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Scale 3541: "Racryllic"

Scale 3541: Racryllic, 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

41161837294116105072918310504116183
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
Racryllic

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

{0,2,4,6,7,8,10,11}

Forte Number

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

8-24

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.

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.

4

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: 1399

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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, 6, 4, 7, 4, 3]

Interval Spectrum

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

p4m7n4s6d4t3

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> = {4,5,6}
<4> = {5,6,7}
<5> = {6,7,8}
<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.

1.5

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".

Improper

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}242.1
E{4,8,11}341.9
G{7,11,2}431.5
Minor Triadsem{4,7,11}431.5
gm{7,10,2}341.9
bm{11,2,6}242.1
Augmented TriadsC+{0,4,8}252.5
D+{2,6,10}252.5
Diminished Triads{4,7,10}231.9
g♯°{8,11,2}231.9
Parsimonious Voice Leading Between Common Triads of Scale 3541. Created by Ian Ring ©2019 C C C+ C+ C->C+ em em C->em E E C+->E D+ D+ gm gm D+->gm bm bm D+->bm e°->em e°->gm em->E Parsimonious Voice Leading Between Common Triads of Scale 3541. Created by Ian Ring ©2019 G em->G g#° g#° E->g#° gm->G G->g#° G->bm

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.

Diameter5
Radius3
Self-Centeredno
Central Verticese°, em, G, g♯°
Peripheral VerticesC+, D+

Modes

Modes are the rotational transformation of this scale. Scale 3541 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1909
Scale 1909: Epicryllic, Ian Ring Music TheoryEpicryllic
3rd mode:
Scale 1501
Scale 1501: Stygyllic, Ian Ring Music TheoryStygyllic
4th mode:
Scale 1399
Scale 1399: Syryllic, Ian Ring Music TheorySyryllicThis is the prime mode
5th mode:
Scale 2747
Scale 2747: Stythyllic, Ian Ring Music TheoryStythyllic
6th mode:
Scale 3421
Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
7th mode:
Scale 1879
Scale 1879: Mixoryllic, Ian Ring Music TheoryMixoryllic
8th mode:
Scale 2987
Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed

Prime

The prime form of this scale is Scale 1399

Scale 1399Scale 1399: Syryllic, Ian Ring Music TheorySyryllic

Complement

The octatonic modal family [3541, 1909, 1501, 1399, 2747, 3421, 1879, 2987] (Forte: 8-24) is the complement of the tetratonic modal family [277, 337, 1093, 1297] (Forte: 4-24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3541 is 1399

Scale 1399Scale 1399: Syryllic, Ian Ring Music TheorySyryllic

Transformations:

T0 3541  T0I 1399
T1 2987  T1I 2798
T2 1879  T2I 1501
T3 3758  T3I 3002
T4 3421  T4I 1909
T5 2747  T5I 3818
T6 1399  T6I 3541
T7 2798  T7I 2987
T8 1501  T8I 1879
T9 3002  T9I 3758
T10 1909  T10I 3421
T11 3818  T11I 2747

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 3543Scale 3543: Aeolonygic, Ian Ring Music TheoryAeolonygic
Scale 3537Scale 3537: Katogian, Ian Ring Music TheoryKatogian
Scale 3539Scale 3539: Aeoryllic, Ian Ring Music TheoryAeoryllic
Scale 3545Scale 3545: Thyptyllic, Ian Ring Music TheoryThyptyllic
Scale 3549Scale 3549: Messiaen Mode 3 Inverse, Ian Ring Music TheoryMessiaen Mode 3 Inverse
Scale 3525Scale 3525: Zocrian, Ian Ring Music TheoryZocrian
Scale 3533Scale 3533: Thadyllic, Ian Ring Music TheoryThadyllic
Scale 3557Scale 3557, Ian Ring Music Theory
Scale 3573Scale 3573: Kaptygic, Ian Ring Music TheoryKaptygic
Scale 3477Scale 3477: Kyptian, Ian Ring Music TheoryKyptian
Scale 3509Scale 3509: Stogyllic, Ian Ring Music TheoryStogyllic
Scale 3413Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone
Scale 3285Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
Scale 3797Scale 3797: Rocryllic, Ian Ring Music TheoryRocryllic
Scale 4053Scale 4053: Kyrygic, Ian Ring Music TheoryKyrygic
Scale 2517Scale 2517: Harmonic Lydian, Ian Ring Music TheoryHarmonic Lydian
Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
Scale 1493Scale 1493: Lydian Minor, Ian Ring Music TheoryLydian Minor

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.