The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 869: "Kothimic"

Scale 869: Kothimic, 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

41161837294116105072918310504116183
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
Kothimic

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

{0,2,5,6,8,9}

Forte Number

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

6-Z49

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.

[1]

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.

2 (dihemitonic)

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: 667

Deep Scale

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

no

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.

[2, 2, 4, 3, 2, 2]

Interval Spectrum

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

p2m3n4s2d2t2

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

Spectra Variation

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

2

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.

2.366

Polygon Perimeter

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

5.864

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.

[2]

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

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}231.5
F{5,9,0}321.17
Minor Triadsdm{2,5,9}321.17
fm{5,8,0}231.5
Diminished Triads{2,5,8}231.5
f♯°{6,9,0}231.5
Parsimonious Voice Leading Between Common Triads of Scale 869. Created by Ian Ring ©2019 dm dm d°->dm fm fm d°->fm D D dm->D F F dm->F f#° f#° D->f#° fm->F F->f#°

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 Verticesdm, F
Peripheral Verticesd°, D, fm, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 869 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1241
Scale 1241: Pygimic, Ian Ring Music TheoryPygimic
3rd mode:
Scale 667
Scale 667: Rodimic, Ian Ring Music TheoryRodimicThis is the prime mode
4th mode:
Scale 2381
Scale 2381: Takemitsu Linea Mode 1, Ian Ring Music TheoryTakemitsu Linea Mode 1
5th mode:
Scale 1619
Scale 1619: Prometheus Neapolitan, Ian Ring Music TheoryPrometheus Neapolitan
6th mode:
Scale 2857
Scale 2857: Stythimic, Ian Ring Music TheoryStythimic

Prime

The prime form of this scale is Scale 667

Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic

Complement

The hexatonic modal family [869, 1241, 667, 2381, 1619, 2857] (Forte: 6-Z49) is the complement of the hexatonic modal family [619, 857, 1427, 1613, 2357, 2761] (Forte: 6-Z28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 869 is 1241

Scale 1241Scale 1241: Pygimic, Ian Ring Music TheoryPygimic

Transformations:

T0 869  T0I 1241
T1 1738  T1I 2482
T2 3476  T2I 869
T3 2857  T3I 1738
T4 1619  T4I 3476
T5 3238  T5I 2857
T6 2381  T6I 1619
T7 667  T7I 3238
T8 1334  T8I 2381
T9 2668  T9I 667
T10 1241  T10I 1334
T11 2482  T11I 2668

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 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7
Scale 865Scale 865, Ian Ring Music Theory
Scale 867Scale 867: Phrocrimic, Ian Ring Music TheoryPhrocrimic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 877Scale 877: Moravian Pistalkova, Ian Ring Music TheoryMoravian Pistalkova
Scale 885Scale 885: Sathian, Ian Ring Music TheorySathian
Scale 837Scale 837: Epaditonic, Ian Ring Music TheoryEpaditonic
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 805Scale 805: Rothitonic, Ian Ring Music TheoryRothitonic
Scale 933Scale 933: Dadimic, Ian Ring Music TheoryDadimic
Scale 997Scale 997: Rycrian, Ian Ring Music TheoryRycrian
Scale 613Scale 613: Phralitonic, Ian Ring Music TheoryPhralitonic
Scale 741Scale 741: Gathimic, Ian Ring Music TheoryGathimic
Scale 357Scale 357: Banitonic, Ian Ring Music TheoryBanitonic
Scale 1381Scale 1381: Padimic, Ian Ring Music TheoryPadimic
Scale 1893Scale 1893: Ionylian, Ian Ring Music TheoryIonylian
Scale 2917Scale 2917: Nohkan Flute Scale, Ian Ring Music TheoryNohkan Flute Scale

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.