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Scale 655: "Kataptimic"

Scale 655: Kataptimic, 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
Kataptimic

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

{0,1,2,3,7,9}

Forte Number

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

6-Z41

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: 3625

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.

2 (dicohemitonic)

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: 335

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 1, 4, 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.

<3, 3, 2, 2, 3, 2>

Interval Spectrum

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

p3m2n2s3d3t2

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

(20, 17, 64)

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
Minor Triadscm{0,3,7}110.5
Diminished Triads{9,0,3}110.5

The following pitch classes are not present in any of the common triads: {1,2}

Parsimonious Voice Leading Between Common Triads of Scale 655. Created by Ian Ring ©2019 cm cm cm->a°

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 655 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2375
Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic
3rd mode:
Scale 3235
Scale 3235: Pothimic, Ian Ring Music TheoryPothimic
4th mode:
Scale 3665
Scale 3665: Stalimic, Ian Ring Music TheoryStalimic
5th mode:
Scale 485
Scale 485: Stoptimic, Ian Ring Music TheoryStoptimic
6th mode:
Scale 1145
Scale 1145: Zygimic, Ian Ring Music TheoryZygimic

Prime

The prime form of this scale is Scale 335

Scale 335Scale 335: Zanimic, Ian Ring Music TheoryZanimic

Complement

The hexatonic modal family [655, 2375, 3235, 3665, 485, 1145] (Forte: 6-Z41) is the complement of the hexatonic modal family [215, 1475, 1805, 2155, 2785, 3125] (Forte: 6-Z12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 655 is 3625

Scale 3625Scale 3625: Podimic, Ian Ring Music TheoryPodimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 655 is chiral, and its enantiomorph is scale 3625

Scale 3625Scale 3625: Podimic, Ian Ring Music TheoryPodimic

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> 655       T0I <11,0> 3625
T1 <1,1> 1310      T1I <11,1> 3155
T2 <1,2> 2620      T2I <11,2> 2215
T3 <1,3> 1145      T3I <11,3> 335
T4 <1,4> 2290      T4I <11,4> 670
T5 <1,5> 485      T5I <11,5> 1340
T6 <1,6> 970      T6I <11,6> 2680
T7 <1,7> 1940      T7I <11,7> 1265
T8 <1,8> 3880      T8I <11,8> 2530
T9 <1,9> 3665      T9I <11,9> 965
T10 <1,10> 3235      T10I <11,10> 1930
T11 <1,11> 2375      T11I <11,11> 3860
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3625      T0MI <7,0> 655
T1M <5,1> 3155      T1MI <7,1> 1310
T2M <5,2> 2215      T2MI <7,2> 2620
T3M <5,3> 335      T3MI <7,3> 1145
T4M <5,4> 670      T4MI <7,4> 2290
T5M <5,5> 1340      T5MI <7,5> 485
T6M <5,6> 2680      T6MI <7,6> 970
T7M <5,7> 1265      T7MI <7,7> 1940
T8M <5,8> 2530      T8MI <7,8> 3880
T9M <5,9> 965      T9MI <7,9> 3665
T10M <5,10> 1930      T10MI <7,10> 3235
T11M <5,11> 3860      T11MI <7,11> 2375

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 653Scale 653: Dorian Pentatonic, Ian Ring Music TheoryDorian Pentatonic
Scale 651Scale 651: Golitonic, Ian Ring Music TheoryGolitonic
Scale 647Scale 647, Ian Ring Music Theory
Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic
Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian
Scale 687Scale 687: Aeolythian, Ian Ring Music TheoryAeolythian
Scale 719Scale 719: Kanian, Ian Ring Music TheoryKanian
Scale 527Scale 527, Ian Ring Music Theory
Scale 591Scale 591: Gaptimic, Ian Ring Music TheoryGaptimic
Scale 783Scale 783, Ian Ring Music Theory
Scale 911Scale 911: Radian, Ian Ring Music TheoryRadian
Scale 143Scale 143, Ian Ring Music Theory
Scale 399Scale 399: Zynimic, Ian Ring Music TheoryZynimic
Scale 1167Scale 1167: Aerodimic, Ian Ring Music TheoryAerodimic
Scale 1679Scale 1679: Kydian, Ian Ring Music TheoryKydian
Scale 2703Scale 2703: Galian, Ian Ring Music TheoryGalian

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.