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Scale 1109: "Kataditonic"

Scale 1109: Kataditonic, 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
Kataditonic
Dozenal
Gupian

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

{0,2,4,6,10}

Forte Number

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

5-33

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.

[2]

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.

0 (anhemitonic)

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.

5

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: 341

Generator

Indicates if the scale can be constructed using a generator, and an origin.

generator: 2
origin: 10

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.

[2, 2, 2, 4, 2]

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.

<0, 4, 0, 4, 0, 2>

Interval Spectrum

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

m4s4t2

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> = {2,4}
<2> = {4,6}
<3> = {6,8}
<4> = {8,10}

Spectra Variation

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

1.6

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.

2.165

Polygon Perimeter

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

5.732

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

yes

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.

[4]

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, 10, 20)

Generator

This scale has a generator of 2, originating on 10.

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
Augmented TriadsD+{2,6,10}000

The following pitch classes are not present in any of the common triads: {0,4}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 1109 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1301
Scale 1301: Koditonic, Ian Ring Music TheoryKoditonic
3rd mode:
Scale 1349
Scale 1349: Tholitonic, Ian Ring Music TheoryTholitonic
4th mode:
Scale 1361
Scale 1361: Bolitonic, Ian Ring Music TheoryBolitonic
5th mode:
Scale 341
Scale 341: Bothitonic, Ian Ring Music TheoryBothitonicThis is the prime mode

Prime

The prime form of this scale is Scale 341

Scale 341Scale 341: Bothitonic, Ian Ring Music TheoryBothitonic

Complement

The pentatonic modal family [1109, 1301, 1349, 1361, 341] (Forte: 5-33) is the complement of the heptatonic modal family [1367, 1373, 1397, 1493, 1877, 2731, 3413] (Forte: 7-33)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1109 is 1349

Scale 1349Scale 1349: Tholitonic, Ian Ring Music TheoryTholitonic

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> 1109       T0I <11,0> 1349
T1 <1,1> 2218      T1I <11,1> 2698
T2 <1,2> 341      T2I <11,2> 1301
T3 <1,3> 682      T3I <11,3> 2602
T4 <1,4> 1364      T4I <11,4> 1109
T5 <1,5> 2728      T5I <11,5> 2218
T6 <1,6> 1361      T6I <11,6> 341
T7 <1,7> 2722      T7I <11,7> 682
T8 <1,8> 1349      T8I <11,8> 1364
T9 <1,9> 2698      T9I <11,9> 2728
T10 <1,10> 1301      T10I <11,10> 1361
T11 <1,11> 2602      T11I <11,11> 2722
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1349      T0MI <7,0> 1109
T1M <5,1> 2698      T1MI <7,1> 2218
T2M <5,2> 1301      T2MI <7,2> 341
T3M <5,3> 2602      T3MI <7,3> 682
T4M <5,4> 1109       T4MI <7,4> 1364
T5M <5,5> 2218      T5MI <7,5> 2728
T6M <5,6> 341      T6MI <7,6> 1361
T7M <5,7> 682      T7MI <7,7> 2722
T8M <5,8> 1364      T8MI <7,8> 1349
T9M <5,9> 2728      T9MI <7,9> 2698
T10M <5,10> 1361      T10MI <7,10> 1301
T11M <5,11> 2722      T11MI <7,11> 2602

The transformations that map this set to itself are: T0, T4I, T4M, 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 1111Scale 1111: Sycrimic, Ian Ring Music TheorySycrimic
Scale 1105Scale 1105: Messiaen Truncated Mode 6 Inverse, Ian Ring Music TheoryMessiaen Truncated Mode 6 Inverse
Scale 1107Scale 1107: Mogitonic, Ian Ring Music TheoryMogitonic
Scale 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2
Scale 1117Scale 1117: Raptimic, Ian Ring Music TheoryRaptimic
Scale 1093Scale 1093: Lydic, Ian Ring Music TheoryLydic
Scale 1101Scale 1101: Stothitonic, Ian Ring Music TheoryStothitonic
Scale 1125Scale 1125: Ionaritonic, Ian Ring Music TheoryIonaritonic
Scale 1141Scale 1141: Rynimic, Ian Ring Music TheoryRynimic
Scale 1045Scale 1045: Gibian, Ian Ring Music TheoryGibian
Scale 1077Scale 1077: Govian, Ian Ring Music TheoryGovian
Scale 1173Scale 1173: Dominant Pentatonic, Ian Ring Music TheoryDominant Pentatonic
Scale 1237Scale 1237: Salimic, Ian Ring Music TheorySalimic
Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone
Scale 1621Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus
Scale 85Scale 85: Segian, Ian Ring Music TheorySegian
Scale 597Scale 597: Kung, Ian Ring Music TheoryKung
Scale 2133Scale 2133: Raga Kumurdaki, Ian Ring Music TheoryRaga Kumurdaki
Scale 3157Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic

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.