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Scale 3667: "Kaptian"

Scale 3667: Kaptian, 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
Kaptian
Dozenal
Wuzian

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

{0,1,4,6,9,10,11}

Forte Number

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

7-Z12

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.

[5]

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.

no

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.

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.

6

Prime Form

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

no
prime: 671

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.

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

<4, 4, 4, 3, 4, 2>

Interval Spectrum

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

p4m3n4s4d4t2

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

Spectra Variation

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

2.857

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.

[10]

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.

(41, 30, 96)

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♯{6,10,1}231.5
A{9,1,4}321.17
Minor Triadsf♯m{6,9,1}321.17
am{9,0,4}231.5
Diminished Triadsf♯°{6,9,0}231.5
a♯°{10,1,4}231.5

The following pitch classes are not present in any of the common triads: {11}

Parsimonious Voice Leading Between Common Triads of Scale 3667. Created by Ian Ring ©2019 f#° f#° f#m f#m f#°->f#m am am f#°->am F# F# f#m->F# A A f#m->A a#° a#° F#->a#° am->A A->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.

Diameter3
Radius2
Self-Centeredno
Central Verticesf♯m, A
Peripheral Verticesf♯°, F♯, am, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 3667 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3881
Scale 3881: Morian, Ian Ring Music TheoryMorian
3rd mode:
Scale 997
Scale 997: Rycrian, Ian Ring Music TheoryRycrian
4th mode:
Scale 1273
Scale 1273: Ronian, Ian Ring Music TheoryRonian
5th mode:
Scale 671
Scale 671: Stycrian, Ian Ring Music TheoryStycrianThis is the prime mode
6th mode:
Scale 2383
Scale 2383: Katorian, Ian Ring Music TheoryKatorian
7th mode:
Scale 3239
Scale 3239: Mela Tanarupi, Ian Ring Music TheoryMela Tanarupi

Prime

The prime form of this scale is Scale 671

Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian

Complement

The heptatonic modal family [3667, 3881, 997, 1273, 671, 2383, 3239] (Forte: 7-Z12) is the complement of the pentatonic modal family [107, 1411, 1549, 2101, 2753] (Forte: 5-Z12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3667 is 2383

Scale 2383Scale 2383: Katorian, Ian Ring Music TheoryKatorian

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> 3667       T0I <11,0> 2383
T1 <1,1> 3239      T1I <11,1> 671
T2 <1,2> 2383      T2I <11,2> 1342
T3 <1,3> 671      T3I <11,3> 2684
T4 <1,4> 1342      T4I <11,4> 1273
T5 <1,5> 2684      T5I <11,5> 2546
T6 <1,6> 1273      T6I <11,6> 997
T7 <1,7> 2546      T7I <11,7> 1994
T8 <1,8> 997      T8I <11,8> 3988
T9 <1,9> 1994      T9I <11,9> 3881
T10 <1,10> 3988      T10I <11,10> 3667
T11 <1,11> 3881      T11I <11,11> 3239
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 997      T0MI <7,0> 1273
T1M <5,1> 1994      T1MI <7,1> 2546
T2M <5,2> 3988      T2MI <7,2> 997
T3M <5,3> 3881      T3MI <7,3> 1994
T4M <5,4> 3667       T4MI <7,4> 3988
T5M <5,5> 3239      T5MI <7,5> 3881
T6M <5,6> 2383      T6MI <7,6> 3667
T7M <5,7> 671      T7MI <7,7> 3239
T8M <5,8> 1342      T8MI <7,8> 2383
T9M <5,9> 2684      T9MI <7,9> 671
T10M <5,10> 1273      T10MI <7,10> 1342
T11M <5,11> 2546      T11MI <7,11> 2684

The transformations that map this set to itself are: T0, T10I, T4M, T6MI

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 3665Scale 3665: Stalimic, Ian Ring Music TheoryStalimic
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 3675Scale 3675: Monyllic, Ian Ring Music TheoryMonyllic
Scale 3651Scale 3651: Wuqian, Ian Ring Music TheoryWuqian
Scale 3659Scale 3659: Polian, Ian Ring Music TheoryPolian
Scale 3683Scale 3683: Dycrian, Ian Ring Music TheoryDycrian
Scale 3699Scale 3699: Galyllic, Ian Ring Music TheoryGalyllic
Scale 3603Scale 3603: Womian, Ian Ring Music TheoryWomian
Scale 3635Scale 3635: Katygian, Ian Ring Music TheoryKatygian
Scale 3731Scale 3731: Aeryrian, Ian Ring Music TheoryAeryrian
Scale 3795Scale 3795: Epothyllic, Ian Ring Music TheoryEpothyllic
Scale 3923Scale 3923: Stoptyllic, Ian Ring Music TheoryStoptyllic
Scale 3155Scale 3155: Ladimic, Ian Ring Music TheoryLadimic
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 2643Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi
Scale 1619Scale 1619: Prometheus Neapolitan, Ian Ring Music TheoryPrometheus Neapolitan

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.