Xenharmonic Wiki

Interleaving: Difference between revisions

← Older editNewer edit →
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,668 edits
m Inthar moved page Flought scale to Interleaved scale
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,668 edits
No edit summary
Line 1: Line 1:
A scale is (''k''-)'''flought''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', and ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls strictly between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the ''polyoffset'', which is determined up to inversion and [[equave]]-equivalence. A flought scale is thus a [[cross-set]] with a little additional structure. One can '''floughten''' or '''flought''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''floughtens'' ''s''" or "''s'' is ''floughtenable'' with Δ") if ''s'' is the strand scale of a flought scale with polyoffset Δ. Such a scale is denoted Flought(''s''; Δ). The concept of flought scales is a generalization of [[bipentatonic scale]]s and (even-length) [[generator-offset]] scales.  
A scale is (''k''-)'''interleaved''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', and ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls strictly between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the ''polyoffset'', which is determined up to inversion and [[equave]]-equivalence. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''interleaved''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' with Δ") if ''s'' is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted Interleaved(''s''; Δ). The concept of interleaved scales is a generalization of [[bipentatonic scale]]s and (even-length) [[generator-offset]] scales.  


[[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of flought scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale.
[[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of interleaved scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale.


The term ''flought'' was coined by [[Inthar]] by evolving the Old English past participle ''(ġe)flohten'' of the verb ''fleohtan'' 'to weave; to plait' into a hypothetical native Modern English cognate to the words ''plait'' and ''plexus''.
== Some interleaved scales ==
== Some flought scales ==
Interleaved scales can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:
Flought scales can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:
* Interleaved(12:14:16:18:21:24; 11:12)
* Flought(12:14:16:18:21:24; 11:12)
* Interleaved(12:14:16:18:21:24; 12:13:22)
* Flought(12:14:16:18:21:24; 12:13:22)
* Interleaved(12:14:16:18:21:24; 8:10:11)
* Flought(12:14:16:18:21:24; 8:10:11)
** [[User:Userminusone/Userminusone's_11_limit_15_tone_scale]]  
** [[User:Userminusone/Userminusone's_11_limit_15_tone_scale]]  
* Flought(12:14:16:18:21:24; 9:10:11)  
* Interleaved(12:14:16:18:21:24; 9:10:11)  
** Note: detempered 11-limit Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11, 10/9, 11/10, 12/11, 10/9, 11/10, 189/176)
** Note: detempered 11-limit Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11, 10/9, 11/10, 12/11, 10/9, 11/10, 189/176)
* Flought(Pyth[5]; 8:10:11)
* Interleaved(Pyth[5]; 8:10:11)
* Flought(Pyth[5]; 9:10:11)  
* Interleaved(Pyth[5]; 9:10:11)  
** Note: detempered 2.3.5.11 Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11)
** Note: detempered 2.3.5.11 Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11)
* Flought(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)
* Interleaved(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)
== Properties ==
== Properties ==
# The following is a necessary and sufficient condition for floughtenability. Let ''S'' be a scale with equave E, <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, E). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' ∈ {0, ... len(''S'') - 1}.
# The following is a necessary and sufficient condition for interleavability. Let ''S'' be a scale with equave E, <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, E). Then the polyoffset chord Δ interleaves ''S'' if and only if no nonunison (positive) dyad in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' ∈ {0, ... len(''S'') - 1}.
# For any periodic scale ''S'' with equave E, if δ is an offset and Flought(''S''; δ) exists, then Flought(''S''; δ) = Flought(''S''; E - δ) = Flought(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.
# For any periodic scale ''S'' with equave E, if δ is an offset and Interleaved(''S''; δ) exists, then Interleaved(''S''; δ) = Interleaved(''S''; E - δ) = Interleaved(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the interleaved scale, nor does shifting any individual offset by equaves.
# Given an E-equivalent scale ''S'', offsets δ within the open interval (0, min({step sizes in ''S''})) are called ''small'' in the context of floughtening ''S''. Small offsets are significant because the resulting flought scale has a structure that closely mimics the underlying scale structure: if ''S'' is a circular word <math>w(a_1, a_2, ..., a_n)</math> then Flought(''s''; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely <math>w(\delta b_1, \delta b_2, ..., \delta b_n)</math> where <math>b_i = a_i - \delta</math>.
# Given an E-equivalent scale ''S'', offsets δ within the open interval (0, min({step sizes in ''S''})) are called ''small'' in the context of interleaveing ''S''. Small offsets are significant because the resulting interleaved scale has a structure that closely mimics the underlying scale structure: if ''S'' is a circular word <math>w(a_1, a_2, ..., a_n)</math> then Interleaved(''s''; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely <math>w(\delta b_1, \delta b_2, ..., \delta b_n)</math> where <math>b_i = a_i - \delta</math>.
# A flought scale is not always CS, even when the strand is CS and the scale has a [[generator sequence]] where every generator subtends the same number of steps. One such scale is Flought(Zarlino; 32/25) = 25/24 9/8 75/64 5/4 125/96 4/3 375/256 3/2 25/16 5/3 225/128 15/8 125/64 2/1 which has [[GS]](32/25 125/96 32/25 5/4).
# An interleaved scale is not always CS, even when the strand is CS and the scale has a [[generator sequence]] where every generator subtends the same number of steps. One such scale is Interleaved(Zarlino; 32/25) = 25/24 9/8 75/64 5/4 125/96 4/3 375/256 3/2 25/16 5/3 225/128 15/8 125/64 2/1 which has [[GS]](32/25 125/96 32/25 5/4).
{{proof|title=Proof of the offset constraints|contents=
{{proof|title=Proof of the offset constraints|contents=
If the polyoffset has more than two notes, the interleaving condition only needs to hold for ''pairs'' of distinct strands, and hence the above property only needs to hold for pairs of notes in the polyoffset. This reduces the proof to the case of one offset δ.
If the polyoffset has more than two notes, the interleaving condition only needs to hold for ''pairs'' of distinct strands, and hence the above property only needs to hold for pairs of notes in the polyoffset. This reduces the proof to the case of one offset δ.
Line 33: Line 32:


== Generalizations ==
== Generalizations ==
=== Co-floughtenability (?) ===
=== Co-interleavability (?) ===
Periodic scales <math>S, T : \mathbb{Z} \to \mathbb{R}</math> of the same length and equave are ''co-floughtenable'' (the ''co-'' is not meant to suggest any kind of duality) if there exists <math>\delta\in\mathbb{R}</math> such that ''S'' and ''T'' + δ are interleaved. Note that though a given 2''n''-note scale being a co-floughtened result of some pair of scales may be trivial, a given pair of scales being co-floughtenable is less so: for example, '''MMMM''' and '''Lsss''' are not co-floughtenable when '''s''' is too small.
Periodic scales <math>S, T : \mathbb{Z} \to \mathbb{R}</math> of the same length and equave are ''co-interleavable'' (the ''co-'' is not meant to suggest any kind of duality) if there exists <math>\delta\in\mathbb{R}</math> such that ''S'' and ''T'' + δ are interleaved. Note that though a given 2''n''-note scale being a co-interleaveed result of some pair of scales may be trivial, a given pair of scales being co-interleavable is less so: for example, '''MMMM''' and '''Lsss''' are not co-interleavable when '''s''' is too small.


A ''contraflought'' scale is a co-floughtened pair of the two chiralities of a [[chiral scale]].
A ''contrainterleaved'' scale is a co-interleaveed pair of the two chiralities of a [[chiral scale]].
[[Category:Scale]]
[[Category:Scale]]
[[Category:Terms]]
[[Category:Terms]]
Retrieved from "https://en.xen.wiki/w/Interleaving"