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Scale 1731: "Koxian"

Scale 1731: Koxian, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names

Dozenal
Koxian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,6,7,9,10}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z13

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

[3.5]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

no

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

4

Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

5

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 219

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 5, 1, 2, 1, 2]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<3, 2, 4, 2, 2, 2>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p2m2n4s2d3t2

Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,5}
<2> = {3,6}
<3> = {4,5,7,8}
<4> = {6,9}
<5> = {7,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

3

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

1.866

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.485

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.

[7]

Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Improper

Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(24, 6, 51)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsF♯{6,10,1}221
Minor Triadsf♯m{6,9,1}221
Diminished Triadsf♯°{6,9,0}131.5
{7,10,1}131.5
Parsimonious Voice Leading Between Common Triads of Scale 1731. Created by Ian Ring ©2019 f#° f#° f#m f#m f#°->f#m F# F# f#m->F# F#->g°

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter3
Radius2
Self-Centeredno
Central Verticesf♯m, F♯
Peripheral Verticesf♯°, g°

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Diminished: {6, 9, 0}
Diminished: {7, 10, 1}

Modes

Modes are the rotational transformation of this scale. Scale 1731 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2913
Scale 2913: Senian, Ian Ring Music TheorySenian
3rd mode:
Scale 219
Scale 219: Istrian, Ian Ring Music TheoryIstrianThis is the prime mode
4th mode:
Scale 2157
Scale 2157: Nexian, Ian Ring Music TheoryNexian
5th mode:
Scale 1563
Scale 1563: Joyian, Ian Ring Music TheoryJoyian
6th mode:
Scale 2829
Scale 2829: Rupian, Ian Ring Music TheoryRupian

Prime

The prime form of this scale is Scale 219

Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian

Complement

The hexatonic modal family [1731, 2913, 219, 2157, 1563, 2829] (Forte: 6-Z13) is the complement of the hexatonic modal family [591, 633, 969, 2343, 3219, 3657] (Forte: 6-Z42)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1731 is 2157

Scale 2157Scale 2157: Nexian, Ian Ring Music TheoryNexian

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 1731       T0I <11,0> 2157
T1 <1,1> 3462      T1I <11,1> 219
T2 <1,2> 2829      T2I <11,2> 438
T3 <1,3> 1563      T3I <11,3> 876
T4 <1,4> 3126      T4I <11,4> 1752
T5 <1,5> 2157      T5I <11,5> 3504
T6 <1,6> 219      T6I <11,6> 2913
T7 <1,7> 438      T7I <11,7> 1731
T8 <1,8> 876      T8I <11,8> 3462
T9 <1,9> 1752      T9I <11,9> 2829
T10 <1,10> 3504      T10I <11,10> 1563
T11 <1,11> 2913      T11I <11,11> 3126
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2661      T0MI <7,0> 1227
T1M <5,1> 1227      T1MI <7,1> 2454
T2M <5,2> 2454      T2MI <7,2> 813
T3M <5,3> 813      T3MI <7,3> 1626
T4M <5,4> 1626      T4MI <7,4> 3252
T5M <5,5> 3252      T5MI <7,5> 2409
T6M <5,6> 2409      T6MI <7,6> 723
T7M <5,7> 723      T7MI <7,7> 1446
T8M <5,8> 1446      T8MI <7,8> 2892
T9M <5,9> 2892      T9MI <7,9> 1689
T10M <5,10> 1689      T10MI <7,10> 3378
T11M <5,11> 3378      T11MI <7,11> 2661

The transformations that map this set to itself are: T0, T7I

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 1729Scale 1729: Kowian, Ian Ring Music TheoryKowian
Scale 1733Scale 1733: Raga Sarasvati, Ian Ring Music TheoryRaga Sarasvati
Scale 1735Scale 1735: Mela Navanitam, Ian Ring Music TheoryMela Navanitam
Scale 1739Scale 1739: Mela Sadvidhamargini, Ian Ring Music TheoryMela Sadvidhamargini
Scale 1747Scale 1747: Mela Ramapriya, Ian Ring Music TheoryMela Ramapriya
Scale 1763Scale 1763: Katalian, Ian Ring Music TheoryKatalian
Scale 1667Scale 1667: Kekian, Ian Ring Music TheoryKekian
Scale 1699Scale 1699: Raga Rasavali, Ian Ring Music TheoryRaga Rasavali
Scale 1603Scale 1603: Juxian, Ian Ring Music TheoryJuxian
Scale 1859Scale 1859: Lixian, Ian Ring Music TheoryLixian
Scale 1987Scale 1987: Mexian, Ian Ring Music TheoryMexian
Scale 1219Scale 1219: Hidian, Ian Ring Music TheoryHidian
Scale 1475Scale 1475: Uffian, Ian Ring Music TheoryUffian
Scale 707Scale 707: Ehoian, Ian Ring Music TheoryEhoian
Scale 2755Scale 2755: Rivian, Ian Ring Music TheoryRivian
Scale 3779Scale 3779, Ian Ring Music Theory

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.