Arseniiv: linkify, add an isoharmonic clarification

2025-07-24T16:34:00Z

linkify, add an isoharmonic clarification

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	'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord\|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this chord can be represented with integer <math>a_k</math> values.		'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord\|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this (then ''isoharmonic'') chord can be represented with integer <math>a_k</math> values.

	Isodifferential subdivision of <math>s</math> into two intervals <math>a, h</math> is already quite notable: <math>a</math> is the arithmetic mean of <math>s</math> and 1, the unison —this is just by definition chosen above,— and <math>h</math> is the harmonic mean of <math>s</math> and 1.		Isodifferential subdivision of <math>s</math> into two intervals <math>a, h</math> is already quite notable: <math>a</math> is the arithmetic mean of <math>s</math> and 1, the unison —this is just by definition chosen above,— and <math>h</math> is the harmonic mean of <math>s</math> and 1.

	Dividing a superparticular interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree		Dividing a [[superparticular]] interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree
	<pre> 1:2		<pre> 1:2
	/ \		/ \
Line 13:		Line 13:
	/ \ / \ / \ / \		/ \ / \ / \ / \
	8: 9:10:11:12:13:14:15:16</pre>		8: 9:10:11:12:13:14:15:16</pre>
	which generates every superparticular interval by linearly dividing an octave into 2, 4, 8 parts and so on.		which generates every superparticular interval by linearly dividing an [[octave]] into 2, 4, 8 parts and so on.

	One can also successively divide a tritave into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).		One can also successively divide a [[tritave]] into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).

	== Properties ==		== Properties ==

Arseniiv: Created page with "{{stub}} '''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval $s$ into $d$ parts is an isodifferential chord of $d+1$ notes, spanning $s$ —that is, a chord $a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d$ with $\frac{a_d}{a_0} = s$ , where frequency differences between consecutive notes are the same: $a_1 - a_0 = a_2 - a_1 = \ldot..."$

2025-07-24T16:22:26Z

Created page with "{{stub}} '''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an isodifferential chord of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldot..."

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{{stub}}

'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this chord can be represented with integer <math>a_k</math> values.

Isodifferential subdivision of <math>s</math> into two intervals <math>a, h</math> is already quite notable: <math>a</math> is the arithmetic mean of <math>s</math> and 1, the unison —this is just by definition chosen above,— and <math>h</math> is the harmonic mean of <math>s</math> and 1.

Dividing a superparticular interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree
<pre> 1:2
/ \
2:3 3:4
/ \ / \
4:5 5:6 6:7 7:8
/ \ / \ / \ / \
8: 9:10:11:12:13:14:15:16</pre>
which generates every superparticular interval by linearly dividing an octave into 2, 4, 8 parts and so on.

One can also successively divide a tritave into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).

== Properties ==

1. Subdividing an interval into <math>d</math> parts and then each of those into <math>d'</math> parts is the same as doing it in the opposite order or dividing into <math>d d'</math> parts at once.
: This is obvious from looking at isoharmonic chords like 5:7:9:11:13:15:17: such a chord is a linear subdivision of both 5:11:17 into three and 5:9:13:17 into two, which are divisions of 5:17 themselves. Dividing irrational intervals works exactly the same, which can be visualized by dividing segments on the frequency line <math>\mathbb R_{>0}</math> into equal parts, segments themselves representing intervals.

2. The first part is a weighted arithmetic mean of 1 and <math>s</math> with weights <math>d - 1</math> and 1.
: Example: in 3:5:7:9, '''5/3''' is 2 parts '''1/1''' and 1 part '''3/1''': <math>\frac{2\cdot\mathbf1 + 1\cdot\mathbf3}{2 + 1} = \frac53</math>.

3. The last part is a weighted harmonic mean of 1 and <math>s</math> with weights <math>d - 1</math> and 1.
: Example: in 3:5:7:9, '''7/9''' is 2 parts '''1/1''' and 1 part '''1/3''' (NB harmonic mean): <math>\frac{2\cdot\mathbf1 + 1\cdot\mathbf{1/3}}{2 + 1} = \frac{7/3}3 = \frac79</math>.

4. Other parts can be realized as weighted harmonic and arithmetic means taken one after another with correct weights to get a suffix of a prefix (or vice versa).
: Example: in 3:5:7:9, we can first select the prefix '''3:'''(5)''':7''' as AM with weights 1 and 2: <math>\frac{1\cdot\mathbf1 + 2\cdot\mathbf3}{1 + 2} = \frac73</math>, and then take the last part of 3:5:7 as HM with weights 1 and 1 (the common one): <math>\frac{\mathbf1 + \mathbf{3/7}}2 = \frac{10/7}2 = \frac57</math>. We can do it in reverse order: first focus on '''5:'''(7)''':9''' taking HM with weights 1 and 2, then take AM with weights 1 and 1 to get the first part.

== Formula ==
We obtain the formula for <math>k</math>-th part <math>a_k / a_{k-1}</math>.

For simplicity, choose <math>a_0 = 1, a_d = s</math> and thus get <math>s - 1 = d\Delta</math> for <math>\Delta</math> the difference between consecutive <math>a_k</math> here, which gives <math>\Delta = \tfrac{s - 1}d</math> and <math>a_k = 1 + k\Delta = \tfrac{d + k(s - 1)}d</math>. So the part is <math>\frac{a_k}{a_{k-1}} = \frac{d + k(s - 1)}{d + (k - 1)(s - 1)}</math>.

If <math>s</math> is rational <math>\tfrac{m + D}m</math>, an intuitive approach is as follows:

Let <math>L = \operatorname{lcm}(D, d)</math>. If <math>D</math> isn't already divisible by <math>d</math>, multiply the fraction with <math>a = L / D</math> to get <math>\tfrac{a m + L}{a m}</math>. Then integer <math>\Delta = L / d</math> and we get the <math>k</math>-th part being <math>\frac{a m + k \Delta}{a m + (k-1) \Delta}</math> or, writing everything out, <math>\frac{m d + k D}{m d + (k-1) D}</math> (if you work mentally, this one can have unnecessary cancellations after working with larger numbers).

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Arseniiv: linkify, add an isoharmonic clarification