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Scale 3695: "Kodygic"

Scale 3695: Kodygic, 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

Zeitler
Kodygic
Dozenal
Xeqian

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,1,2,3,5,6,9,10,11}

Forte Number

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

9-3

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.

none

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.

yes
enantiomorph: 3791

Hemitonia

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

7 (multihemitonic)

Cohemitonia

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

5 (multicohemitonic)

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.

3

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.

8

Prime Form

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

no
prime: 895

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, 1, 1, 2, 1, 3, 1, 1, 1]

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.

<7, 6, 7, 7, 6, 3>

Interval Spectrum

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

p6m7n7s6d7t3

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

Spectra Variation

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

2.222

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.

2.683

Polygon Perimeter

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

6.038

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.

none

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.

(64, 107, 194)

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 TriadsD{2,6,9}442.12
F{5,9,0}342.53
F♯{6,10,1}342.24
A♯{10,2,5}442.24
B{11,3,6}342.53
Minor Triadsdm{2,5,9}342.24
d♯m{3,6,10}342.35
f♯m{6,9,1}442.24
a♯m{10,1,5}342.35
bm{11,2,6}342.35
Augmented TriadsC♯+{1,5,9}442.24
D+{2,6,10}542
Diminished Triads{0,3,6}242.76
d♯°{3,6,9}242.59
f♯°{6,9,0}252.71
{9,0,3}242.76
{11,2,5}252.71
Parsimonious Voice Leading Between Common Triads of Scale 3695. Created by Ian Ring ©2019 c°->a° B B c°->B C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m a#m a#m C#+->a#m D D dm->D A# A# dm->A# D+ D+ D->D+ d#° d#° D->d#° D->f#m d#m d#m D+->d#m F# F# D+->F# D+->A# bm bm D+->bm d#°->d#m d#m->B f#° f#° F->f#° F->a° f#°->f#m f#m->F# F#->a#m a#m->A# A#->b° b°->bm bm->B

view full size

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.

Diameter5
Radius4
Self-Centeredno
Central Verticesc°, C♯+, dm, D, D+, d♯°, d♯m, F, f♯m, F♯, a°, a♯m, A♯, bm, B
Peripheral Verticesf♯°, b°

Modes

Modes are the rotational transformation of this scale. Scale 3695 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3895
Scale 3895: Eparygic, Ian Ring Music TheoryEparygic
3rd mode:
Scale 3995
Scale 3995: Ionygic, Ian Ring Music TheoryIonygic
4th mode:
Scale 4045
Scale 4045: Gyptygic, Ian Ring Music TheoryGyptygic
5th mode:
Scale 2035
Scale 2035: Aerythygic, Ian Ring Music TheoryAerythygic
6th mode:
Scale 3065
Scale 3065: Zothygic, Ian Ring Music TheoryZothygic
7th mode:
Scale 895
Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygicThis is the prime mode
8th mode:
Scale 2495
Scale 2495: Aeolocrygic, Ian Ring Music TheoryAeolocrygic
9th mode:
Scale 3295
Scale 3295: Phroptygic, Ian Ring Music TheoryPhroptygic

Prime

The prime form of this scale is Scale 895

Scale 895Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygic

Complement

The enneatonic modal family [3695, 3895, 3995, 4045, 2035, 3065, 895, 2495, 3295] (Forte: 9-3) is the complement of the tritonic modal family [19, 769, 2057] (Forte: 3-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3695 is 3791

Scale 3791Scale 3791: Stodygic, Ian Ring Music TheoryStodygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3695 is chiral, and its enantiomorph is scale 3791

Scale 3791Scale 3791: Stodygic, Ian Ring Music TheoryStodygic

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> 3695       T0I <11,0> 3791
T1 <1,1> 3295      T1I <11,1> 3487
T2 <1,2> 2495      T2I <11,2> 2879
T3 <1,3> 895      T3I <11,3> 1663
T4 <1,4> 1790      T4I <11,4> 3326
T5 <1,5> 3580      T5I <11,5> 2557
T6 <1,6> 3065      T6I <11,6> 1019
T7 <1,7> 2035      T7I <11,7> 2038
T8 <1,8> 4070      T8I <11,8> 4076
T9 <1,9> 4045      T9I <11,9> 4057
T10 <1,10> 3995      T10I <11,10> 4019
T11 <1,11> 3895      T11I <11,11> 3943
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1775      T0MI <7,0> 3821
T1M <5,1> 3550      T1MI <7,1> 3547
T2M <5,2> 3005      T2MI <7,2> 2999
T3M <5,3> 1915      T3MI <7,3> 1903
T4M <5,4> 3830      T4MI <7,4> 3806
T5M <5,5> 3565      T5MI <7,5> 3517
T6M <5,6> 3035      T6MI <7,6> 2939
T7M <5,7> 1975      T7MI <7,7> 1783
T8M <5,8> 3950      T8MI <7,8> 3566
T9M <5,9> 3805      T9MI <7,9> 3037
T10M <5,10> 3515      T10MI <7,10> 1979
T11M <5,11> 2935      T11MI <7,11> 3958

The transformations that map this set to itself are: T0

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 3693Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic
Scale 3691Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic
Scale 3687Scale 3687: Zonyllic, Ian Ring Music TheoryZonyllic
Scale 3703Scale 3703: Katalygic, Ian Ring Music TheoryKatalygic
Scale 3711Scale 3711: Decatonic Chromatic 4, Ian Ring Music TheoryDecatonic Chromatic 4
Scale 3663Scale 3663: Sonyllic, Ian Ring Music TheorySonyllic
Scale 3679Scale 3679: Rycrygic, Ian Ring Music TheoryRycrygic
Scale 3631Scale 3631: Gydyllic, Ian Ring Music TheoryGydyllic
Scale 3759Scale 3759: Darygic, Ian Ring Music TheoryDarygic
Scale 3823Scale 3823: Epinyllian, Ian Ring Music TheoryEpinyllian
Scale 3951Scale 3951: Mathyllian, Ian Ring Music TheoryMathyllian
Scale 3183Scale 3183: Mixonyllic, Ian Ring Music TheoryMixonyllic
Scale 3439Scale 3439: Lythygic, Ian Ring Music TheoryLythygic
Scale 2671Scale 2671: Aerolyllic, Ian Ring Music TheoryAerolyllic
Scale 1647Scale 1647: Polyllic, Ian Ring Music TheoryPolyllic

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.