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Scale 1993: "Katoptian"

Scale 1993: Katoptian, 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
Katoptian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,3,6,7,8,9,10}

Forte Number

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

7-10

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: 637

Hemitonia

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

4 (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.

4

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.

6

Prime Form

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

no
prime: 607

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.

[4, 4, 5, 3, 3, 2]

Interval Spectrum

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

p3m3n5s4d4t2

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

Spectra Variation

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

3.143

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.

2.433

Polygon Perimeter

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

5.899

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

Harmonic Chords

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 TriadsD♯{3,7,10}231.75
G♯{8,0,3}231.75
Minor Triadscm{0,3,7}331.63
d♯m{3,6,10}331.63
Diminished Triads{0,3,6}231.75
d♯°{3,6,9}231.75
f♯°{6,9,0}231.88
{9,0,3}231.88
Parsimonious Voice Leading Between Common Triads of Scale 1993. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m D# D# cm->D# G# G# cm->G# d#° d#° d#°->d#m f#° f#° d#°->f#° d#m->D# f#°->a° G#->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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1993 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 761
Scale 761: Ponian, Ian Ring Music TheoryPonian
3rd mode:
Scale 607
Scale 607: Kadian, Ian Ring Music TheoryKadianThis is the prime mode
4th mode:
Scale 2351
Scale 2351: Gynian, Ian Ring Music TheoryGynian
5th mode:
Scale 3223
Scale 3223: Thyphian, Ian Ring Music TheoryThyphian
6th mode:
Scale 3659
Scale 3659: Polian, Ian Ring Music TheoryPolian
7th mode:
Scale 3877
Scale 3877: Thanian, Ian Ring Music TheoryThanian

Prime

The prime form of this scale is Scale 607

Scale 607Scale 607: Kadian, Ian Ring Music TheoryKadian

Complement

The heptatonic modal family [1993, 761, 607, 2351, 3223, 3659, 3877] (Forte: 7-10) is the complement of the pentatonic modal family [91, 1547, 1729, 2093, 2821] (Forte: 5-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1993 is 637

Scale 637Scale 637: Debussy's Heptatonic, Ian Ring Music TheoryDebussy's Heptatonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1993 is chiral, and its enantiomorph is scale 637

Scale 637Scale 637: Debussy's Heptatonic, Ian Ring Music TheoryDebussy's Heptatonic

Transformations:

T0 1993  T0I 637
T1 3986  T1I 1274
T2 3877  T2I 2548
T3 3659  T3I 1001
T4 3223  T4I 2002
T5 2351  T5I 4004
T6 607  T6I 3913
T7 1214  T7I 3731
T8 2428  T8I 3367
T9 761  T9I 2639
T10 1522  T10I 1183
T11 3044  T11I 2366

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 1995Scale 1995: Aeolacryllic, Ian Ring Music TheoryAeolacryllic
Scale 1997Scale 1997: Raga Cintamani, Ian Ring Music TheoryRaga Cintamani
Scale 1985Scale 1985, Ian Ring Music Theory
Scale 1989Scale 1989: Dydian, Ian Ring Music TheoryDydian
Scale 2001Scale 2001: Gydian, Ian Ring Music TheoryGydian
Scale 2009Scale 2009: Stacryllic, Ian Ring Music TheoryStacryllic
Scale 2025Scale 2025, Ian Ring Music Theory
Scale 1929Scale 1929: Aeolycrimic, Ian Ring Music TheoryAeolycrimic
Scale 1961Scale 1961: Soptian, Ian Ring Music TheorySoptian
Scale 1865Scale 1865: Thagimic, Ian Ring Music TheoryThagimic
Scale 1737Scale 1737: Raga Madhukauns, Ian Ring Music TheoryRaga Madhukauns
Scale 1481Scale 1481: Zagimic, Ian Ring Music TheoryZagimic
Scale 969Scale 969: Ionogimic, Ian Ring Music TheoryIonogimic
Scale 3017Scale 3017: Gacrian, Ian Ring Music TheoryGacrian
Scale 4041Scale 4041: Zaryllic, Ian Ring Music TheoryZaryllic

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.