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Scale 3409: "Katanimic"

Scale 3409: Katanimic, 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
Katanimic

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

{0,4,6,8,10,11}

Forte Number

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

6-22

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: 343

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.

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.

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: 343

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.

[4, 2, 2, 2, 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.

<2, 4, 1, 4, 2, 2>

Interval Spectrum

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

p2m4ns4d2t2

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

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.

2.232

Polygon Perimeter

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

5.767

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 TriadsE{4,8,11}110.5
Augmented TriadsC+{0,4,8}110.5

The following pitch classes are not present in any of the common triads: {6,10}

Parsimonious Voice Leading Between Common Triads of Scale 3409. Created by Ian Ring ©2019 C+ C+ E E C+->E

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3409 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 469
Scale 469: Katyrimic, Ian Ring Music TheoryKatyrimic
3rd mode:
Scale 1141
Scale 1141: Rynimic, Ian Ring Music TheoryRynimic
4th mode:
Scale 1309
Scale 1309: Pogimic, Ian Ring Music TheoryPogimic
5th mode:
Scale 1351
Scale 1351: Aeraptimic, Ian Ring Music TheoryAeraptimic
6th mode:
Scale 2723
Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika

Prime

The prime form of this scale is Scale 343

Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic

Complement

The hexatonic modal family [3409, 469, 1141, 1309, 1351, 2723] (Forte: 6-22) is the complement of the hexatonic modal family [343, 1393, 1477, 1813, 2219, 3157] (Forte: 6-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3409 is 343

Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3409 is chiral, and its enantiomorph is scale 343

Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic

Transformations:

T0 3409  T0I 343
T1 2723  T1I 686
T2 1351  T2I 1372
T3 2702  T3I 2744
T4 1309  T4I 1393
T5 2618  T5I 2786
T6 1141  T6I 1477
T7 2282  T7I 2954
T8 469  T8I 1813
T9 938  T9I 3626
T10 1876  T10I 3157
T11 3752  T11I 2219

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 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3413Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone
Scale 3417Scale 3417: Golian, Ian Ring Music TheoryGolian
Scale 3393Scale 3393, Ian Ring Music Theory
Scale 3401Scale 3401: Palimic, Ian Ring Music TheoryPalimic
Scale 3425Scale 3425, Ian Ring Music Theory
Scale 3441Scale 3441: Thacrian, Ian Ring Music TheoryThacrian
Scale 3345Scale 3345: Zylitonic, Ian Ring Music TheoryZylitonic
Scale 3377Scale 3377: Phralimic, Ian Ring Music TheoryPhralimic
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 3537Scale 3537: Katogian, Ian Ring Music TheoryKatogian
Scale 3153Scale 3153: Zathitonic, Ian Ring Music TheoryZathitonic
Scale 3281Scale 3281: Raga Vijayavasanta, Ian Ring Music TheoryRaga Vijayavasanta
Scale 3665Scale 3665: Stalimic, Ian Ring Music TheoryStalimic
Scale 3921Scale 3921: Pythian, Ian Ring Music TheoryPythian
Scale 2385Scale 2385: Aeolanitonic, Ian Ring Music TheoryAeolanitonic
Scale 2897Scale 2897: Rycrimic, Ian Ring Music TheoryRycrimic
Scale 1361Scale 1361: Bolitonic, Ian Ring Music TheoryBolitonic

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.