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Scale 1529: "Kataryllic"

Scale 1529: Kataryllic, 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
Kataryllic
Dozenal
Jidian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

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

Forte Number

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

8-11

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: 1013

Hemitonia

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

5 (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.

7

Prime Form

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

no
prime: 703

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, 1, 1, 1, 2, 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.

<5, 6, 5, 5, 5, 2>

Interval Spectrum

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

p5m5n5s6d5t2

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

Spectra Variation

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

2.75

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.

2.616

Polygon Perimeter

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

6.002

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.

(59, 69, 149)

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}331.56
D♯{3,7,10}341.78
G♯{8,0,3}231.78
Minor Triadscm{0,3,7}431.44
d♯m{3,6,10}252.33
fm{5,8,0}152.67
Augmented TriadsC+{0,4,8}341.89
Diminished Triads{0,3,6}242
{4,7,10}231.89
Parsimonious Voice Leading Between Common Triads of Scale 1529. 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# d#m->D# D#->e°

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
Radius3
Self-Centeredno
Central Verticescm, C, e°, G♯
Peripheral Verticesd♯m, fm

Modes

Modes are the rotational transformation of this scale. Scale 1529 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 703
Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllicThis is the prime mode
3rd mode:
Scale 2399
Scale 2399: Zanyllic, Ian Ring Music TheoryZanyllic
4th mode:
Scale 3247
Scale 3247: Aeolonyllic, Ian Ring Music TheoryAeolonyllic
5th mode:
Scale 3671
Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
6th mode:
Scale 3883
Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
7th mode:
Scale 3989
Scale 3989: Sythyllic, Ian Ring Music TheorySythyllic
8th mode:
Scale 2021
Scale 2021: Katycryllic, Ian Ring Music TheoryKatycryllic

Prime

The prime form of this scale is Scale 703

Scale 703Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllic

Complement

The octatonic modal family [1529, 703, 2399, 3247, 3671, 3883, 3989, 2021] (Forte: 8-11) is the complement of the tetratonic modal family [43, 1409, 1541, 2069] (Forte: 4-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1529 is 1013

Scale 1013Scale 1013: Stydyllic, Ian Ring Music TheoryStydyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1529 is chiral, and its enantiomorph is scale 1013

Scale 1013Scale 1013: Stydyllic, Ian Ring Music TheoryStydyllic

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> 1529       T0I <11,0> 1013
T1 <1,1> 3058      T1I <11,1> 2026
T2 <1,2> 2021      T2I <11,2> 4052
T3 <1,3> 4042      T3I <11,3> 4009
T4 <1,4> 3989      T4I <11,4> 3923
T5 <1,5> 3883      T5I <11,5> 3751
T6 <1,6> 3671      T6I <11,6> 3407
T7 <1,7> 3247      T7I <11,7> 2719
T8 <1,8> 2399      T8I <11,8> 1343
T9 <1,9> 703      T9I <11,9> 2686
T10 <1,10> 1406      T10I <11,10> 1277
T11 <1,11> 2812      T11I <11,11> 2554
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2399      T0MI <7,0> 3923
T1M <5,1> 703      T1MI <7,1> 3751
T2M <5,2> 1406      T2MI <7,2> 3407
T3M <5,3> 2812      T3MI <7,3> 2719
T4M <5,4> 1529       T4MI <7,4> 1343
T5M <5,5> 3058      T5MI <7,5> 2686
T6M <5,6> 2021      T6MI <7,6> 1277
T7M <5,7> 4042      T7MI <7,7> 2554
T8M <5,8> 3989      T8MI <7,8> 1013
T9M <5,9> 3883      T9MI <7,9> 2026
T10M <5,10> 3671      T10MI <7,10> 4052
T11M <5,11> 3247      T11MI <7,11> 4009

The transformations that map this set to itself are: T0, T4M

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 1531Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic
Scale 1533Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
Scale 1521Scale 1521: Stanian, Ian Ring Music TheoryStanian
Scale 1525Scale 1525: Sodyllic, Ian Ring Music TheorySodyllic
Scale 1513Scale 1513: Stathian, Ian Ring Music TheoryStathian
Scale 1497Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini
Scale 1465Scale 1465: Mela Ragavardhani, Ian Ring Music TheoryMela Ragavardhani
Scale 1401Scale 1401: Pagian, Ian Ring Music TheoryPagian
Scale 1273Scale 1273: Ronian, Ian Ring Music TheoryRonian
Scale 1785Scale 1785: Tharyllic, Ian Ring Music TheoryTharyllic
Scale 2041Scale 2041: Aeolacrygic, Ian Ring Music TheoryAeolacrygic
Scale 505Scale 505: Sanian, Ian Ring Music TheorySanian
Scale 1017Scale 1017: Dythyllic, Ian Ring Music TheoryDythyllic
Scale 2553Scale 2553: Aeolaptyllic, Ian Ring Music TheoryAeolaptyllic
Scale 3577Scale 3577: Loptygic, Ian Ring Music TheoryLoptygic

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.