The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1959: "Katolyllic"

Scale 1959: Katolyllic, 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
Katolyllic

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

{0,1,2,5,7,8,9,10}

Forte Number

A code assigned by theorist Alan 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: 3261

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.

no
prime: 759

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.

[5, 5, 5, 5, 6, 2]

Interval Spectrum

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

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

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}341.9
F{5,9,0}242.1
A♯{10,2,5}341.9
Minor Triadsdm{2,5,9}331.7
fm{5,8,0}252.5
gm{7,10,2}252.5
a♯m{10,1,5}331.7
Augmented TriadsC♯+{1,5,9}431.5
Diminished Triads{2,5,8}242.1
{7,10,1}242.3
Parsimonious Voice Leading Between Common Triads of Scale 1959. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F a#m a#m C#+->a#m d°->dm A# A# dm->A# fm->F gm gm g°->gm g°->a#m gm->A# a#m->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♯+, dm, a♯m
Peripheral Verticesfm, gm

Modes

Modes are the rotational transformation of this scale. Scale 1959 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3027
Scale 3027: Rythyllic, Ian Ring Music TheoryRythyllic
3rd mode:
Scale 3561
Scale 3561: Pothyllic, Ian Ring Music TheoryPothyllic
4th mode:
Scale 957
Scale 957: Phronyllic, Ian Ring Music TheoryPhronyllic
5th mode:
Scale 1263
Scale 1263: Stynyllic, Ian Ring Music TheoryStynyllic
6th mode:
Scale 2679
Scale 2679: Rathyllic, Ian Ring Music TheoryRathyllic
7th mode:
Scale 3387
Scale 3387: Aeryptyllic, Ian Ring Music TheoryAeryptyllic
8th mode:
Scale 3741
Scale 3741: Zydyllic, Ian Ring Music TheoryZydyllic

Prime

The prime form of this scale is Scale 759

Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic

Complement

The octatonic modal family [1959, 3027, 3561, 957, 1263, 2679, 3387, 3741] (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 1959 is 3261

Scale 3261Scale 3261: Dodyllic, Ian Ring Music TheoryDodyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1959 is chiral, and its enantiomorph is scale 3261

Scale 3261Scale 3261: Dodyllic, Ian Ring Music TheoryDodyllic

Transformations:

T0 1959  T0I 3261
T1 3918  T1I 2427
T2 3741  T2I 759
T3 3387  T3I 1518
T4 2679  T4I 3036
T5 1263  T5I 1977
T6 2526  T6I 3954
T7 957  T7I 3813
T8 1914  T8I 3531
T9 3828  T9I 2967
T10 3561  T10I 1839
T11 3027  T11I 3678

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 1957Scale 1957: Pyrian, Ian Ring Music TheoryPyrian
Scale 1955Scale 1955: Sonian, Ian Ring Music TheorySonian
Scale 1963Scale 1963: Epocryllic, Ian Ring Music TheoryEpocryllic
Scale 1967Scale 1967: Diatonic Dorian Mixed, Ian Ring Music TheoryDiatonic Dorian Mixed
Scale 1975Scale 1975: Ionocrygic, Ian Ring Music TheoryIonocrygic
Scale 1927Scale 1927, Ian Ring Music Theory
Scale 1943Scale 1943, Ian Ring Music Theory
Scale 1991Scale 1991: Phryptyllic, Ian Ring Music TheoryPhryptyllic
Scale 2023Scale 2023: Zodygic, Ian Ring Music TheoryZodygic
Scale 1831Scale 1831: Pothian, Ian Ring Music TheoryPothian
Scale 1895Scale 1895: Salyllic, Ian Ring Music TheorySalyllic
Scale 1703Scale 1703: Mela Vanaspati, Ian Ring Music TheoryMela Vanaspati
Scale 1447Scale 1447: Mela Ratnangi, Ian Ring Music TheoryMela Ratnangi
Scale 935Scale 935: Chromatic Dorian, Ian Ring Music TheoryChromatic Dorian
Scale 2983Scale 2983: Zythyllic, Ian Ring Music TheoryZythyllic
Scale 4007Scale 4007: Doptygic, Ian Ring Music TheoryDoptygic

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.