The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3685: "Kodian"

Scale 3685: Kodian, 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
Kodian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

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

Forte Number

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

7-Z18

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: 1231

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

6

Prime Form

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

no
prime: 755

Deep Scale

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

no

Interval Formula

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

[2, 3, 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.

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

Interval Spectrum

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

p4m4n4s3d4t2

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

Spectra Variation

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

2.571

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.

2.433

Polygon Perimeter

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

5.899

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

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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}331.5
F{5,9,0}242
A♯{10,2,5}331.5
Minor Triadsdm{2,5,9}331.5
bm{11,2,6}242
Augmented TriadsD+{2,6,10}331.5
Diminished Triadsf♯°{6,9,0}242
{11,2,5}242
Parsimonious Voice Leading Between Common Triads of Scale 3685. Created by Ian Ring ©2019 dm dm D D dm->D F F dm->F A# A# dm->A# D+ D+ D->D+ f#° f#° D->f#° D+->A# bm bm D+->bm F->f#° A#->b° b°->bm

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.

Diameter4
Radius3
Self-Centeredno
Central Verticesdm, D, D+, A♯
Peripheral VerticesF, f♯°, b°, bm

Modes

Modes are the rotational transformation of this scale. Scale 3685 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1945
Scale 1945: Zarian, Ian Ring Music TheoryZarian
3rd mode:
Scale 755
Scale 755: Phrythian, Ian Ring Music TheoryPhrythianThis is the prime mode
4th mode:
Scale 2425
Scale 2425: Rorian, Ian Ring Music TheoryRorian
5th mode:
Scale 815
Scale 815: Bolian, Ian Ring Music TheoryBolian
6th mode:
Scale 2455
Scale 2455: Bothian, Ian Ring Music TheoryBothian
7th mode:
Scale 3275
Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani

Prime

The prime form of this scale is Scale 755

Scale 755Scale 755: Phrythian, Ian Ring Music TheoryPhrythian

Complement

The heptatonic modal family [3685, 1945, 755, 2425, 815, 2455, 3275] (Forte: 7-Z18) is the complement of the pentatonic modal family [179, 779, 1633, 2137, 2437] (Forte: 5-Z18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3685 is 1231

Scale 1231Scale 1231: Logian, Ian Ring Music TheoryLogian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3685 is chiral, and its enantiomorph is scale 1231

Scale 1231Scale 1231: Logian, Ian Ring Music TheoryLogian

Transformations:

T0 3685  T0I 1231
T1 3275  T1I 2462
T2 2455  T2I 829
T3 815  T3I 1658
T4 1630  T4I 3316
T5 3260  T5I 2537
T6 2425  T6I 979
T7 755  T7I 1958
T8 1510  T8I 3916
T9 3020  T9I 3737
T10 1945  T10I 3379
T11 3890  T11I 2663

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 3687Scale 3687: Zonyllic, Ian Ring Music TheoryZonyllic
Scale 3681Scale 3681, Ian Ring Music Theory
Scale 3683Scale 3683: Dycrian, Ian Ring Music TheoryDycrian
Scale 3689Scale 3689: Katocrian, Ian Ring Music TheoryKatocrian
Scale 3693Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic
Scale 3701Scale 3701: Bagyllic, Ian Ring Music TheoryBagyllic
Scale 3653Scale 3653: Sathimic, Ian Ring Music TheorySathimic
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3621Scale 3621: Gylimic, Ian Ring Music TheoryGylimic
Scale 3749Scale 3749: Raga Sorati, Ian Ring Music TheoryRaga Sorati
Scale 3813Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
Scale 3941Scale 3941: Stathyllic, Ian Ring Music TheoryStathyllic
Scale 3173Scale 3173: Zarimic, Ian Ring Music TheoryZarimic
Scale 3429Scale 3429: Marian, Ian Ring Music TheoryMarian
Scale 2661Scale 2661: Stydimic, Ian Ring Music TheoryStydimic
Scale 1637Scale 1637: Syptimic, Ian Ring Music TheorySyptimic

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. 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.