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Scale 3261: "Dodyllic"

Scale 3261: Dodyllic, 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
Dodyllic
Dozenal
Uloian

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

{0,2,3,4,5,7,10,11}

Forte Number

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

8-14

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: 1959

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 Structure

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

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

<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 TriadsC{0,4,7}252.5
D♯{3,7,10}331.7
G{7,11,2}331.7
A♯{10,2,5}252.5
Minor Triadscm{0,3,7}242.1
em{4,7,11}341.9
gm{7,10,2}341.9
Augmented TriadsD♯+{3,7,11}431.5
Diminished Triads{4,7,10}242.1
{11,2,5}242.3
Parsimonious Voice Leading Between Common Triads of Scale 3261. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ em em C->em D# D# D#->D#+ D#->e° gm gm D#->gm D#+->em Parsimonious Voice Leading Between Common Triads of Scale 3261. Created by Ian Ring ©2019 G D#+->G e°->em gm->G A# A# gm->A# G->b° A#->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.

Diameter5
Radius3
Self-Centeredno
Central VerticesD♯, D♯+, G
Peripheral VerticesC, A♯

Modes

Modes are the rotational transformation of this scale. Scale 3261 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1839
Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
3rd mode:
Scale 2967
Scale 2967: Madyllic, Ian Ring Music TheoryMadyllic
4th mode:
Scale 3531
Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
5th mode:
Scale 3813
Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
6th mode:
Scale 1977
Scale 1977: Dagyllic, Ian Ring Music TheoryDagyllic
7th mode:
Scale 759
Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllicThis is the prime mode
8th mode:
Scale 2427
Scale 2427: Katoryllic, Ian Ring Music TheoryKatoryllic

Prime

The prime form of this scale is Scale 759

Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic

Complement

The octatonic modal family [3261, 1839, 2967, 3531, 3813, 1977, 759, 2427] (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 3261 is 1959

Scale 1959Scale 1959: Katolyllic, Ian Ring Music TheoryKatolyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3261 is chiral, and its enantiomorph is scale 1959

Scale 1959Scale 1959: Katolyllic, Ian Ring Music TheoryKatolyllic

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

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 3263Scale 3263: Pyrygic, Ian Ring Music TheoryPyrygic
Scale 3257Scale 3257: Mela Calanata, Ian Ring Music TheoryMela Calanata
Scale 3259Scale 3259: Ulian, Ian Ring Music TheoryUlian
Scale 3253Scale 3253: Mela Naganandini, Ian Ring Music TheoryMela Naganandini
Scale 3245Scale 3245: Mela Varunapriya, Ian Ring Music TheoryMela Varunapriya
Scale 3229Scale 3229: Aeolaptian, Ian Ring Music TheoryAeolaptian
Scale 3293Scale 3293: Saryllic, Ian Ring Music TheorySaryllic
Scale 3325Scale 3325: Mixolygic, Ian Ring Music TheoryMixolygic
Scale 3133Scale 3133: Tosian, Ian Ring Music TheoryTosian
Scale 3197Scale 3197: Gylyllic, Ian Ring Music TheoryGylyllic
Scale 3389Scale 3389: Socryllic, Ian Ring Music TheorySocryllic
Scale 3517Scale 3517: Epocrygic, Ian Ring Music TheoryEpocrygic
Scale 3773Scale 3773: Raga Malgunji, Ian Ring Music TheoryRaga Malgunji
Scale 2237Scale 2237: Epothian, Ian Ring Music TheoryEpothian
Scale 2749Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
Scale 1213Scale 1213: Gyrian, Ian Ring Music TheoryGyrian

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.