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Scale 2861: "Katothian"

Scale 2861: Katothian, 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
Katothian
Dozenal
Sahian

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

{0,2,3,5,8,9,11}

Forte Number

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

7-31

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: 1691

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

0 (ancohemitonic)

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: 731

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.

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

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

Interval Spectrum

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

p3m3n6s3d3t3

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

Spectra Variation

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

1.714

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.

2.549

Polygon Perimeter

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

5.967

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.

(4, 27, 84)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsF{5,9,0}331.7
G♯{8,0,3}331.7
Minor Triadsdm{2,5,9}331.8
fm{5,8,0}431.6
g♯m{8,11,3}331.8
Diminished Triads{2,5,8}231.9
{5,8,11}231.9
g♯°{8,11,2}232
{9,0,3}232
{11,2,5}232
Parsimonious Voice Leading Between Common Triads of Scale 2861. Created by Ian Ring ©2019 dm dm d°->dm fm fm d°->fm F F dm->F dm->b° f°->fm g#m g#m f°->g#m fm->F G# G# fm->G# F->a° g#° g#° g#°->g#m g#°->b° g#m->G# 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 2861 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1739
Scale 1739: Mela Sadvidhamargini, Ian Ring Music TheoryMela Sadvidhamargini
3rd mode:
Scale 2917
Scale 2917: Nohkan Flute Scale, Ian Ring Music TheoryNohkan Flute Scale
4th mode:
Scale 1753
Scale 1753: Hungarian Major, Ian Ring Music TheoryHungarian Major
5th mode:
Scale 731
Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating HeptamodeThis is the prime mode
6th mode:
Scale 2413
Scale 2413: Locrian Natural 2, Ian Ring Music TheoryLocrian Natural 2
7th mode:
Scale 1627
Scale 1627: Zyptian, Ian Ring Music TheoryZyptian

Prime

The prime form of this scale is Scale 731

Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode

Complement

The heptatonic modal family [2861, 1739, 2917, 1753, 731, 2413, 1627] (Forte: 7-31) is the complement of the pentatonic modal family [587, 601, 713, 1609, 2341] (Forte: 5-31)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2861 is 1691

Scale 1691Scale 1691: Kathian, Ian Ring Music TheoryKathian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2861 is chiral, and its enantiomorph is scale 1691

Scale 1691Scale 1691: Kathian, Ian Ring Music TheoryKathian

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> 2861       T0I <11,0> 1691
T1 <1,1> 1627      T1I <11,1> 3382
T2 <1,2> 3254      T2I <11,2> 2669
T3 <1,3> 2413      T3I <11,3> 1243
T4 <1,4> 731      T4I <11,4> 2486
T5 <1,5> 1462      T5I <11,5> 877
T6 <1,6> 2924      T6I <11,6> 1754
T7 <1,7> 1753      T7I <11,7> 3508
T8 <1,8> 3506      T8I <11,8> 2921
T9 <1,9> 2917      T9I <11,9> 1747
T10 <1,10> 1739      T10I <11,10> 3494
T11 <1,11> 3478      T11I <11,11> 2893
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1691      T0MI <7,0> 2861
T1M <5,1> 3382      T1MI <7,1> 1627
T2M <5,2> 2669      T2MI <7,2> 3254
T3M <5,3> 1243      T3MI <7,3> 2413
T4M <5,4> 2486      T4MI <7,4> 731
T5M <5,5> 877      T5MI <7,5> 1462
T6M <5,6> 1754      T6MI <7,6> 2924
T7M <5,7> 3508      T7MI <7,7> 1753
T8M <5,8> 2921      T8MI <7,8> 3506
T9M <5,9> 1747      T9MI <7,9> 2917
T10M <5,10> 3494      T10MI <7,10> 1739
T11M <5,11> 2893      T11MI <7,11> 3478

The transformations that map this set to itself are: T0, T0MI

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 2863Scale 2863: Aerogyllic, Ian Ring Music TheoryAerogyllic
Scale 2857Scale 2857: Stythimic, Ian Ring Music TheoryStythimic
Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
Scale 2853Scale 2853: Baptimic, Ian Ring Music TheoryBaptimic
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
Scale 2877Scale 2877: Phrylyllic, Ian Ring Music TheoryPhrylyllic
Scale 2829Scale 2829: Rupian, Ian Ring Music TheoryRupian
Scale 2845Scale 2845: Baptian, Ian Ring Music TheoryBaptian
Scale 2893Scale 2893: Lylian, Ian Ring Music TheoryLylian
Scale 2925Scale 2925: Diminished, Ian Ring Music TheoryDiminished
Scale 2989Scale 2989: Bebop Minor, Ian Ring Music TheoryBebop Minor
Scale 2605Scale 2605: Rylimic, Ian Ring Music TheoryRylimic
Scale 2733Scale 2733: Melodic Minor Ascending, Ian Ring Music TheoryMelodic Minor Ascending
Scale 2349Scale 2349: Raga Ghantana, Ian Ring Music TheoryRaga Ghantana
Scale 3373Scale 3373: Lodian, Ian Ring Music TheoryLodian
Scale 3885Scale 3885: Styryllic, Ian Ring Music TheoryStyryllic
Scale 813Scale 813: Larimic, Ian Ring Music TheoryLarimic
Scale 1837Scale 1837: Dalian, Ian Ring Music TheoryDalian

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.