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Scale 743: "Chromatic Hypophrygian Inverse"

Scale 743: Chromatic Hypophrygian Inverse, 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 Chromatic
Chromatic Hypophrygian Inverse
Dozenal
Enuian
Zeitler
Lanian

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

{0,1,2,5,6,7,9}

Forte Number

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

7-20

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: 3305

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.

2 (dicohemitonic)

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.

2

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.

yes

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, 1, 3, 1, 1, 2, 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, 3, 3, 4, 5, 2>

Interval Spectrum

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

p5m4n3s3d4t2

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

Spectra Variation

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

2

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.

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.

(10, 26, 90)

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 TriadsD{2,6,9}231.5
F{5,9,0}231.5
Minor Triadsdm{2,5,9}231.5
f♯m{6,9,1}321.17
Augmented TriadsC♯+{1,5,9}321.17
Diminished Triadsf♯°{6,9,0}231.5

The following pitch classes are not present in any of the common triads: {7}

Parsimonious Voice Leading Between Common Triads of Scale 743. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m D D dm->D D->f#m f#° f#° F->f#° f#°->f#m

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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC♯+, f♯m
Peripheral Verticesdm, D, F, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 743 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2419
Scale 2419: Raga Lalita, Ian Ring Music TheoryRaga Lalita
3rd mode:
Scale 3257
Scale 3257: Mela Calanata, Ian Ring Music TheoryMela Calanata
4th mode:
Scale 919
Scale 919: Chromatic Phrygian Inverse, Ian Ring Music TheoryChromatic Phrygian Inverse
5th mode:
Scale 2507
Scale 2507: Todi That, Ian Ring Music TheoryTodi That
6th mode:
Scale 3301
Scale 3301: Chromatic Mixolydian Inverse, Ian Ring Music TheoryChromatic Mixolydian Inverse
7th mode:
Scale 1849
Scale 1849: Chromatic Hypodorian Inverse, Ian Ring Music TheoryChromatic Hypodorian Inverse

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [743, 2419, 3257, 919, 2507, 3301, 1849] (Forte: 7-20) is the complement of the pentatonic modal family [355, 395, 1585, 2225, 2245] (Forte: 5-20)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 743 is 3305

Scale 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 743 is chiral, and its enantiomorph is scale 3305

Scale 3305Scale 3305: Chromatic Hypophrygian, Ian Ring Music TheoryChromatic Hypophrygian

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> 743       T0I <11,0> 3305
T1 <1,1> 1486      T1I <11,1> 2515
T2 <1,2> 2972      T2I <11,2> 935
T3 <1,3> 1849      T3I <11,3> 1870
T4 <1,4> 3698      T4I <11,4> 3740
T5 <1,5> 3301      T5I <11,5> 3385
T6 <1,6> 2507      T6I <11,6> 2675
T7 <1,7> 919      T7I <11,7> 1255
T8 <1,8> 1838      T8I <11,8> 2510
T9 <1,9> 3676      T9I <11,9> 925
T10 <1,10> 3257      T10I <11,10> 1850
T11 <1,11> 2419      T11I <11,11> 3700
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3683      T0MI <7,0> 2255
T1M <5,1> 3271      T1MI <7,1> 415
T2M <5,2> 2447      T2MI <7,2> 830
T3M <5,3> 799      T3MI <7,3> 1660
T4M <5,4> 1598      T4MI <7,4> 3320
T5M <5,5> 3196      T5MI <7,5> 2545
T6M <5,6> 2297      T6MI <7,6> 995
T7M <5,7> 499      T7MI <7,7> 1990
T8M <5,8> 998      T8MI <7,8> 3980
T9M <5,9> 1996      T9MI <7,9> 3865
T10M <5,10> 3992      T10MI <7,10> 3635
T11M <5,11> 3889      T11MI <7,11> 3175

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 741Scale 741: Gathimic, Ian Ring Music TheoryGathimic
Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic
Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian
Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian
Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic
Scale 711Scale 711: Raga Chandrajyoti, Ian Ring Music TheoryRaga Chandrajyoti
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 615Scale 615: Schoenberg Hexachord, Ian Ring Music TheorySchoenberg Hexachord
Scale 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7
Scale 999Scale 999: Ionodyllic, Ian Ring Music TheoryIonodyllic
Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian
Scale 487Scale 487: Dynian, Ian Ring Music TheoryDynian
Scale 1255Scale 1255: Chromatic Mixolydian, Ian Ring Music TheoryChromatic Mixolydian
Scale 1767Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic
Scale 2791Scale 2791: Mixothyllic, Ian Ring Music TheoryMixothyllic

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.