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Scale 369: "Laditonic"

Scale 369: Laditonic, 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
Laditonic
Dozenal
Celian

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

{0,4,5,6,8}

Forte Number

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

5-13

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: 465

Hemitonia

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

2 (dihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

1 (uncohemitonic)

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.

4

Prime Form

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

no
prime: 279

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.

[4, 1, 1, 2, 4]

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.

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

Interval Spectrum

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

pm3ns2d2t

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> = {1,2,4}
<2> = {2,3,5,6,8}
<3> = {4,6,7,9,10}
<4> = {8,10,11}

Spectra Variation

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

3.6

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.

1.799

Polygon Perimeter

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

5.499

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.

(13, 7, 36)

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 Triadsfm{5,8,0}110.5
Augmented TriadsC+{0,4,8}110.5

The following pitch classes are not present in any of the common triads: {6}

Parsimonious Voice Leading Between Common Triads of Scale 369. Created by Ian Ring ©2019 C+ C+ fm fm C+->fm

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

2nd mode:
Scale 279
Scale 279: Poditonic, Ian Ring Music TheoryPoditonicThis is the prime mode
3rd mode:
Scale 2187
Scale 2187: Ionothitonic, Ian Ring Music TheoryIonothitonic
4th mode:
Scale 3141
Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
5th mode:
Scale 1809
Scale 1809: Ranitonic, Ian Ring Music TheoryRanitonic

Prime

The prime form of this scale is Scale 279

Scale 279Scale 279: Poditonic, Ian Ring Music TheoryPoditonic

Complement

The pentatonic modal family [369, 279, 2187, 3141, 1809] (Forte: 5-13) is the complement of the heptatonic modal family [375, 1815, 1905, 2235, 2955, 3165, 3525] (Forte: 7-13)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 369 is 465

Scale 465Scale 465: Zoditonic, Ian Ring Music TheoryZoditonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 369 is chiral, and its enantiomorph is scale 465

Scale 465Scale 465: Zoditonic, Ian Ring Music TheoryZoditonic

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

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 371Scale 371: Rythimic, Ian Ring Music TheoryRythimic
Scale 373Scale 373: Epagimic, Ian Ring Music TheoryEpagimic
Scale 377Scale 377: Kathimic, Ian Ring Music TheoryKathimic
Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian
Scale 361Scale 361: Bocritonic, Ian Ring Music TheoryBocritonic
Scale 337Scale 337: Koptic, Ian Ring Music TheoryKoptic
Scale 305Scale 305: Gonic, Ian Ring Music TheoryGonic
Scale 433Scale 433: Raga Zilaf, Ian Ring Music TheoryRaga Zilaf
Scale 497Scale 497: Kadimic, Ian Ring Music TheoryKadimic
Scale 113Scale 113, Ian Ring Music Theory
Scale 241Scale 241: Bilian, Ian Ring Music TheoryBilian
Scale 625Scale 625: Ionyptitonic, Ian Ring Music TheoryIonyptitonic
Scale 881Scale 881: Aerothimic, Ian Ring Music TheoryAerothimic
Scale 1393Scale 1393: Mycrimic, Ian Ring Music TheoryMycrimic
Scale 2417Scale 2417: Kanimic, Ian Ring Music TheoryKanimic

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.