The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3453: "Katarygic"

Scale 3453: Katarygic, 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
Katarygic

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,2,3,4,5,6,8,10,11}

Forte Number

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

9-8

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: 2007

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

6 (multihemitonic)

Cohemitonia

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

4 (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.

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.

8

Prime Form

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

no
prime: 1503

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.

[2, 1, 1, 1, 1, 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.

<6, 7, 6, 7, 6, 4>

Interval Spectrum

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

p6m7n6s7d6t4

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> = {6,7,8}
<6> = {7,8,9}
<7> = {8,9,10}
<8> = {10,11}

Spectra Variation

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

1.556

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.

2.799

Polygon Perimeter

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

6.106

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}342.33
G♯{8,0,3}342.33
A♯{10,2,5}342.47
B{11,3,6}442.07
Minor Triadsd♯m{3,6,10}242.47
fm{5,8,0}342.47
g♯m{8,11,3}442.07
bm{11,2,6}442.2
Augmented TriadsC+{0,4,8}342.4
D+{2,6,10}342.4
Diminished Triads{0,3,6}242.47
{2,5,8}242.53
{5,8,11}242.67
g♯°{8,11,2}242.33
{11,2,5}242.53
Parsimonious Voice Leading Between Common Triads of Scale 3453. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ E E C+->E fm fm C+->fm C+->G# d°->fm A# A# d°->A# D+ D+ d#m d#m D+->d#m D+->A# bm bm D+->bm d#m->B E->f° g#m g#m E->g#m f°->fm g#° g#° g#°->g#m g#°->bm g#m->G# g#m->B A#->b° b°->bm bm->B

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3453 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 1887
Scale 1887: Aerocrygic, Ian Ring Music TheoryAerocrygic
3rd mode:
Scale 2991
Scale 2991: Zanygic, Ian Ring Music TheoryZanygic
4th mode:
Scale 3543
Scale 3543: Aeolonygic, Ian Ring Music TheoryAeolonygic
5th mode:
Scale 3819
Scale 3819: Aeolanygic, Ian Ring Music TheoryAeolanygic
6th mode:
Scale 3957
Scale 3957: Porygic, Ian Ring Music TheoryPorygic
7th mode:
Scale 2013
Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic
8th mode:
Scale 1527
Scale 1527: Aeolyrigic, Ian Ring Music TheoryAeolyrigic
9th mode:
Scale 2811
Scale 2811: Barygic, Ian Ring Music TheoryBarygic

Prime

The prime form of this scale is Scale 1503

Scale 1503Scale 1503: Padygic, Ian Ring Music TheoryPadygic

Complement

The enneatonic modal family [3453, 1887, 2991, 3543, 3819, 3957, 2013, 1527, 2811] (Forte: 9-8) is the complement of the tritonic modal family [69, 321, 1041] (Forte: 3-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3453 is 2007

Scale 2007Scale 2007: Stonygic, Ian Ring Music TheoryStonygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3453 is chiral, and its enantiomorph is scale 2007

Scale 2007Scale 2007: Stonygic, Ian Ring Music TheoryStonygic

Transformations:

T0 3453  T0I 2007
T1 2811  T1I 4014
T2 1527  T2I 3933
T3 3054  T3I 3771
T4 2013  T4I 3447
T5 4026  T5I 2799
T6 3957  T6I 1503
T7 3819  T7I 3006
T8 3543  T8I 1917
T9 2991  T9I 3834
T10 1887  T10I 3573
T11 3774  T11I 3051

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 3455Scale 3455: Ryptyllian, Ian Ring Music TheoryRyptyllian
Scale 3449Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 3445Scale 3445: Messiaen Mode 6 Inverse, Ian Ring Music TheoryMessiaen Mode 6 Inverse
Scale 3437Scale 3437, Ian Ring Music Theory
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 3389Scale 3389: Socryllic, Ian Ring Music TheorySocryllic
Scale 3517Scale 3517: Epocrygic, Ian Ring Music TheoryEpocrygic
Scale 3581Scale 3581: Epocryllian, Ian Ring Music TheoryEpocryllian
Scale 3197Scale 3197: Gylyllic, Ian Ring Music TheoryGylyllic
Scale 3325Scale 3325: Mixolygic, Ian Ring Music TheoryMixolygic
Scale 3709Scale 3709: Katynygic, Ian Ring Music TheoryKatynygic
Scale 3965Scale 3965: Messiaen Mode 7 Inverse, Ian Ring Music TheoryMessiaen Mode 7 Inverse
Scale 2429Scale 2429: Kadyllic, Ian Ring Music TheoryKadyllic
Scale 2941Scale 2941: Laptygic, Ian Ring Music TheoryLaptygic
Scale 1405Scale 1405: Goryllic, Ian Ring Music TheoryGoryllic

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.