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Scale 2187: "Ionothitonic"

Scale 2187: Ionothitonic, 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
Ionothitonic
Dozenal
Neqian

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

{0,1,3,7,11}

Forte Number

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

5-13

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: 2595

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.

[1, 2, 4, 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.

<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 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: {1}

Parsimonious Voice Leading Between Common Triads of Scale 2187. 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 2187 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3141
Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
3rd mode:
Scale 1809
Scale 1809: Ranitonic, Ian Ring Music TheoryRanitonic
4th mode:
Scale 369
Scale 369: Laditonic, Ian Ring Music TheoryLaditonic
5th mode:
Scale 279
Scale 279: Poditonic, Ian Ring Music TheoryPoditonicThis is the prime mode

Prime

The prime form of this scale is Scale 279

Scale 279Scale 279: Poditonic, Ian Ring Music TheoryPoditonic

Complement

The pentatonic modal family [2187, 3141, 1809, 369, 279] (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 2187 is 2595

Scale 2595Scale 2595: Rolitonic, Ian Ring Music TheoryRolitonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2187 is chiral, and its enantiomorph is scale 2595

Scale 2595Scale 2595: Rolitonic, Ian Ring Music TheoryRolitonic

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

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 2185Scale 2185: Dygic, Ian Ring Music TheoryDygic
Scale 2189Scale 2189: Zagitonic, Ian Ring Music TheoryZagitonic
Scale 2191Scale 2191: Thydimic, Ian Ring Music TheoryThydimic
Scale 2179Scale 2179, Ian Ring Music Theory
Scale 2183Scale 2183: Nenian, Ian Ring Music TheoryNenian
Scale 2195Scale 2195: Zalitonic, Ian Ring Music TheoryZalitonic
Scale 2203Scale 2203: Dorimic, Ian Ring Music TheoryDorimic
Scale 2219Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic
Scale 2251Scale 2251: Zodimic, Ian Ring Music TheoryZodimic
Scale 2059Scale 2059: Moqian, Ian Ring Music TheoryMoqian
Scale 2123Scale 2123: Nacian, Ian Ring Music TheoryNacian
Scale 2315Scale 2315: Orkian, Ian Ring Music TheoryOrkian
Scale 2443Scale 2443: Panimic, Ian Ring Music TheoryPanimic
Scale 2699Scale 2699: Sythimic, Ian Ring Music TheorySythimic
Scale 3211Scale 3211: Epacrimic, Ian Ring Music TheoryEpacrimic
Scale 139Scale 139: Ayoian, Ian Ring Music TheoryAyoian
Scale 1163Scale 1163: Raga Rukmangi, Ian Ring Music TheoryRaga Rukmangi

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.