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Scale 2873: "Ionian Augmented Sharp 2"

Scale 2873: Ionian Augmented Sharp 2, 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

Western Altered
Ionian Augmented Sharp 2
Western
Hungarian Romani Minor 3rd Mode
Zeitler
Docrian
Dozenal
SAPian

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

{0,3,4,5,8,9,11}

Forte Number

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

7-22

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.

[4]

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.

4 (multihemitonic)

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.

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.

6

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 871

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, an indicator of maximum hierarchization.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

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

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

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.5, 0, 0.5, 0.667, 0.5, 0.5>

Interval Spectrum

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

p4m5n4s2d4t2

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}
<3> = {4,5,6}
<4> = {6,7,8}
<5> = {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.

1.714

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 Distribution Spectra have 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.

yes

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.

[8]

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.

(4, 22, 84)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.814

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.333

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

Generator

This scale has no generator.

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 TriadsE{4,8,11}331.67
F{5,9,0}242
G♯{8,0,3}331.67
Minor Triadsfm{5,8,0}331.67
g♯m{8,11,3}242
am{9,0,4}331.67
Augmented TriadsC+{0,4,8}421.33
Diminished Triads{5,8,11}242
{9,0,3}242
Parsimonious Voice Leading Between Common Triads of Scale 2873. Created by Ian Ring ©2019 C+ C+ E E C+->E fm fm C+->fm G# G# C+->G# am am C+->am E->f° g#m g#m E->g#m f°->fm F F fm->F F->am g#m->G# G#->a° a°->am

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesC+
Peripheral Verticesf°, F, g♯m, a°

Modes

Modes are the rotational transformation of this scale. Scale 2873 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 871
Scale 871: Hungarian Romani Minor 4th Mode, Ian Ring Music TheoryHungarian Romani Minor 4th ModeThis is the prime mode
3rd mode:
Scale 2483
Scale 2483: Double Harmonic, Ian Ring Music TheoryDouble Harmonic
4th mode:
Scale 3289
Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6
5th mode:
Scale 923
Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian
6th mode:
Scale 2509
Scale 2509: Double Harmonic Minor, Ian Ring Music TheoryDouble Harmonic Minor
7th mode:
Scale 1651
Scale 1651: Asian, Ian Ring Music TheoryAsian

Prime

The prime form of this scale is Scale 871

Scale 871Scale 871: Hungarian Romani Minor 4th Mode, Ian Ring Music TheoryHungarian Romani Minor 4th Mode

Complement

The heptatonic modal family [2873, 871, 2483, 3289, 923, 2509, 1651] (Forte: 7-22) is the complement of the pentatonic modal family [403, 611, 793, 2249, 2353] (Forte: 5-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2873 is 923

Scale 923Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian

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> 2873       T0I <11,0> 923
T1 <1,1> 1651      T1I <11,1> 1846
T2 <1,2> 3302      T2I <11,2> 3692
T3 <1,3> 2509      T3I <11,3> 3289
T4 <1,4> 923      T4I <11,4> 2483
T5 <1,5> 1846      T5I <11,5> 871
T6 <1,6> 3692      T6I <11,6> 1742
T7 <1,7> 3289      T7I <11,7> 3484
T8 <1,8> 2483      T8I <11,8> 2873
T9 <1,9> 871      T9I <11,9> 1651
T10 <1,10> 1742      T10I <11,10> 3302
T11 <1,11> 3484      T11I <11,11> 2509
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 923      T0MI <7,0> 2873
T1M <5,1> 1846      T1MI <7,1> 1651
T2M <5,2> 3692      T2MI <7,2> 3302
T3M <5,3> 3289      T3MI <7,3> 2509
T4M <5,4> 2483      T4MI <7,4> 923
T5M <5,5> 871      T5MI <7,5> 1846
T6M <5,6> 1742      T6MI <7,6> 3692
T7M <5,7> 3484      T7MI <7,7> 3289
T8M <5,8> 2873       T8MI <7,8> 2483
T9M <5,9> 1651      T9MI <7,9> 871
T10M <5,10> 3302      T10MI <7,10> 1742
T11M <5,11> 2509      T11MI <7,11> 3484

The transformations that map this set to itself are: T0, T8I, T8M, 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 2875Scale 2875: Ganyllic, Ian Ring Music TheoryGanyllic
Scale 2877Scale 2877: Phrylyllic, Ian Ring Music TheoryPhrylyllic
Scale 2865Scale 2865: Solimic, Ian Ring Music TheorySolimic
Scale 2869Scale 2869: Major Augmented, Ian Ring Music TheoryMajor Augmented
Scale 2857Scale 2857: Stythimic, Ian Ring Music TheoryStythimic
Scale 2841Scale 2841: African Pentatonic 3, Ian Ring Music TheoryAfrican Pentatonic 3
Scale 2905Scale 2905: Lydian Augmented Sharp 2, Ian Ring Music TheoryLydian Augmented Sharp 2
Scale 2937Scale 2937: Aeolathyllic, Ian Ring Music TheoryAeolathyllic
Scale 3001Scale 3001: Lonyllic, Ian Ring Music TheoryLonyllic
Scale 2617Scale 2617: Pylimic, Ian Ring Music TheoryPylimic
Scale 2745Scale 2745: Mela Sulini, Ian Ring Music TheoryMela Sulini
Scale 2361Scale 2361: Docrimic, Ian Ring Music TheoryDocrimic
Scale 3385Scale 3385: Chromatic Phrygian, Ian Ring Music TheoryChromatic Phrygian
Scale 3897Scale 3897: Locryllic, Ian Ring Music TheoryLocryllic
Scale 825Scale 825: Thyptimic, Ian Ring Music TheoryThyptimic
Scale 1849Scale 1849: Chromatic Hypodorian Inverse, Ian Ring Music TheoryChromatic Hypodorian Inverse

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.