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Scale 2045: "Katogyllian"

Scale 2045: Katogyllian, 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
Katogyllian
Dozenal
Mihian

Analysis

Cardinality

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

10 (decatonic)

Pitch Class Set

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

{0,2,3,4,5,6,7,8,9,10}

Forte Number

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

10-2

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.

[0]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

yes

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.

8 (multihemitonic)

Cohemitonia

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

7 (multicohemitonic)

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.

2

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.

9

Prime Form

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

no
prime: 1535

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, 1, 1, 1, 1, 1, 1, 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.

<8, 9, 8, 8, 8, 4>

Interval Spectrum

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

p8m8n8s9d8t4

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

Spectra Variation

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

1.6

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.

yes

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.

2.866

Polygon Perimeter

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

6.141

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.

[0]

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.

(56, 147, 228)

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 TriadsC{0,4,7}352.7
D{2,6,9}452.6
D♯{3,7,10}452.6
F{5,9,0}452.6
G♯{8,0,3}352.7
A♯{10,2,5}252.8
Minor Triadscm{0,3,7}452.6
dm{2,5,9}452.6
d♯m{3,6,10}452.6
fm{5,8,0}352.7
gm{7,10,2}252.8
am{9,0,4}352.7
Augmented TriadsC+{0,4,8}452.6
D+{2,6,10}452.6
Diminished Triads{0,3,6}252.8
{2,5,8}252.9
d♯°{3,6,9}252.8
{4,7,10}252.9
f♯°{6,9,0}252.8
{9,0,3}253
Parsimonious Voice Leading Between Common Triads of Scale 2045. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m C C cm->C D# D# cm->D# G# G# cm->G# C+ C+ C->C+ C->e° fm fm C+->fm C+->G# am am C+->am dm dm d°->dm d°->fm D D dm->D F F dm->F A# A# dm->A# D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° D+->d#m gm gm D+->gm D+->A# d#°->d#m d#m->D# D#->e° D#->gm fm->F F->f#° F->am G#->a° a°->am

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.

Diameter5
Radius5
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2045 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 1535
Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllianThis is the prime mode
3rd mode:
Scale 2815
Scale 2815: Aeradyllian, Ian Ring Music TheoryAeradyllian
4th mode:
Scale 3455
Scale 3455: Ryptyllian, Ian Ring Music TheoryRyptyllian
5th mode:
Scale 3775
Scale 3775: Loptyllian, Ian Ring Music TheoryLoptyllian
6th mode:
Scale 3935
Scale 3935: Kataphyllian, Ian Ring Music TheoryKataphyllian
7th mode:
Scale 4015
Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
8th mode:
Scale 4055
Scale 4055: Dagyllian, Ian Ring Music TheoryDagyllian
9th mode:
Scale 4075
Scale 4075: Katyllian, Ian Ring Music TheoryKatyllian
10th mode:
Scale 4085
Scale 4085: Rechberger's Decamode, Ian Ring Music TheoryRechberger's Decamode

Prime

The prime form of this scale is Scale 1535

Scale 1535Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllian

Complement

The decatonic modal family [2045, 1535, 2815, 3455, 3775, 3935, 4015, 4055, 4075, 4085] (Forte: 10-2) is the complement of the ditonic modal family [5, 1025] (Forte: 2-2)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2045 is itself, because it is a palindromic scale!

Scale 2045Scale 2045: Katogyllian, Ian Ring Music TheoryKatogyllian

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> 2045       T0I <11,0> 2045
T1 <1,1> 4090      T1I <11,1> 4090
T2 <1,2> 4085      T2I <11,2> 4085
T3 <1,3> 4075      T3I <11,3> 4075
T4 <1,4> 4055      T4I <11,4> 4055
T5 <1,5> 4015      T5I <11,5> 4015
T6 <1,6> 3935      T6I <11,6> 3935
T7 <1,7> 3775      T7I <11,7> 3775
T8 <1,8> 3455      T8I <11,8> 3455
T9 <1,9> 2815      T9I <11,9> 2815
T10 <1,10> 1535      T10I <11,10> 1535
T11 <1,11> 3070      T11I <11,11> 3070
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3935      T0MI <7,0> 3935
T1M <5,1> 3775      T1MI <7,1> 3775
T2M <5,2> 3455      T2MI <7,2> 3455
T3M <5,3> 2815      T3MI <7,3> 2815
T4M <5,4> 1535      T4MI <7,4> 1535
T5M <5,5> 3070      T5MI <7,5> 3070
T6M <5,6> 2045       T6MI <7,6> 2045
T7M <5,7> 4090      T7MI <7,7> 4090
T8M <5,8> 4085      T8MI <7,8> 4085
T9M <5,9> 4075      T9MI <7,9> 4075
T10M <5,10> 4055      T10MI <7,10> 4055
T11M <5,11> 4015      T11MI <7,11> 4015

The transformations that map this set to itself are: T0, T0I, T6M, T6MI

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 2047Scale 2047: Chromatic Undecamode, Ian Ring Music TheoryChromatic Undecamode
Scale 2041Scale 2041: Aeolacrygic, Ian Ring Music TheoryAeolacrygic
Scale 2043Scale 2043: Maqam Tarzanuyn, Ian Ring Music TheoryMaqam Tarzanuyn
Scale 2037Scale 2037: Sythygic, Ian Ring Music TheorySythygic
Scale 2029Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi
Scale 2013Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic
Scale 1981Scale 1981: Houseini, Ian Ring Music TheoryHouseini
Scale 1917Scale 1917: Sacrygic, Ian Ring Music TheorySacrygic
Scale 1789Scale 1789: Blues Enneatonic II, Ian Ring Music TheoryBlues Enneatonic II
Scale 1533Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
Scale 1021Scale 1021: Ladygic, Ian Ring Music TheoryLadygic
Scale 3069Scale 3069: Maqam Shawq Afza, Ian Ring Music TheoryMaqam Shawq Afza
Scale 4093Scale 4093: Chromatic Undecamode 11, Ian Ring Music TheoryChromatic Undecamode 11

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.