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Scale 2217: "Kagitonic"

Scale 2217: Kagitonic, 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
Kagitonic
Dozenal
Nijian

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,3,5,7,11}

Forte Number

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

5-30

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: 675

Hemitonia

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

1 (unhemitonic)

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.

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.

4

Prime Form

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

no
prime: 339

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, 2, 2, 4, 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.

<1, 2, 1, 3, 2, 1>

Interval Spectrum

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

p2m3ns2dt

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

Spectra Variation

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

2

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.

2.049

Polygon Perimeter

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

5.664

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".

Proper

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.

(0, 5, 34)

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
Augmented TriadsD♯+{3,7,11}110.5

The following pitch classes are not present in any of the common triads: {5}

Parsimonious Voice Leading Between Common Triads of Scale 2217. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+

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 2217 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 789
Scale 789: Zogitonic, Ian Ring Music TheoryZogitonic
3rd mode:
Scale 1221
Scale 1221: Epyritonic, Ian Ring Music TheoryEpyritonic
4th mode:
Scale 1329
Scale 1329: Epygitonic, Ian Ring Music TheoryEpygitonic
5th mode:
Scale 339
Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonicThis is the prime mode

Prime

The prime form of this scale is Scale 339

Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic

Complement

The pentatonic modal family [2217, 789, 1221, 1329, 339] (Forte: 5-30) is the complement of the heptatonic modal family [855, 1395, 1485, 1845, 2475, 2745, 3285] (Forte: 7-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2217 is 675

Scale 675Scale 675: Altered Pentatonic, Ian Ring Music TheoryAltered Pentatonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2217 is chiral, and its enantiomorph is scale 675

Scale 675Scale 675: Altered Pentatonic, Ian Ring Music TheoryAltered Pentatonic

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> 2217       T0I <11,0> 675
T1 <1,1> 339      T1I <11,1> 1350
T2 <1,2> 678      T2I <11,2> 2700
T3 <1,3> 1356      T3I <11,3> 1305
T4 <1,4> 2712      T4I <11,4> 2610
T5 <1,5> 1329      T5I <11,5> 1125
T6 <1,6> 2658      T6I <11,6> 2250
T7 <1,7> 1221      T7I <11,7> 405
T8 <1,8> 2442      T8I <11,8> 810
T9 <1,9> 789      T9I <11,9> 1620
T10 <1,10> 1578      T10I <11,10> 3240
T11 <1,11> 3156      T11I <11,11> 2385
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2187      T0MI <7,0> 2595
T1M <5,1> 279      T1MI <7,1> 1095
T2M <5,2> 558      T2MI <7,2> 2190
T3M <5,3> 1116      T3MI <7,3> 285
T4M <5,4> 2232      T4MI <7,4> 570
T5M <5,5> 369      T5MI <7,5> 1140
T6M <5,6> 738      T6MI <7,6> 2280
T7M <5,7> 1476      T7MI <7,7> 465
T8M <5,8> 2952      T8MI <7,8> 930
T9M <5,9> 1809      T9MI <7,9> 1860
T10M <5,10> 3618      T10MI <7,10> 3720
T11M <5,11> 3141      T11MI <7,11> 3345

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 2219Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic
Scale 2221Scale 2221: Raga Sindhura Kafi, Ian Ring Music TheoryRaga Sindhura Kafi
Scale 2209Scale 2209: Nidian, Ian Ring Music TheoryNidian
Scale 2213Scale 2213: Raga Desh, Ian Ring Music TheoryRaga Desh
Scale 2225Scale 2225: Ionian Pentatonic, Ian Ring Music TheoryIonian Pentatonic
Scale 2233Scale 2233: Donimic, Ian Ring Music TheoryDonimic
Scale 2185Scale 2185: Dygic, Ian Ring Music TheoryDygic
Scale 2201Scale 2201: Ionagitonic, Ian Ring Music TheoryIonagitonic
Scale 2249Scale 2249: Raga Multani, Ian Ring Music TheoryRaga Multani
Scale 2281Scale 2281: Rathimic, Ian Ring Music TheoryRathimic
Scale 2089Scale 2089: Mujian, Ian Ring Music TheoryMujian
Scale 2153Scale 2153: Navian, Ian Ring Music TheoryNavian
Scale 2345Scale 2345: Raga Chandrakauns, Ian Ring Music TheoryRaga Chandrakauns
Scale 2473Scale 2473: Raga Takka, Ian Ring Music TheoryRaga Takka
Scale 2729Scale 2729: Aeragimic, Ian Ring Music TheoryAeragimic
Scale 3241Scale 3241: Dalimic, Ian Ring Music TheoryDalimic
Scale 169Scale 169: Vietnamese Tetratonic, Ian Ring Music TheoryVietnamese Tetratonic
Scale 1193Scale 1193: Minor Pentatonic, Ian Ring Music TheoryMinor Pentatonic

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.