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Scale 607: "Kadian"

Scale 607: Kadian, 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
Kadian
Dozenal
Dobian

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

{0,1,2,3,4,6,9}

Forte Number

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

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: 3913

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.

yes

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

<4, 4, 5, 3, 3, 2>

Interval Spectrum

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

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

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.

(46, 40, 104)

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{2,6,9}231.75
A{9,1,4}231.75
Minor Triadsf♯m{6,9,1}331.63
am{9,0,4}331.63
Diminished Triads{0,3,6}231.88
d♯°{3,6,9}231.88
f♯°{6,9,0}231.75
{9,0,3}231.75
Parsimonious Voice Leading Between Common Triads of Scale 607. Created by Ian Ring ©2019 d#° d#° c°->d#° c°->a° D D D->d#° f#m f#m D->f#m f#° f#° f#°->f#m am am f#°->am A A f#m->A a°->am am->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 607 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2351
Scale 2351: Gynian, Ian Ring Music TheoryGynian
3rd mode:
Scale 3223
Scale 3223: Thyphian, Ian Ring Music TheoryThyphian
4th mode:
Scale 3659
Scale 3659: Polian, Ian Ring Music TheoryPolian
5th mode:
Scale 3877
Scale 3877: Thanian, Ian Ring Music TheoryThanian
6th mode:
Scale 1993
Scale 1993: Katoptian, Ian Ring Music TheoryKatoptian
7th mode:
Scale 761
Scale 761: Ponian, Ian Ring Music TheoryPonian

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [607, 2351, 3223, 3659, 3877, 1993, 761] (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 607 is 3913

Scale 3913Scale 3913: Bonian, Ian Ring Music TheoryBonian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 607 is chiral, and its enantiomorph is scale 3913

Scale 3913Scale 3913: Bonian, Ian Ring Music TheoryBonian

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> 607       T0I <11,0> 3913
T1 <1,1> 1214      T1I <11,1> 3731
T2 <1,2> 2428      T2I <11,2> 3367
T3 <1,3> 761      T3I <11,3> 2639
T4 <1,4> 1522      T4I <11,4> 1183
T5 <1,5> 3044      T5I <11,5> 2366
T6 <1,6> 1993      T6I <11,6> 637
T7 <1,7> 3986      T7I <11,7> 1274
T8 <1,8> 3877      T8I <11,8> 2548
T9 <1,9> 3659      T9I <11,9> 1001
T10 <1,10> 3223      T10I <11,10> 2002
T11 <1,11> 2351      T11I <11,11> 4004
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1897      T0MI <7,0> 733
T1M <5,1> 3794      T1MI <7,1> 1466
T2M <5,2> 3493      T2MI <7,2> 2932
T3M <5,3> 2891      T3MI <7,3> 1769
T4M <5,4> 1687      T4MI <7,4> 3538
T5M <5,5> 3374      T5MI <7,5> 2981
T6M <5,6> 2653      T6MI <7,6> 1867
T7M <5,7> 1211      T7MI <7,7> 3734
T8M <5,8> 2422      T8MI <7,8> 3373
T9M <5,9> 749      T9MI <7,9> 2651
T10M <5,10> 1498      T10MI <7,10> 1207
T11M <5,11> 2996      T11MI <7,11> 2414

The transformations that map this set to itself are: T0

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 605Scale 605: Dycrimic, Ian Ring Music TheoryDycrimic
Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic
Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic
Scale 591Scale 591: Gaptimic, Ian Ring Music TheoryGaptimic
Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian
Scale 639Scale 639: Ionaryllic, Ian Ring Music TheoryIonaryllic
Scale 543Scale 543: Denian, Ian Ring Music TheoryDenian
Scale 575Scale 575: Ionydian, Ian Ring Music TheoryIonydian
Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian
Scale 735Scale 735: Sylyllic, Ian Ring Music TheorySylyllic
Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic
Scale 95Scale 95: Arkian, Ian Ring Music TheoryArkian
Scale 351Scale 351: Epanian, Ian Ring Music TheoryEpanian
Scale 1119Scale 1119: Rarian, Ian Ring Music TheoryRarian
Scale 1631Scale 1631: Rynyllic, Ian Ring Music TheoryRynyllic
Scale 2655Scale 2655: Qojian, Ian Ring Music TheoryQojian

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.