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Scale 2617: "Pylimic"

Scale 2617: Pylimic, 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
Pylimic
Dozenal
Qelian

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

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

Forte Number

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

6-Z43

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: 907

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: 359

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, 1, 4, 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.

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

(10, 17, 64)

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}121
Minor Triadsam{9,0,4}210.67
Diminished Triads{9,0,3}121

The following pitch classes are not present in any of the common triads: {11}

Parsimonious Voice Leading Between Common Triads of Scale 2617. Created by Ian Ring ©2019 F F am am F->am 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
Radius1
Self-Centeredno
Central Verticesam
Peripheral VerticesF, a°

Modes

Modes are the rotational transformation of this scale. Scale 2617 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 839
Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
3rd mode:
Scale 2467
Scale 2467: Raga Padi, Ian Ring Music TheoryRaga Padi
4th mode:
Scale 3281
Scale 3281: Raga Vijayavasanta, Ian Ring Music TheoryRaga Vijayavasanta
5th mode:
Scale 461
Scale 461: Raga Syamalam, Ian Ring Music TheoryRaga Syamalam
6th mode:
Scale 1139
Scale 1139: Aerygimic, Ian Ring Music TheoryAerygimic

Prime

The prime form of this scale is Scale 359

Scale 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic

Complement

The hexatonic modal family [2617, 839, 2467, 3281, 461, 1139] (Forte: 6-Z43) is the complement of the hexatonic modal family [407, 739, 1817, 2251, 2417, 3173] (Forte: 6-Z17)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2617 is 907

Scale 907Scale 907: Tholimic, Ian Ring Music TheoryTholimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2617 is chiral, and its enantiomorph is scale 907

Scale 907Scale 907: Tholimic, Ian Ring Music TheoryTholimic

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> 2617       T0I <11,0> 907
T1 <1,1> 1139      T1I <11,1> 1814
T2 <1,2> 2278      T2I <11,2> 3628
T3 <1,3> 461      T3I <11,3> 3161
T4 <1,4> 922      T4I <11,4> 2227
T5 <1,5> 1844      T5I <11,5> 359
T6 <1,6> 3688      T6I <11,6> 718
T7 <1,7> 3281      T7I <11,7> 1436
T8 <1,8> 2467      T8I <11,8> 2872
T9 <1,9> 839      T9I <11,9> 1649
T10 <1,10> 1678      T10I <11,10> 3298
T11 <1,11> 3356      T11I <11,11> 2501
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 907      T0MI <7,0> 2617
T1M <5,1> 1814      T1MI <7,1> 1139
T2M <5,2> 3628      T2MI <7,2> 2278
T3M <5,3> 3161      T3MI <7,3> 461
T4M <5,4> 2227      T4MI <7,4> 922
T5M <5,5> 359      T5MI <7,5> 1844
T6M <5,6> 718      T6MI <7,6> 3688
T7M <5,7> 1436      T7MI <7,7> 3281
T8M <5,8> 2872      T8MI <7,8> 2467
T9M <5,9> 1649      T9MI <7,9> 839
T10M <5,10> 3298      T10MI <7,10> 1678
T11M <5,11> 2501      T11MI <7,11> 3356

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 2619Scale 2619: Ionyrian, Ian Ring Music TheoryIonyrian
Scale 2621Scale 2621: Ionogian, Ian Ring Music TheoryIonogian
Scale 2609Scale 2609: Raga Bhinna Shadja, Ian Ring Music TheoryRaga Bhinna Shadja
Scale 2613Scale 2613: Raga Hamsa Vinodini, Ian Ring Music TheoryRaga Hamsa Vinodini
Scale 2601Scale 2601: Raga Chandrakauns, Ian Ring Music TheoryRaga Chandrakauns
Scale 2585Scale 2585: Otlian, Ian Ring Music TheoryOtlian
Scale 2649Scale 2649: Aeolythimic, Ian Ring Music TheoryAeolythimic
Scale 2681Scale 2681: Aerycrian, Ian Ring Music TheoryAerycrian
Scale 2745Scale 2745: Mela Sulini, Ian Ring Music TheoryMela Sulini
Scale 2873Scale 2873: Ionian Augmented Sharp 2, Ian Ring Music TheoryIonian Augmented Sharp 2
Scale 2105Scale 2105: Rigian, Ian Ring Music TheoryRigian
Scale 2361Scale 2361: Docrimic, Ian Ring Music TheoryDocrimic
Scale 3129Scale 3129: Toqian, Ian Ring Music TheoryToqian
Scale 3641Scale 3641: Thocrian, Ian Ring Music TheoryThocrian
Scale 569Scale 569: Mothitonic, Ian Ring Music TheoryMothitonic
Scale 1593Scale 1593: Zogimic, Ian Ring Music TheoryZogimic

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.