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Scale 561: "Phratic"

Scale 561: Phratic, 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
Phratic
Dozenal
Diyian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,4,5,9}

Forte Number

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

4-20

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.

[4.5]

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.

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

3

Prime Form

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

no
prime: 291

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, 4, 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.

<1, 0, 1, 2, 2, 0>

Interval Spectrum

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

p2m2nd

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

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.

1.616

Polygon Perimeter

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

5.396

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.

[9]

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".

Strictly Proper

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.

(0, 0, 14)

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 TriadsF{5,9,0}110.5
Minor Triadsam{9,0,4}110.5
Parsimonious Voice Leading Between Common Triads of Scale 561. Created by Ian Ring ©2019 F F am am F->am

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 561 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 291
Scale 291: Raga Lavangi, Ian Ring Music TheoryRaga LavangiThis is the prime mode
3rd mode:
Scale 2193
Scale 2193: Major Seventh, Ian Ring Music TheoryMajor Seventh
4th mode:
Scale 393
Scale 393: Lothic, Ian Ring Music TheoryLothic

Prime

The prime form of this scale is Scale 291

Scale 291Scale 291: Raga Lavangi, Ian Ring Music TheoryRaga Lavangi

Complement

The tetratonic modal family [561, 291, 2193, 393] (Forte: 4-20) is the complement of the octatonic modal family [951, 1767, 1851, 2523, 2931, 2973, 3309, 3513] (Forte: 8-20)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 561 is 393

Scale 393Scale 393: Lothic, Ian Ring Music TheoryLothic

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> 561       T0I <11,0> 393
T1 <1,1> 1122      T1I <11,1> 786
T2 <1,2> 2244      T2I <11,2> 1572
T3 <1,3> 393      T3I <11,3> 3144
T4 <1,4> 786      T4I <11,4> 2193
T5 <1,5> 1572      T5I <11,5> 291
T6 <1,6> 3144      T6I <11,6> 582
T7 <1,7> 2193      T7I <11,7> 1164
T8 <1,8> 291      T8I <11,8> 2328
T9 <1,9> 582      T9I <11,9> 561
T10 <1,10> 1164      T10I <11,10> 1122
T11 <1,11> 2328      T11I <11,11> 2244
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 771      T0MI <7,0> 2073
T1M <5,1> 1542      T1MI <7,1> 51
T2M <5,2> 3084      T2MI <7,2> 102
T3M <5,3> 2073      T3MI <7,3> 204
T4M <5,4> 51      T4MI <7,4> 408
T5M <5,5> 102      T5MI <7,5> 816
T6M <5,6> 204      T6MI <7,6> 1632
T7M <5,7> 408      T7MI <7,7> 3264
T8M <5,8> 816      T8MI <7,8> 2433
T9M <5,9> 1632      T9MI <7,9> 771
T10M <5,10> 3264      T10MI <7,10> 1542
T11M <5,11> 2433      T11MI <7,11> 3084

The transformations that map this set to itself are: T0, T9I

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 563Scale 563: Thacritonic, Ian Ring Music TheoryThacritonic
Scale 565Scale 565: Aeolyphritonic, Ian Ring Music TheoryAeolyphritonic
Scale 569Scale 569: Mothitonic, Ian Ring Music TheoryMothitonic
Scale 545Scale 545: Dewian, Ian Ring Music TheoryDewian
Scale 553Scale 553: Rothic 2, Ian Ring Music TheoryRothic 2
Scale 529Scale 529: Raga Bilwadala, Ian Ring Music TheoryRaga Bilwadala
Scale 593Scale 593: Saric, Ian Ring Music TheorySaric
Scale 625Scale 625: Ionyptitonic, Ian Ring Music TheoryIonyptitonic
Scale 689Scale 689: Raga Nagasvaravali, Ian Ring Music TheoryRaga Nagasvaravali
Scale 817Scale 817: Zothitonic, Ian Ring Music TheoryZothitonic
Scale 49Scale 49: Aguian, Ian Ring Music TheoryAguian
Scale 305Scale 305: Gonic, Ian Ring Music TheoryGonic
Scale 1073Scale 1073: Gosian, Ian Ring Music TheoryGosian
Scale 1585Scale 1585: Raga Khamaji Durga, Ian Ring Music TheoryRaga Khamaji Durga
Scale 2609Scale 2609: Raga Bhinna Shadja, Ian Ring Music TheoryRaga Bhinna Shadja

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.