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Scale 2679: "Rathyllic"

Scale 2679: Rathyllic, 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
Rathyllic

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

{0,1,2,4,5,6,9,11}

Forte Number

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

8-14

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: 3531

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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: 759

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 2, 1, 1, 3, 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.

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

Interval Spectrum

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

p6m5n5s5d5t2

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

Spectra Variation

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

2.25

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.

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.

2.616

Polygon Perimeter

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

6.002

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.

(34, 56, 136)

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 TriadsD{2,6,9}341.9
F{5,9,0}341.9
A{9,1,4}242.1
Minor Triadsdm{2,5,9}331.7
f♯m{6,9,1}331.7
am{9,0,4}252.5
bm{11,2,6}252.5
Augmented TriadsC♯+{1,5,9}431.5
Diminished Triadsf♯°{6,9,0}242.1
{11,2,5}242.3
Parsimonious Voice Leading Between Common Triads of Scale 2679. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F f#m f#m C#+->f#m A A C#+->A D D dm->D dm->b° D->f#m bm bm D->bm f#° f#° F->f#° am am F->am f#°->f#m am->A b°->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 VerticesC♯+, dm, f♯m
Peripheral Verticesam, bm

Modes

Modes are the rotational transformation of this scale. Scale 2679 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3387
Scale 3387: Aeryptyllic, Ian Ring Music TheoryAeryptyllic
3rd mode:
Scale 3741
Scale 3741: Zydyllic, Ian Ring Music TheoryZydyllic
4th mode:
Scale 1959
Scale 1959: Katolyllic, Ian Ring Music TheoryKatolyllic
5th mode:
Scale 3027
Scale 3027: Rythyllic, Ian Ring Music TheoryRythyllic
6th mode:
Scale 3561
Scale 3561: Pothyllic, Ian Ring Music TheoryPothyllic
7th mode:
Scale 957
Scale 957: Phronyllic, Ian Ring Music TheoryPhronyllic
8th mode:
Scale 1263
Scale 1263: Stynyllic, Ian Ring Music TheoryStynyllic

Prime

The prime form of this scale is Scale 759

Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic

Complement

The octatonic modal family [2679, 3387, 3741, 1959, 3027, 3561, 957, 1263] (Forte: 8-14) is the complement of the tetratonic modal family [141, 417, 1059, 2577] (Forte: 4-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2679 is 3531

Scale 3531Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2679 is chiral, and its enantiomorph is scale 3531

Scale 3531Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri

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> 2679       T0I <11,0> 3531
T1 <1,1> 1263      T1I <11,1> 2967
T2 <1,2> 2526      T2I <11,2> 1839
T3 <1,3> 957      T3I <11,3> 3678
T4 <1,4> 1914      T4I <11,4> 3261
T5 <1,5> 3828      T5I <11,5> 2427
T6 <1,6> 3561      T6I <11,6> 759
T7 <1,7> 3027      T7I <11,7> 1518
T8 <1,8> 1959      T8I <11,8> 3036
T9 <1,9> 3918      T9I <11,9> 1977
T10 <1,10> 3741      T10I <11,10> 3954
T11 <1,11> 3387      T11I <11,11> 3813
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2019      T0MI <7,0> 2301
T1M <5,1> 4038      T1MI <7,1> 507
T2M <5,2> 3981      T2MI <7,2> 1014
T3M <5,3> 3867      T3MI <7,3> 2028
T4M <5,4> 3639      T4MI <7,4> 4056
T5M <5,5> 3183      T5MI <7,5> 4017
T6M <5,6> 2271      T6MI <7,6> 3939
T7M <5,7> 447      T7MI <7,7> 3783
T8M <5,8> 894      T8MI <7,8> 3471
T9M <5,9> 1788      T9MI <7,9> 2847
T10M <5,10> 3576      T10MI <7,10> 1599
T11M <5,11> 3057      T11MI <7,11> 3198

The transformations that map this set to itself are: T0

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 2677Scale 2677: Thodian, Ian Ring Music TheoryThodian
Scale 2675Scale 2675: Chromatic Lydian, Ian Ring Music TheoryChromatic Lydian
Scale 2683Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
Scale 2687Scale 2687: Thacrygic, Ian Ring Music TheoryThacrygic
Scale 2663Scale 2663: Lalian, Ian Ring Music TheoryLalian
Scale 2671Scale 2671: Aerolyllic, Ian Ring Music TheoryAerolyllic
Scale 2647Scale 2647: Dadian, Ian Ring Music TheoryDadian
Scale 2615Scale 2615: Thoptian, Ian Ring Music TheoryThoptian
Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic
Scale 2807Scale 2807: Zylygic, Ian Ring Music TheoryZylygic
Scale 2935Scale 2935: Modygic, Ian Ring Music TheoryModygic
Scale 2167Scale 2167, Ian Ring Music Theory
Scale 2423Scale 2423, Ian Ring Music Theory
Scale 3191Scale 3191: Bynyllic, Ian Ring Music TheoryBynyllic
Scale 3703Scale 3703: Katalygic, Ian Ring Music TheoryKatalygic
Scale 631Scale 631: Zygian, Ian Ring Music TheoryZygian
Scale 1655Scale 1655: Katygyllic, Ian Ring Music TheoryKatygyllic

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.