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Scale 763: "Doryllic"

Scale 763: Doryllic, 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
Doryllic
Dozenal
Eruian

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

{0,1,3,4,5,6,7,9}

Forte Number

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

8-12

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: 3049

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.

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.

yes

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, 1, 2, 3]

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, 6, 5, 4, 3>

Interval Spectrum

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

p4m5n6s5d5t3

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

2.5

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.

2.616

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.

(46, 63, 142)

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}342
F{5,9,0}342
A{9,1,4}342
Minor Triadscm{0,3,7}342.17
f♯m{6,9,1}342.17
am{9,0,4}441.83
Augmented TriadsC♯+{1,5,9}342
Diminished Triads{0,3,6}242.33
c♯°{1,4,7}242.33
d♯°{3,6,9}242.33
f♯°{6,9,0}242.33
{9,0,3}242.17
Parsimonious Voice Leading Between Common Triads of Scale 763. Created by Ian Ring ©2019 cm cm c°->cm d#° d#° c°->d#° C C cm->C cm->a° c#° c#° C->c#° am am C->am A A c#°->A C#+ C#+ F F C#+->F f#m f#m C#+->f#m C#+->A d#°->f#m f#° f#° F->f#° F->am f#°->f#m a°->am am->A

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 763 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2429
Scale 2429: Kadyllic, Ian Ring Music TheoryKadyllic
3rd mode:
Scale 1631
Scale 1631: Rynyllic, Ian Ring Music TheoryRynyllic
4th mode:
Scale 2863
Scale 2863: Aerogyllic, Ian Ring Music TheoryAerogyllic
5th mode:
Scale 3479
Scale 3479: Rothyllic, Ian Ring Music TheoryRothyllic
6th mode:
Scale 3787
Scale 3787: Kagyllic, Ian Ring Music TheoryKagyllic
7th mode:
Scale 3941
Scale 3941: Stathyllic, Ian Ring Music TheoryStathyllic
8th mode:
Scale 2009
Scale 2009: Stacryllic, Ian Ring Music TheoryStacryllic

Prime

This is the prime form of this scale.

Complement

The octatonic modal family [763, 2429, 1631, 2863, 3479, 3787, 3941, 2009] (Forte: 8-12) is the complement of the tetratonic modal family [77, 833, 1043, 2569] (Forte: 4-12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 763 is 3049

Scale 3049Scale 3049: Phrydyllic, Ian Ring Music TheoryPhrydyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 763 is chiral, and its enantiomorph is scale 3049

Scale 3049Scale 3049: Phrydyllic, Ian Ring Music TheoryPhrydyllic

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> 763       T0I <11,0> 3049
T1 <1,1> 1526      T1I <11,1> 2003
T2 <1,2> 3052      T2I <11,2> 4006
T3 <1,3> 2009      T3I <11,3> 3917
T4 <1,4> 4018      T4I <11,4> 3739
T5 <1,5> 3941      T5I <11,5> 3383
T6 <1,6> 3787      T6I <11,6> 2671
T7 <1,7> 3479      T7I <11,7> 1247
T8 <1,8> 2863      T8I <11,8> 2494
T9 <1,9> 1631      T9I <11,9> 893
T10 <1,10> 3262      T10I <11,10> 1786
T11 <1,11> 2429      T11I <11,11> 3572
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2923      T0MI <7,0> 2779
T1M <5,1> 1751      T1MI <7,1> 1463
T2M <5,2> 3502      T2MI <7,2> 2926
T3M <5,3> 2909      T3MI <7,3> 1757
T4M <5,4> 1723      T4MI <7,4> 3514
T5M <5,5> 3446      T5MI <7,5> 2933
T6M <5,6> 2797      T6MI <7,6> 1771
T7M <5,7> 1499      T7MI <7,7> 3542
T8M <5,8> 2998      T8MI <7,8> 2989
T9M <5,9> 1901      T9MI <7,9> 1883
T10M <5,10> 3802      T10MI <7,10> 3766
T11M <5,11> 3509      T11MI <7,11> 3437

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 761Scale 761: Ponian, Ian Ring Music TheoryPonian
Scale 765Scale 765: Erkian, Ian Ring Music TheoryErkian
Scale 767Scale 767: Raptygic, Ian Ring Music TheoryRaptygic
Scale 755Scale 755: Phrythian, Ian Ring Music TheoryPhrythian
Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic
Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian
Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode
Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian
Scale 635Scale 635: Epolian, Ian Ring Music TheoryEpolian
Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic
Scale 1019Scale 1019: Aeranygic, Ian Ring Music TheoryAeranygic
Scale 251Scale 251: Borian, Ian Ring Music TheoryBorian
Scale 507Scale 507: Moryllic, Ian Ring Music TheoryMoryllic
Scale 1275Scale 1275: Stagyllic, Ian Ring Music TheoryStagyllic
Scale 1787Scale 1787: Mycrygic, Ian Ring Music TheoryMycrygic
Scale 2811Scale 2811: Barygic, Ian Ring Music TheoryBarygic

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.