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Scale 1817: "Phrythimic"

Scale 1817: Phrythimic, 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
Phrythimic
Dozenal
Lexian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,3,4,8,9,10}

Forte Number

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

6-Z17

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: 797

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

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.

5

Prime Form

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

no
prime: 407

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.

[3, 1, 4, 1, 1, 2]

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.

<3, 2, 2, 3, 3, 2>

Interval Spectrum

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

p3m3n2s2d3t2

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> = {1,2,3,4}
<2> = {2,3,4,5}
<3> = {4,6,8}
<4> = {7,8,9,10}
<5> = {8,9,10,11}

Spectra Variation

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

2.667

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.

2.116

Polygon Perimeter

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

5.699

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.

(12, 10, 57)

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 TriadsG♯{8,0,3}221
Minor Triadsam{9,0,4}221
Augmented TriadsC+{0,4,8}221
Diminished Triads{9,0,3}221

The following pitch classes are not present in any of the common triads: {10}

Parsimonious Voice Leading Between Common Triads of Scale 1817. Created by Ian Ring ©2019 C+ C+ G# G# C+->G# am am C+->am G#->a° a°->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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1817 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 739
Scale 739: Rorimic, Ian Ring Music TheoryRorimic
3rd mode:
Scale 2417
Scale 2417: Kanimic, Ian Ring Music TheoryKanimic
4th mode:
Scale 407
Scale 407: All-Trichord Hexachord, Ian Ring Music TheoryAll-Trichord HexachordThis is the prime mode
5th mode:
Scale 2251
Scale 2251: Zodimic, Ian Ring Music TheoryZodimic
6th mode:
Scale 3173
Scale 3173: Zarimic, Ian Ring Music TheoryZarimic

Prime

The prime form of this scale is Scale 407

Scale 407Scale 407: All-Trichord Hexachord, Ian Ring Music TheoryAll-Trichord Hexachord

Complement

The hexatonic modal family [1817, 739, 2417, 407, 2251, 3173] (Forte: 6-Z17) is the complement of the hexatonic modal family [359, 907, 1649, 2227, 2501, 3161] (Forte: 6-Z43)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1817 is 797

Scale 797Scale 797: Katocrimic, Ian Ring Music TheoryKatocrimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1817 is chiral, and its enantiomorph is scale 797

Scale 797Scale 797: Katocrimic, Ian Ring Music TheoryKatocrimic

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> 1817       T0I <11,0> 797
T1 <1,1> 3634      T1I <11,1> 1594
T2 <1,2> 3173      T2I <11,2> 3188
T3 <1,3> 2251      T3I <11,3> 2281
T4 <1,4> 407      T4I <11,4> 467
T5 <1,5> 814      T5I <11,5> 934
T6 <1,6> 1628      T6I <11,6> 1868
T7 <1,7> 3256      T7I <11,7> 3736
T8 <1,8> 2417      T8I <11,8> 3377
T9 <1,9> 739      T9I <11,9> 2659
T10 <1,10> 1478      T10I <11,10> 1223
T11 <1,11> 2956      T11I <11,11> 2446
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 797      T0MI <7,0> 1817
T1M <5,1> 1594      T1MI <7,1> 3634
T2M <5,2> 3188      T2MI <7,2> 3173
T3M <5,3> 2281      T3MI <7,3> 2251
T4M <5,4> 467      T4MI <7,4> 407
T5M <5,5> 934      T5MI <7,5> 814
T6M <5,6> 1868      T6MI <7,6> 1628
T7M <5,7> 3736      T7MI <7,7> 3256
T8M <5,8> 3377      T8MI <7,8> 2417
T9M <5,9> 2659      T9MI <7,9> 739
T10M <5,10> 1223      T10MI <7,10> 1478
T11M <5,11> 2446      T11MI <7,11> 2956

The transformations that map this set to itself are: T0, 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 1819Scale 1819: Pydian, Ian Ring Music TheoryPydian
Scale 1821Scale 1821: Aeradian, Ian Ring Music TheoryAeradian
Scale 1809Scale 1809: Ranitonic, Ian Ring Music TheoryRanitonic
Scale 1813Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
Scale 1801Scale 1801: Lanian, Ian Ring Music TheoryLanian
Scale 1833Scale 1833: Ionacrimic, Ian Ring Music TheoryIonacrimic
Scale 1849Scale 1849: Chromatic Hypodorian Inverse, Ian Ring Music TheoryChromatic Hypodorian Inverse
Scale 1881Scale 1881: Katorian, Ian Ring Music TheoryKatorian
Scale 1945Scale 1945: Zarian, Ian Ring Music TheoryZarian
Scale 1561Scale 1561: Joxian, Ian Ring Music TheoryJoxian
Scale 1689Scale 1689: Lorimic, Ian Ring Music TheoryLorimic
Scale 1305Scale 1305: Dynitonic, Ian Ring Music TheoryDynitonic
Scale 793Scale 793: Mocritonic, Ian Ring Music TheoryMocritonic
Scale 2841Scale 2841: Sothimic, Ian Ring Music TheorySothimic
Scale 3865Scale 3865: Starian, Ian Ring Music TheoryStarian

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.