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Scale 2781: "Gycryllic"

Scale 2781: Gycryllic, 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
Gycryllic
Dozenal
Rilian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,2,3,4,6,7,9,11}

Forte Number

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

8-26

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.

[3]

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.

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.

7

Prime Form

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

no
prime: 1467

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.

[2, 1, 1, 2, 1, 2, 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.

<4, 5, 6, 5, 6, 2>

Interval Spectrum

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

p6m5n6s5d4t2

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

Spectra Variation

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

1.25

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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.

[6]

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".

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, 32, 110)

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}342.15
D{2,6,9}342.23
G{7,11,2}242.23
B{11,3,6}441.92
Minor Triadscm{0,3,7}441.92
em{4,7,11}242.23
am{9,0,4}342.23
bm{11,2,6}342.15
Augmented TriadsD♯+{3,7,11}441.85
Diminished Triads{0,3,6}242.15
d♯°{3,6,9}242.31
f♯°{6,9,0}242.31
{9,0,3}242.31
Parsimonious Voice Leading Between Common Triads of Scale 2781. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ cm->a° em em C->em am am C->am D D d#° d#° D->d#° f#° f#° D->f#° bm bm D->bm d#°->B D#+->em Parsimonious Voice Leading Between Common Triads of Scale 2781. Created by Ian Ring ©2019 G D#+->G D#+->B f#°->am G->bm a°->am bm->B

view full size

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2781 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1719
Scale 1719: Lyryllic, Ian Ring Music TheoryLyryllic
3rd mode:
Scale 2907
Scale 2907: Magen Abot 2, Ian Ring Music TheoryMagen Abot 2
4th mode:
Scale 3501
Scale 3501: Maqam Nahawand, Ian Ring Music TheoryMaqam Nahawand
5th mode:
Scale 1899
Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic
6th mode:
Scale 2997
Scale 2997: Major Bebop, Ian Ring Music TheoryMajor Bebop
7th mode:
Scale 1773
Scale 1773: Blues Scale II, Ian Ring Music TheoryBlues Scale II
8th mode:
Scale 1467
Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish PhrygianThis is the prime mode

Prime

The prime form of this scale is Scale 1467

Scale 1467Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish Phrygian

Complement

The octatonic modal family [2781, 1719, 2907, 3501, 1899, 2997, 1773, 1467] (Forte: 8-26) is the complement of the tetratonic modal family [297, 549, 657, 1161] (Forte: 4-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2781 is 1899

Scale 1899Scale 1899: Moptyllic, Ian Ring Music TheoryMoptyllic

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> 2781       T0I <11,0> 1899
T1 <1,1> 1467      T1I <11,1> 3798
T2 <1,2> 2934      T2I <11,2> 3501
T3 <1,3> 1773      T3I <11,3> 2907
T4 <1,4> 3546      T4I <11,4> 1719
T5 <1,5> 2997      T5I <11,5> 3438
T6 <1,6> 1899      T6I <11,6> 2781
T7 <1,7> 3798      T7I <11,7> 1467
T8 <1,8> 3501      T8I <11,8> 2934
T9 <1,9> 2907      T9I <11,9> 1773
T10 <1,10> 1719      T10I <11,10> 3546
T11 <1,11> 3438      T11I <11,11> 2997
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4041      T0MI <7,0> 639
T1M <5,1> 3987      T1MI <7,1> 1278
T2M <5,2> 3879      T2MI <7,2> 2556
T3M <5,3> 3663      T3MI <7,3> 1017
T4M <5,4> 3231      T4MI <7,4> 2034
T5M <5,5> 2367      T5MI <7,5> 4068
T6M <5,6> 639      T6MI <7,6> 4041
T7M <5,7> 1278      T7MI <7,7> 3987
T8M <5,8> 2556      T8MI <7,8> 3879
T9M <5,9> 1017      T9MI <7,9> 3663
T10M <5,10> 2034      T10MI <7,10> 3231
T11M <5,11> 4068      T11MI <7,11> 2367

The transformations that map this set to itself are: T0, T6I

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 2783Scale 2783: Gothygic, Ian Ring Music TheoryGothygic
Scale 2777Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic
Scale 2779Scale 2779: Shostakovich, Ian Ring Music TheoryShostakovich
Scale 2773Scale 2773: Lydian, Ian Ring Music TheoryLydian
Scale 2765Scale 2765: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
Scale 2797Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
Scale 2813Scale 2813: Zolygic, Ian Ring Music TheoryZolygic
Scale 2717Scale 2717: Epygian, Ian Ring Music TheoryEpygian
Scale 2749Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
Scale 2653Scale 2653: Sygian, Ian Ring Music TheorySygian
Scale 2909Scale 2909: Mocryllic, Ian Ring Music TheoryMocryllic
Scale 3037Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale
Scale 2269Scale 2269: Pygian, Ian Ring Music TheoryPygian
Scale 2525Scale 2525: Aeolaryllic, Ian Ring Music TheoryAeolaryllic
Scale 3293Scale 3293: Saryllic, Ian Ring Music TheorySaryllic
Scale 3805Scale 3805: Moptygic, Ian Ring Music TheoryMoptygic
Scale 733Scale 733: Donian, Ian Ring Music TheoryDonian
Scale 1757Scale 1757: Kunian, Ian Ring Music TheoryKunian

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.