The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 759: "Katalyllic"

Scale 759: Katalyllic, 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
Katalyllic
Dozenal
Enkian

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

{0,1,2,4,5,6,7,9}

Forte Number

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

8-14

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: 3561

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.

3 (tricohemitonic)

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.

7

Prime Form

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

yes

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.

[1, 1, 2, 1, 1, 1, 2, 3]

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, 5, 5, 5, 6, 2>

Interval Spectrum

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

p6m5n5s5d5t2

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}
<4> = {5,7}
<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.

2.25

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.

(34, 56, 136)

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}252.5
D{2,6,9}252.5
F{5,9,0}331.7
A{9,1,4}331.7
Minor Triadsdm{2,5,9}242.1
f♯m{6,9,1}341.9
am{9,0,4}341.9
Augmented TriadsC♯+{1,5,9}431.5
Diminished Triadsc♯°{1,4,7}242.3
f♯°{6,9,0}242.1
Parsimonious Voice Leading Between Common Triads of Scale 759. Created by Ian Ring ©2019 C C c#° c#° C->c#° am am C->am A A c#°->A C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m C#+->A D D dm->D D->f#m f#° f#° F->f#° F->am f#°->f#m am->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
Radius3
Self-Centeredno
Central VerticesC♯+, F, A
Peripheral VerticesC, D

Modes

Modes are the rotational transformation of this scale. Scale 759 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2427
Scale 2427: Katoryllic, Ian Ring Music TheoryKatoryllic
3rd mode:
Scale 3261
Scale 3261: Dodyllic, Ian Ring Music TheoryDodyllic
4th mode:
Scale 1839
Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
5th mode:
Scale 2967
Scale 2967: Madyllic, Ian Ring Music TheoryMadyllic
6th mode:
Scale 3531
Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
7th mode:
Scale 3813
Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
8th mode:
Scale 1977
Scale 1977: Dagyllic, Ian Ring Music TheoryDagyllic

Prime

This is the prime form of this scale.

Complement

The octatonic modal family [759, 2427, 3261, 1839, 2967, 3531, 3813, 1977] (Forte: 8-14) is the complement of the tetratonic modal family [141, 417, 1059, 2577] (Forte: 4-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 759 is 3561

Scale 3561Scale 3561: Pothyllic, Ian Ring Music TheoryPothyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 759 is chiral, and its enantiomorph is scale 3561

Scale 3561Scale 3561: Pothyllic, Ian Ring Music TheoryPothyllic

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> 759       T0I <11,0> 3561
T1 <1,1> 1518      T1I <11,1> 3027
T2 <1,2> 3036      T2I <11,2> 1959
T3 <1,3> 1977      T3I <11,3> 3918
T4 <1,4> 3954      T4I <11,4> 3741
T5 <1,5> 3813      T5I <11,5> 3387
T6 <1,6> 3531      T6I <11,6> 2679
T7 <1,7> 2967      T7I <11,7> 1263
T8 <1,8> 1839      T8I <11,8> 2526
T9 <1,9> 3678      T9I <11,9> 957
T10 <1,10> 3261      T10I <11,10> 1914
T11 <1,11> 2427      T11I <11,11> 3828
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3939      T0MI <7,0> 2271
T1M <5,1> 3783      T1MI <7,1> 447
T2M <5,2> 3471      T2MI <7,2> 894
T3M <5,3> 2847      T3MI <7,3> 1788
T4M <5,4> 1599      T4MI <7,4> 3576
T5M <5,5> 3198      T5MI <7,5> 3057
T6M <5,6> 2301      T6MI <7,6> 2019
T7M <5,7> 507      T7MI <7,7> 4038
T8M <5,8> 1014      T8MI <7,8> 3981
T9M <5,9> 2028      T9MI <7,9> 3867
T10M <5,10> 4056      T10MI <7,10> 3639
T11M <5,11> 4017      T11MI <7,11> 3183

The transformations that map this set to itself are: T0

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 757Scale 757: Ionyptian, Ian Ring Music TheoryIonyptian
Scale 755Scale 755: Phrythian, Ian Ring Music TheoryPhrythian
Scale 763Scale 763: Doryllic, Ian Ring Music TheoryDoryllic
Scale 767Scale 767: Raptygic, Ian Ring Music TheoryRaptygic
Scale 743Scale 743: Chromatic Hypophrygian Inverse, Ian Ring Music TheoryChromatic Hypophrygian Inverse
Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 695Scale 695: Sarian, Ian Ring Music TheorySarian
Scale 631Scale 631: Zygian, Ian Ring Music TheoryZygian
Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic
Scale 1015Scale 1015: Ionodygic, Ian Ring Music TheoryIonodygic
Scale 247Scale 247: Bopian, Ian Ring Music TheoryBopian
Scale 503Scale 503: Thoptyllic, Ian Ring Music TheoryThoptyllic
Scale 1271Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic
Scale 1783Scale 1783: Youlan Scale, Ian Ring Music TheoryYoulan Scale
Scale 2807Scale 2807: Zylygic, Ian Ring Music TheoryZylygic

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.