings

[math] I_1 \langle a_1 \ a_2 \ \cdots \ a_n] \subseteq I_2 \langle m_1a_1 \ m_1a_2 \ \cdots \ m_1a_n \ a_{n+1}] \subseteq I_3 \langle m_1m_2a_1 \ m_1m_2a_2 \ \cdots \ m_1m_2a_n \ m_2a_{n+1} \ a_{n+2}] \subseteq \cdots [/math]

where [math] I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots [/math] are proper inclusions of JI subgroups. We denote the "limit" of the tower with I, which is the minimal JI subgroup containing all the I_i-s.

Thus, as long as we don't multiply the edos by integers greater than 1 only a finite amount of times, we obtain an ∞edo JI mapping by means of the tower of inclusions, making ∞edo into a temperament. The axiom of temperament is satisfied — since a JI interval has only a finite amount of nonzero coordinates in its monzo, it is possible to truncate the tower to I_i until both JI intervals α, β are in I_i, and use the temperament at that level.

Since we can multiply by any positive integer upon each extension, and there are countable infinity such extensions, there are [math]|\mathbb{Z}^\mathbb{Z}|[/math] possible temperament mappings for ∞edo.

Operations

Given a mapping of ∞edo, the intervals of ∞edo can be specified by that of a JI interval α, and some nedo, defined as the least edo with its associated subgroup containing α. The actual number of scale steps in nedo can be inferred from the val tower. The good news: there is now a natural algebraic structure on ∞edo with respect to JI intervals! Suppose we have two scale steps of ∞edo, (α, m) and (β, n) (with m ≤ n), and we want to stack them. Suppose we have also calculated the scale steps in their respective edos as s and t. Then the result is simply (αβ, n), and it is readily verified that the number of scale steps of this interval is (n/m)s + t.

[More operations at your request.]

Other properties

∞edo by this construction is a flexible object. Not all ∞edos are the same, even if the temperament structure is discarded. Some have defined ∞edo as simply the union of all edos, which is actually supported by this construction. At the h-th level of the tower with medo, we only need to adjoin (mh)edo to obtain the (h+1)-th level, and we would have encompassed all integer factors along the tower and hence all edos. (Though, the intervals of an arbitrary subset edo do not follow a val mapping.)

On the other hand, ∞edo can also be built from, say 5ⁿedos. Then it would not contain 2edo, among other edos that are not powers of 5. This tuning is practically isomorphic to the Prüfer p-group.

Implementation

∞edo is readily implemented by calculators by the above definitions and operations. The only downside is that an infinite stream of JI basis intervals and another infinite stream of edos are to be read for the algorithms to work. However, the calculations are guaranteed to be finitary.

[I'll make a program here if I've got the time.]