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Scale 4075: "Katyllian"

Scale 4075: Katyllian, 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
Katyllian

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

{0,1,3,5,6,7,8,9,10,11}

Forte Number

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

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.

[3]

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.

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

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.

[8, 9, 8, 8, 8, 4]

Interval Spectrum

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

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.

[6]

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 TriadsC♯{1,5,8}252.8
D♯{3,7,10}352.7
F{5,9,0}452.6
F♯{6,10,1}452.6
G♯{8,0,3}452.6
B{11,3,6}352.7
Minor Triadscm{0,3,7}352.7
d♯m{3,6,10}452.6
fm{5,8,0}452.6
f♯m{6,9,1}452.6
g♯m{8,11,3}352.7
a♯m{10,1,5}252.8
Augmented TriadsC♯+{1,5,9}452.6
D♯+{3,7,11}452.6
Diminished Triads{0,3,6}253
d♯°{3,6,9}252.8
{5,8,11}252.9
f♯°{6,9,0}252.8
{7,10,1}252.9
{9,0,3}252.8
Parsimonious Voice Leading Between Common Triads of Scale 4075. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ G# G# cm->G# C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F f#m f#m C#+->f#m a#m a#m C#+->a#m d#° d#° d#m d#m d#°->d#m d#°->f#m D# D# d#m->D# F# F# d#m->F# d#m->B D#->D#+ D#->g° g#m g#m D#+->g#m D#+->B f°->fm f°->g#m fm->F fm->G# f#° f#° F->f#° F->a° f#°->f#m f#m->F# F#->g° F#->a#m g#m->G# G#->a°

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 4075 can be rotated to make 9 other scales. The 1st mode is itself.

2nd mode:
Scale 4085
Scale 4085: Rechberger's Decamode, Ian Ring Music TheoryRechberger's Decamode
3rd mode:
Scale 2045
Scale 2045: Katogyllian, Ian Ring Music TheoryKatogyllian
4th mode:
Scale 1535
Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllianThis is the prime mode
5th mode:
Scale 2815
Scale 2815: Aeradyllian, Ian Ring Music TheoryAeradyllian
6th mode:
Scale 3455
Scale 3455: Ryptyllian, Ian Ring Music TheoryRyptyllian
7th mode:
Scale 3775
Scale 3775: Loptyllian, Ian Ring Music TheoryLoptyllian
8th mode:
Scale 3935
Scale 3935: Kataphyllian, Ian Ring Music TheoryKataphyllian
9th mode:
Scale 4015
Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
10th mode:
Scale 4055
Scale 4055: Dagyllian, Ian Ring Music TheoryDagyllian

Prime

The prime form of this scale is Scale 1535

Scale 1535Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllian

Complement

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

Scale 2815Scale 2815: Aeradyllian, Ian Ring Music TheoryAeradyllian

Transformations:

T0 4075  T0I 2815
T1 4055  T1I 1535
T2 4015  T2I 3070
T3 3935  T3I 2045
T4 3775  T4I 4090
T5 3455  T5I 4085
T6 2815  T6I 4075
T7 1535  T7I 4055
T8 3070  T8I 4015
T9 2045  T9I 3935
T10 4090  T10I 3775
T11 4085  T11I 3455

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 4073Scale 4073: Sathygic, Ian Ring Music TheorySathygic
Scale 4077Scale 4077: Gothyllian, Ian Ring Music TheoryGothyllian
Scale 4079Scale 4079: Ionatic, Ian Ring Music TheoryIonatic
Scale 4067Scale 4067: Aeolarygic, Ian Ring Music TheoryAeolarygic
Scale 4071Scale 4071: Rygyllian, Ian Ring Music TheoryRygyllian
Scale 4083Scale 4083: Bathyllian, Ian Ring Music TheoryBathyllian
Scale 4091Scale 4091: Thydatic, Ian Ring Music TheoryThydatic
Scale 4043Scale 4043: Phrocrygic, Ian Ring Music TheoryPhrocrygic
Scale 4059Scale 4059: Zolyllian, Ian Ring Music TheoryZolyllian
Scale 4011Scale 4011: Styrygic, Ian Ring Music TheoryStyrygic
Scale 3947Scale 3947: Ryptygic, Ian Ring Music TheoryRyptygic
Scale 3819Scale 3819: Aeolanygic, Ian Ring Music TheoryAeolanygic
Scale 3563Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic
Scale 3051Scale 3051: Stalygic, Ian Ring Music TheoryStalygic
Scale 2027Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic

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.