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Scale 2659: "Katynimic"

Scale 2659: Katynimic, 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
Katynimic

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

{0,1,5,6,9,11}

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-Z17

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: 2251

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.

1 (uncohemitonic)

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.

5

Prime Form

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

no
prime: 407

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.

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

Interval Spectrum

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

p3m3n2s2d3t2

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

Spectra Variation

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

2.667

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.

2.116

Polygon Perimeter

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

5.699

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

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{5,9,0}221
Minor Triadsf♯m{6,9,1}221
Augmented TriadsC♯+{1,5,9}221
Diminished Triadsf♯°{6,9,0}221

The following pitch classes are not present in any of the common triads: {11}

Parsimonious Voice Leading Between Common Triads of Scale 2659. Created by Ian Ring ©2019 C#+ C#+ F F C#+->F f#m f#m C#+->f#m f#° f#° F->f#° f#°->f#m

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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2659 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3377
Scale 3377: Phralimic, Ian Ring Music TheoryPhralimic
3rd mode:
Scale 467
Scale 467: Raga Dhavalangam, Ian Ring Music TheoryRaga Dhavalangam
4th mode:
Scale 2281
Scale 2281: Rathimic, Ian Ring Music TheoryRathimic
5th mode:
Scale 797
Scale 797: Katocrimic, Ian Ring Music TheoryKatocrimic
6th mode:
Scale 1223
Scale 1223: Phryptimic, Ian Ring Music TheoryPhryptimic

Prime

The prime form of this scale is Scale 407

Scale 407Scale 407: Zylimic, Ian Ring Music TheoryZylimic

Complement

The hexatonic modal family [2659, 3377, 467, 2281, 797, 1223] (Forte: 6-Z17) is the complement of the hexatonic modal family [359, 907, 1649, 2227, 2501, 3161] (Forte: 6-Z43)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2659 is 2251

Scale 2251Scale 2251: Zodimic, Ian Ring Music TheoryZodimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2659 is chiral, and its enantiomorph is scale 2251

Scale 2251Scale 2251: Zodimic, Ian Ring Music TheoryZodimic

Transformations:

T0 2659  T0I 2251
T1 1223  T1I 407
T2 2446  T2I 814
T3 797  T3I 1628
T4 1594  T4I 3256
T5 3188  T5I 2417
T6 2281  T6I 739
T7 467  T7I 1478
T8 934  T8I 2956
T9 1868  T9I 1817
T10 3736  T10I 3634
T11 3377  T11I 3173

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 2657Scale 2657, Ian Ring Music Theory
Scale 2661Scale 2661: Stydimic, Ian Ring Music TheoryStydimic
Scale 2663Scale 2663: Lalian, Ian Ring Music TheoryLalian
Scale 2667Scale 2667: Byrian, Ian Ring Music TheoryByrian
Scale 2675Scale 2675: Chromatic Lydian, Ian Ring Music TheoryChromatic Lydian
Scale 2627Scale 2627, Ian Ring Music Theory
Scale 2643Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi
Scale 2595Scale 2595: Rolitonic, Ian Ring Music TheoryRolitonic
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 2787Scale 2787: Zyrian, Ian Ring Music TheoryZyrian
Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian
Scale 2147Scale 2147, Ian Ring Music Theory
Scale 2403Scale 2403: Lycrimic, Ian Ring Music TheoryLycrimic
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 3683Scale 3683: Dycrian, Ian Ring Music TheoryDycrian
Scale 611Scale 611: Anchihoye, Ian Ring Music TheoryAnchihoye
Scale 1635Scale 1635: Sygimic, Ian Ring Music TheorySygimic

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.