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Scale 2769: "Dyrimic"

Scale 2769: Dyrimic, 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
Dyrimic
Dozenal
Ridian

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

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

Forte Number

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

6-Z25

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: 363

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

5

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 363

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, an indicator of maximum hierarchization.

no

Interval Structure

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

[4, 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.

<2, 3, 3, 2, 4, 1>

Interval Spectrum

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

p4m2n3s3d2t

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

Spectra Variation

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

2.333

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.

2.232

Polygon Perimeter

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

5.767

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(12, 9, 55)

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}221
Minor Triadsem{4,7,11}131.5
am{9,0,4}221
Diminished Triadsf♯°{6,9,0}131.5
Parsimonious Voice Leading Between Common Triads of Scale 2769. Created by Ian Ring ©2019 C C em em C->em am am C->am f#° f#° f#°->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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC, am
Peripheral Verticesem, f♯°

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Minor: {4, 7, 11}
Diminished: {6, 9, 0}

Modes

Modes are the rotational transformation of this scale. Scale 2769 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 429
Scale 429: Koptimic, Ian Ring Music TheoryKoptimic
3rd mode:
Scale 1131
Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
4th mode:
Scale 2613
Scale 2613: Raga Hamsa Vinodini, Ian Ring Music TheoryRaga Hamsa Vinodini
5th mode:
Scale 1677
Scale 1677: Raga Manavi, Ian Ring Music TheoryRaga Manavi
6th mode:
Scale 1443
Scale 1443: Raga Phenadyuti, Ian Ring Music TheoryRaga Phenadyuti

Prime

The prime form of this scale is Scale 363

Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic

Complement

The hexatonic modal family [2769, 429, 1131, 2613, 1677, 1443] (Forte: 6-Z25) is the complement of the hexatonic modal family [663, 741, 1209, 1833, 2379, 3237] (Forte: 6-Z47)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2769 is 363

Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2769 is chiral, and its enantiomorph is scale 363

Scale 363Scale 363: Soptimic, Ian Ring Music TheorySoptimic

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> 2769       T0I <11,0> 363
T1 <1,1> 1443      T1I <11,1> 726
T2 <1,2> 2886      T2I <11,2> 1452
T3 <1,3> 1677      T3I <11,3> 2904
T4 <1,4> 3354      T4I <11,4> 1713
T5 <1,5> 2613      T5I <11,5> 3426
T6 <1,6> 1131      T6I <11,6> 2757
T7 <1,7> 2262      T7I <11,7> 1419
T8 <1,8> 429      T8I <11,8> 2838
T9 <1,9> 858      T9I <11,9> 1581
T10 <1,10> 1716      T10I <11,10> 3162
T11 <1,11> 3432      T11I <11,11> 2229
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3009      T0MI <7,0> 123
T1M <5,1> 1923      T1MI <7,1> 246
T2M <5,2> 3846      T2MI <7,2> 492
T3M <5,3> 3597      T3MI <7,3> 984
T4M <5,4> 3099      T4MI <7,4> 1968
T5M <5,5> 2103      T5MI <7,5> 3936
T6M <5,6> 111      T6MI <7,6> 3777
T7M <5,7> 222      T7MI <7,7> 3459
T8M <5,8> 444      T8MI <7,8> 2823
T9M <5,9> 888      T9MI <7,9> 1551
T10M <5,10> 1776      T10MI <7,10> 3102
T11M <5,11> 3552      T11MI <7,11> 2109

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 2771Scale 2771: Marva That, Ian Ring Music TheoryMarva That
Scale 2773Scale 2773: Lydian, Ian Ring Music TheoryLydian
Scale 2777Scale 2777: Aeolian Harmonic, Ian Ring Music TheoryAeolian Harmonic
Scale 2753Scale 2753: Ritian, Ian Ring Music TheoryRitian
Scale 2761Scale 2761: Dagimic, Ian Ring Music TheoryDagimic
Scale 2785Scale 2785: Ronian, Ian Ring Music TheoryRonian
Scale 2801Scale 2801: Zogian, Ian Ring Music TheoryZogian
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 2737Scale 2737: Raga Hari Nata, Ian Ring Music TheoryRaga Hari Nata
Scale 2641Scale 2641: Raga Hindol, Ian Ring Music TheoryRaga Hindol
Scale 2897Scale 2897: Rycrimic, Ian Ring Music TheoryRycrimic
Scale 3025Scale 3025: Epycrian, Ian Ring Music TheoryEpycrian
Scale 2257Scale 2257: Lydian Pentatonic, Ian Ring Music TheoryLydian Pentatonic
Scale 2513Scale 2513: Aerycrimic, Ian Ring Music TheoryAerycrimic
Scale 3281Scale 3281: Raga Vijayavasanta, Ian Ring Music TheoryRaga Vijayavasanta
Scale 3793Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
Scale 721Scale 721: Raga Dhavalashri, Ian Ring Music TheoryRaga Dhavalashri
Scale 1745Scale 1745: Raga Vutari, Ian Ring Music TheoryRaga Vutari

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.