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Scale 923: "Ultraphrygian"

Scale 923: Ultraphrygian, 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 Modern
Ultraphrygian
Dozenal
Fofian
Zeitler
Ionodian

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

{0,1,3,4,7,8,9}

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.

[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.

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 Rahn/Ring formula.

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.

no

Interval Structure

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

[1, 2, 1, 3, 1, 1, 3]

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>

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 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.

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.

[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".

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)

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.

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 TriadsC{0,4,7}331.67
G♯{8,0,3}331.67
A{9,1,4}242
Minor Triadscm{0,3,7}242
c♯m{1,4,8}331.67
am{9,0,4}331.67
Augmented TriadsC+{0,4,8}421.33
Diminished Triadsc♯°{1,4,7}242
{9,0,3}242
Parsimonious Voice Leading Between Common Triads of Scale 923. Created by Ian Ring ©2019 cm cm C C cm->C G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° c#m c#m C+->c#m C+->G# am am C+->am c#°->c#m A A c#m->A G#->a° a°->am am->A

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 Verticescm, c♯°, a°, A

Modes

Modes are the rotational transformation of this scale. Scale 923 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2509
Scale 2509: Double Harmonic Minor, Ian Ring Music TheoryDouble Harmonic Minor
3rd mode:
Scale 1651
Scale 1651: Asian, Ian Ring Music TheoryAsian
4th mode:
Scale 2873
Scale 2873: Ionian Augmented Sharp 2, Ian Ring Music TheoryIonian Augmented Sharp 2
5th mode:
Scale 871
Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7This is the prime mode
6th mode:
Scale 2483
Scale 2483: Double Harmonic, Ian Ring Music TheoryDouble Harmonic
7th mode:
Scale 3289
Scale 3289: Lydian Sharp 2 Sharp 6, Ian Ring Music TheoryLydian Sharp 2 Sharp 6

Prime

The prime form of this scale is Scale 871

Scale 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7

Complement

The heptatonic modal family [923, 2509, 1651, 2873, 871, 2483, 3289] (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 923 is 2873

Scale 2873Scale 2873: Ionian Augmented Sharp 2, Ian Ring Music TheoryIonian Augmented Sharp 2

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

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 921Scale 921: Bogimic, Ian Ring Music TheoryBogimic
Scale 925Scale 925: Chromatic Hypodorian, Ian Ring Music TheoryChromatic Hypodorian
Scale 927Scale 927: Gaptyllic, Ian Ring Music TheoryGaptyllic
Scale 915Scale 915: Raga Kalagada, Ian Ring Music TheoryRaga Kalagada
Scale 919Scale 919: Chromatic Phrygian Inverse, Ian Ring Music TheoryChromatic Phrygian Inverse
Scale 907Scale 907: Tholimic, Ian Ring Music TheoryTholimic
Scale 939Scale 939: Mela Senavati, Ian Ring Music TheoryMela Senavati
Scale 955Scale 955: Ionogyllic, Ian Ring Music TheoryIonogyllic
Scale 987Scale 987: Aeraptyllic, Ian Ring Music TheoryAeraptyllic
Scale 795Scale 795: Aeologimic, Ian Ring Music TheoryAeologimic
Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic
Scale 411Scale 411: Lygimic, Ian Ring Music TheoryLygimic
Scale 1435Scale 1435: Makam Huzzam, Ian Ring Music TheoryMakam Huzzam
Scale 1947Scale 1947: Byptyllic, Ian Ring Music TheoryByptyllic
Scale 2971Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic

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.