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Scale 1813: "Katothimic"

Scale 1813: Katothimic, 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
Katothimic
Dozenal
Lavian

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

{0,2,4,8,9,10}

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: 1309

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

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.

[2, 2, 4, 1, 1, 2]

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

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.

(10, 23, 64)

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
Minor Triadsam{9,0,4}110.5
Augmented TriadsC+{0,4,8}110.5

The following pitch classes are not present in any of the common triads: {2,10}

Parsimonious Voice Leading Between Common Triads of Scale 1813. Created by Ian Ring ©2019 C+ C+ am am C+->am

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 1813 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1477
Scale 1477: Raga Jaganmohanam, Ian Ring Music TheoryRaga Jaganmohanam
3rd mode:
Scale 1393
Scale 1393: Mycrimic, Ian Ring Music TheoryMycrimic
4th mode:
Scale 343
Scale 343: Ionorimic, Ian Ring Music TheoryIonorimicThis is the prime mode
5th mode:
Scale 2219
Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic
6th mode:
Scale 3157
Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic

Prime

The prime form of this scale is Scale 343

Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic

Complement

The hexatonic modal family [1813, 1477, 1393, 343, 2219, 3157] (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 1813 is 1309

Scale 1309Scale 1309: Pogimic, Ian Ring Music TheoryPogimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1813 is chiral, and its enantiomorph is scale 1309

Scale 1309Scale 1309: Pogimic, Ian Ring Music TheoryPogimic

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> 1813       T0I <11,0> 1309
T1 <1,1> 3626      T1I <11,1> 2618
T2 <1,2> 3157      T2I <11,2> 1141
T3 <1,3> 2219      T3I <11,3> 2282
T4 <1,4> 343      T4I <11,4> 469
T5 <1,5> 686      T5I <11,5> 938
T6 <1,6> 1372      T6I <11,6> 1876
T7 <1,7> 2744      T7I <11,7> 3752
T8 <1,8> 1393      T8I <11,8> 3409
T9 <1,9> 2786      T9I <11,9> 2723
T10 <1,10> 1477      T10I <11,10> 1351
T11 <1,11> 2954      T11I <11,11> 2702
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1813       T0MI <7,0> 1309
T1M <5,1> 3626      T1MI <7,1> 2618
T2M <5,2> 3157      T2MI <7,2> 1141
T3M <5,3> 2219      T3MI <7,3> 2282
T4M <5,4> 343      T4MI <7,4> 469
T5M <5,5> 686      T5MI <7,5> 938
T6M <5,6> 1372      T6MI <7,6> 1876
T7M <5,7> 2744      T7MI <7,7> 3752
T8M <5,8> 1393      T8MI <7,8> 3409
T9M <5,9> 2786      T9MI <7,9> 2723
T10M <5,10> 1477      T10MI <7,10> 1351
T11M <5,11> 2954      T11MI <7,11> 2702

The transformations that map this set to itself are: T0, T0M

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 1815Scale 1815: Godian, Ian Ring Music TheoryGodian
Scale 1809Scale 1809: Ranitonic, Ian Ring Music TheoryRanitonic
Scale 1811Scale 1811: Kyptimic, Ian Ring Music TheoryKyptimic
Scale 1817Scale 1817: Phrythimic, Ian Ring Music TheoryPhrythimic
Scale 1821Scale 1821: Aeradian, Ian Ring Music TheoryAeradian
Scale 1797Scale 1797: Lalian, Ian Ring Music TheoryLalian
Scale 1805Scale 1805: Laqian, Ian Ring Music TheoryLaqian
Scale 1829Scale 1829: Pathimic, Ian Ring Music TheoryPathimic
Scale 1845Scale 1845: Lagian, Ian Ring Music TheoryLagian
Scale 1877Scale 1877: Aeroptian, Ian Ring Music TheoryAeroptian
Scale 1941Scale 1941: Aeranian, Ian Ring Music TheoryAeranian
Scale 1557Scale 1557: Jovian, Ian Ring Music TheoryJovian
Scale 1685Scale 1685: Zeracrimic, Ian Ring Music TheoryZeracrimic
Scale 1301Scale 1301: Koditonic, Ian Ring Music TheoryKoditonic
Scale 789Scale 789: Zogitonic, Ian Ring Music TheoryZogitonic
Scale 2837Scale 2837: Aelothimic, Ian Ring Music TheoryAelothimic
Scale 3861Scale 3861: Phroptian, Ian Ring Music TheoryPhroptian

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.