The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3465: "Katathimic"

Scale 3465: Katathimic, 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
Katathimic
Dozenal
Vogian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,3,7,8,10,11}

Forte Number

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

6-14

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: 567

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.

1 (uncohemitonic)

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.

5

Prime Form

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

no
prime: 315

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, 4, 1, 2, 1, 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, 2, 3, 4, 3, 0>

Interval Spectrum

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

p3m4n3s2d3

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> = {1,2,3,4}
<2> = {2,3,4,5,7}
<3> = {4,5,7,8}
<4> = {5,7,8,9,10}
<5> = {8,9,10,11}

Spectra Variation

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

3.333

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.

2.116

Polygon Perimeter

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

5.699

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.

(19, 22, 65)

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}131.6
G♯{8,0,3}231.4
Minor Triadscm{0,3,7}221.2
g♯m{8,11,3}221.2
Augmented TriadsD♯+{3,7,11}321
Parsimonious Voice Leading Between Common Triads of Scale 3465. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# D# D# D#->D#+ g#m g#m D#+->g#m g#m->G#

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
Radius2
Self-Centeredno
Central Verticescm, D♯+, g♯m
Peripheral VerticesD♯, G♯

Modes

Modes are the rotational transformation of this scale. Scale 3465 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 945
Scale 945: Raga Saravati, Ian Ring Music TheoryRaga Saravati
3rd mode:
Scale 315
Scale 315: Stodimic, Ian Ring Music TheoryStodimicThis is the prime mode
4th mode:
Scale 2205
Scale 2205: Ionocrimic, Ian Ring Music TheoryIonocrimic
5th mode:
Scale 1575
Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic
6th mode:
Scale 2835
Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic

Prime

The prime form of this scale is Scale 315

Scale 315Scale 315: Stodimic, Ian Ring Music TheoryStodimic

Complement

The hexatonic modal family [3465, 945, 315, 2205, 1575, 2835] (Forte: 6-14) is the complement of the hexatonic modal family [315, 945, 1575, 2205, 2835, 3465] (Forte: 6-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3465 is 567

Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3465 is chiral, and its enantiomorph is scale 567

Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic

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> 3465       T0I <11,0> 567
T1 <1,1> 2835      T1I <11,1> 1134
T2 <1,2> 1575      T2I <11,2> 2268
T3 <1,3> 3150      T3I <11,3> 441
T4 <1,4> 2205      T4I <11,4> 882
T5 <1,5> 315      T5I <11,5> 1764
T6 <1,6> 630      T6I <11,6> 3528
T7 <1,7> 1260      T7I <11,7> 2961
T8 <1,8> 2520      T8I <11,8> 1827
T9 <1,9> 945      T9I <11,9> 3654
T10 <1,10> 1890      T10I <11,10> 3213
T11 <1,11> 3780      T11I <11,11> 2331
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2205      T0MI <7,0> 1827
T1M <5,1> 315      T1MI <7,1> 3654
T2M <5,2> 630      T2MI <7,2> 3213
T3M <5,3> 1260      T3MI <7,3> 2331
T4M <5,4> 2520      T4MI <7,4> 567
T5M <5,5> 945      T5MI <7,5> 1134
T6M <5,6> 1890      T6MI <7,6> 2268
T7M <5,7> 3780      T7MI <7,7> 441
T8M <5,8> 3465       T8MI <7,8> 882
T9M <5,9> 2835      T9MI <7,9> 1764
T10M <5,10> 1575      T10MI <7,10> 3528
T11M <5,11> 3150      T11MI <7,11> 2961

The transformations that map this set to itself are: T0, T8M

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 3467Scale 3467: Katonian, Ian Ring Music TheoryKatonian
Scale 3469Scale 3469: Monian, Ian Ring Music TheoryMonian
Scale 3457Scale 3457: Vobian, Ian Ring Music TheoryVobian
Scale 3461Scale 3461: Vodian, Ian Ring Music TheoryVodian
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 3481Scale 3481: Katathian, Ian Ring Music TheoryKatathian
Scale 3497Scale 3497: Phrolian, Ian Ring Music TheoryPhrolian
Scale 3529Scale 3529: Stalian, Ian Ring Music TheoryStalian
Scale 3337Scale 3337: Vafian, Ian Ring Music TheoryVafian
Scale 3401Scale 3401: Palimic, Ian Ring Music TheoryPalimic
Scale 3209Scale 3209: Aeraphitonic, Ian Ring Music TheoryAeraphitonic
Scale 3721Scale 3721: Phragimic, Ian Ring Music TheoryPhragimic
Scale 3977Scale 3977: Kythian, Ian Ring Music TheoryKythian
Scale 2441Scale 2441: Kyritonic, Ian Ring Music TheoryKyritonic
Scale 2953Scale 2953: Ionylimic, Ian Ring Music TheoryIonylimic
Scale 1417Scale 1417: Raga Shailaja, Ian Ring Music TheoryRaga Shailaja

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.