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Scale 4025: "Kalygic"

Scale 4025: Kalygic, 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
Kalygic
Dozenal
Zonian

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

{0,3,4,5,7,8,9,10,11}

Forte Number

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

9-4

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: 959

Hemitonia

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

7 (multihemitonic)

Cohemitonia

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

5 (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.

8

Prime Form

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

no
prime: 959

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.

[3, 1, 1, 2, 1, 1, 1, 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.

<7, 6, 6, 7, 7, 3>

Interval Spectrum

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

p7m7n6s6d7t3

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

Spectra Variation

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

2

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.

2.683

Polygon Perimeter

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

6.038

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.

(41, 103, 190)

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}332
D♯{3,7,10}263
E{4,8,11}431.87
F{5,9,0}263
G♯{8,0,3}442
Minor Triadscm{0,3,7}342.2
em{4,7,11}442.07
fm{5,8,0}352.4
g♯m{8,11,3}342.07
am{9,0,4}352.4
Augmented TriadsC+{0,4,8}541.8
D♯+{3,7,11}452.27
Diminished Triads{4,7,10}252.8
{5,8,11}242.47
{9,0,3}252.6
Parsimonious Voice Leading Between Common Triads of Scale 4025. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ G# G# cm->G# C+ C+ C->C+ em em C->em E E C+->E fm fm C+->fm C+->G# am am C+->am D# D# D#->D#+ D#->e° D#+->em g#m g#m D#+->g#m e°->em em->E E->f° E->g#m f°->fm F F fm->F F->am g#m->G# 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.

Diameter6
Radius3
Self-Centeredno
Central VerticesC, E
Peripheral VerticesD♯, F

Modes

Modes are the rotational transformation of this scale. Scale 4025 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 1015
Scale 1015: Ionodygic, Ian Ring Music TheoryIonodygic
3rd mode:
Scale 2555
Scale 2555: Bythygic, Ian Ring Music TheoryBythygic
4th mode:
Scale 3325
Scale 3325: Mixolygic, Ian Ring Music TheoryMixolygic
5th mode:
Scale 1855
Scale 1855: Gaptygic, Ian Ring Music TheoryGaptygic
6th mode:
Scale 2975
Scale 2975: Aeroptygic, Ian Ring Music TheoryAeroptygic
7th mode:
Scale 3535
Scale 3535: Mylygic, Ian Ring Music TheoryMylygic
8th mode:
Scale 3815
Scale 3815: Galygic, Ian Ring Music TheoryGalygic
9th mode:
Scale 3955
Scale 3955: Pothygic, Ian Ring Music TheoryPothygic

Prime

The prime form of this scale is Scale 959

Scale 959Scale 959: Katylygic, Ian Ring Music TheoryKatylygic

Complement

The enneatonic modal family [4025, 1015, 2555, 3325, 1855, 2975, 3535, 3815, 3955] (Forte: 9-4) is the complement of the tritonic modal family [35, 385, 2065] (Forte: 3-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4025 is 959

Scale 959Scale 959: Katylygic, Ian Ring Music TheoryKatylygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 4025 is chiral, and its enantiomorph is scale 959

Scale 959Scale 959: Katylygic, Ian Ring Music TheoryKatylygic

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> 4025       T0I <11,0> 959
T1 <1,1> 3955      T1I <11,1> 1918
T2 <1,2> 3815      T2I <11,2> 3836
T3 <1,3> 3535      T3I <11,3> 3577
T4 <1,4> 2975      T4I <11,4> 3059
T5 <1,5> 1855      T5I <11,5> 2023
T6 <1,6> 3710      T6I <11,6> 4046
T7 <1,7> 3325      T7I <11,7> 3997
T8 <1,8> 2555      T8I <11,8> 3899
T9 <1,9> 1015      T9I <11,9> 3703
T10 <1,10> 2030      T10I <11,10> 3311
T11 <1,11> 4060      T11I <11,11> 2527
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2975      T0MI <7,0> 3899
T1M <5,1> 1855      T1MI <7,1> 3703
T2M <5,2> 3710      T2MI <7,2> 3311
T3M <5,3> 3325      T3MI <7,3> 2527
T4M <5,4> 2555      T4MI <7,4> 959
T5M <5,5> 1015      T5MI <7,5> 1918
T6M <5,6> 2030      T6MI <7,6> 3836
T7M <5,7> 4060      T7MI <7,7> 3577
T8M <5,8> 4025       T8MI <7,8> 3059
T9M <5,9> 3955      T9MI <7,9> 2023
T10M <5,10> 3815      T10MI <7,10> 4046
T11M <5,11> 3535      T11MI <7,11> 3997

The transformations that map this set to itself are: T0, T8M

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 4027Scale 4027: Ragyllian, Ian Ring Music TheoryRagyllian
Scale 4029Scale 4029: Major/Minor Mixed, Ian Ring Music TheoryMajor/Minor Mixed
Scale 4017Scale 4017: Dolyllic, Ian Ring Music TheoryDolyllic
Scale 4021Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
Scale 4009Scale 4009: Phranyllic, Ian Ring Music TheoryPhranyllic
Scale 3993Scale 3993: Ioniptyllic, Ian Ring Music TheoryIoniptyllic
Scale 4057Scale 4057: Phrygic, Ian Ring Music TheoryPhrygic
Scale 4089Scale 4089: Decatonic Chromatic Descending, Ian Ring Music TheoryDecatonic Chromatic Descending
Scale 3897Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic
Scale 3961Scale 3961: Zathygic, Ian Ring Music TheoryZathygic
Scale 3769Scale 3769: Eponyllic, Ian Ring Music TheoryEponyllic
Scale 3513Scale 3513: Dydyllic, Ian Ring Music TheoryDydyllic
Scale 3001Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
Scale 1977Scale 1977: Dagyllic, Ian Ring Music TheoryDagyllic

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.