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Scale 3195: "Raryllic"

Scale 3195: Raryllic, 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
Raryllic

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

{0,1,3,4,5,6,10,11}

Forte Number

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

8-6

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.

[2]

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.

6 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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: 495

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.

[1, 2, 1, 1, 1, 4, 1, 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.

<6, 5, 4, 4, 6, 3>

Interval Spectrum

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

p6m4n4s5d6t3

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

Spectra Variation

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

2.5

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.

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.

2.366

Polygon Perimeter

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

5.838

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.

[4]

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.

(60, 47, 124)

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♯{6,10,1}231.5
B{11,3,6}241.83
Minor Triadsd♯m{3,6,10}231.5
a♯m{10,1,5}241.83
Diminished Triads{0,3,6}152.5
a♯°{10,1,4}152.5
Parsimonious Voice Leading Between Common Triads of Scale 3195. Created by Ian Ring ©2019 B B c°->B d#m d#m F# F# d#m->F# d#m->B a#m a#m F#->a#m a#° a#° a#°->a#m

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 Verticesd♯m, F♯
Peripheral Verticesc°, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 3195 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3645
Scale 3645: Zycryllic, Ian Ring Music TheoryZycryllic
3rd mode:
Scale 1935
Scale 1935: Mycryllic, Ian Ring Music TheoryMycryllic
4th mode:
Scale 3015
Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic
5th mode:
Scale 3555
Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
6th mode:
Scale 3825
Scale 3825: Pynyllic, Ian Ring Music TheoryPynyllic
7th mode:
Scale 495
Scale 495: Bocryllic, Ian Ring Music TheoryBocryllicThis is the prime mode
8th mode:
Scale 2295
Scale 2295: Kogyllic, Ian Ring Music TheoryKogyllic

Prime

The prime form of this scale is Scale 495

Scale 495Scale 495: Bocryllic, Ian Ring Music TheoryBocryllic

Complement

The octatonic modal family [3195, 3645, 1935, 3015, 3555, 3825, 495, 2295] (Forte: 8-6) is the complement of the tetratonic modal family [135, 225, 2115, 3105] (Forte: 4-6)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3195 is 3015

Scale 3015Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic

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> 3195       T0I <11,0> 3015
T1 <1,1> 2295      T1I <11,1> 1935
T2 <1,2> 495      T2I <11,2> 3870
T3 <1,3> 990      T3I <11,3> 3645
T4 <1,4> 1980      T4I <11,4> 3195
T5 <1,5> 3960      T5I <11,5> 2295
T6 <1,6> 3825      T6I <11,6> 495
T7 <1,7> 3555      T7I <11,7> 990
T8 <1,8> 3015      T8I <11,8> 1980
T9 <1,9> 1935      T9I <11,9> 3960
T10 <1,10> 3870      T10I <11,10> 3825
T11 <1,11> 3645      T11I <11,11> 3555
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 495      T0MI <7,0> 3825
T1M <5,1> 990      T1MI <7,1> 3555
T2M <5,2> 1980      T2MI <7,2> 3015
T3M <5,3> 3960      T3MI <7,3> 1935
T4M <5,4> 3825      T4MI <7,4> 3870
T5M <5,5> 3555      T5MI <7,5> 3645
T6M <5,6> 3015      T6MI <7,6> 3195
T7M <5,7> 1935      T7MI <7,7> 2295
T8M <5,8> 3870      T8MI <7,8> 495
T9M <5,9> 3645      T9MI <7,9> 990
T10M <5,10> 3195       T10MI <7,10> 1980
T11M <5,11> 2295      T11MI <7,11> 3960

The transformations that map this set to itself are: T0, T4I, T10M, T6MI

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 3193Scale 3193: Zathian, Ian Ring Music TheoryZathian
Scale 3197Scale 3197: Gylyllic, Ian Ring Music TheoryGylyllic
Scale 3199Scale 3199: Nonatonic Chromatic 3, Ian Ring Music TheoryNonatonic Chromatic 3
Scale 3187Scale 3187: Koptian, Ian Ring Music TheoryKoptian
Scale 3191Scale 3191: Bynyllic, Ian Ring Music TheoryBynyllic
Scale 3179Scale 3179: Daptian, Ian Ring Music TheoryDaptian
Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian
Scale 3131Scale 3131: Torian, Ian Ring Music TheoryTorian
Scale 3259Scale 3259: Ulian, Ian Ring Music TheoryUlian
Scale 3323Scale 3323: Lacrygic, Ian Ring Music TheoryLacrygic
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 3707Scale 3707: Rynygic, Ian Ring Music TheoryRynygic
Scale 2171Scale 2171: Negian, Ian Ring Music TheoryNegian
Scale 2683Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian

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.