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Scale 3763: "Modyllic"

Scale 3763: Modyllic, 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
Modyllic
Dozenal
YAJian

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

 {0,1,4,5,7,9,10,11}

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-Z15

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: 2479

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.

 3

Modes

Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 8

Prime Form

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

 no
prime: 863

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.

 [1, 3, 1, 2, 2, 1, 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, 5, 3>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.333, 0.333, 0.25, 0.333, 0.333, 0.5>

Interval Spectrum

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

 p5m5n5s5d5t3

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

(30, 59, 140)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.665

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.286

Generator

This scale has no generator.

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}242.18
A{9,1,4}441.82
Minor Triadsem{4,7,11}242.27
am{9,0,4}341.91
a♯m{10,1,5}342
Augmented TriadsC♯+{1,5,9}341.91
Diminished Triadsc♯°{1,4,7}242.09
e°{4,7,10}242.36
g°{7,10,1}242.27
a♯°{10,1,4}242.09
Parsimonious Voice Leading Between Common Triads of Scale 3763. Created by Ian Ring ©2019 C C c#° c#° C->c#° em em C->em am am C->am A A c#°->A C#+ C#+ F F C#+->F C#+->A a#m a#m C#+->a#m e° e° e°->em g° g° e°->g° F->am g°->a#m am->A a#° a#° 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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale.Scale 3763 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3929		Scale 3929: Aeolothyllic, Ian Ring Music TheoryAeolothyllic
3rd mode:
Scale 1003		Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
4th mode:
Scale 2549		Scale 2549: Rydyllic, Ian Ring Music TheoryRydyllic
5th mode:
Scale 1661		Scale 1661: Gonyllic, Ian Ring Music TheoryGonyllic
6th mode:
Scale 1439		Scale 1439: Rolyllic, Ian Ring Music TheoryRolyllic
7th mode:
Scale 2767		Scale 2767: Katydyllic, Ian Ring Music TheoryKatydyllic
8th mode:
Scale 3431		Scale 3431: Zyptyllic, Ian Ring Music TheoryZyptyllic

Prime

The prime form of this scale is Scale 863

Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic

Complement

The octatonic modal family [3763, 3929, 1003, 2549, 1661, 1439, 2767, 3431] (Forte: 8-Z15) is the complement of the tetratonic modal family [83, 773, 1217, 2089] (Forte: 4-Z15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3763 is 2479

Scale 2479Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111100110101113763k = 1h = 1
211100110101113763k = 2h = 1
311100110101113763k = 3h = 1
42(1)(1)00(1)(1)0(1)0(1)(1)(1)3763k = 4h = 2
52(1)(1)00(1)(1)0(1)0(1)(1)(1)3763k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3763 is chiral, and its enantiomorph is scale 2479

Scale 2479Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 3763       T0I <11,0> 2479
T1 <1,1> 3431      T1I <11,1> 863
T2 <1,2> 2767      T2I <11,2> 1726
T3 <1,3> 1439      T3I <11,3> 3452
T4 <1,4> 2878      T4I <11,4> 2809
T5 <1,5> 1661      T5I <11,5> 1523
T6 <1,6> 3322      T6I <11,6> 3046
T7 <1,7> 2549      T7I <11,7> 1997
T8 <1,8> 1003      T8I <11,8> 3994
T9 <1,9> 2006      T9I <11,9> 3893
T10 <1,10> 4012      T10I <11,10> 3691
T11 <1,11> 3929      T11I <11,11> 3287
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2983      T0MI <7,0> 3259
T1M <5,1> 1871      T1MI <7,1> 2423
T2M <5,2> 3742      T2MI <7,2> 751
T3M <5,3> 3389      T3MI <7,3> 1502
T4M <5,4> 2683      T4MI <7,4> 3004
T5M <5,5> 1271      T5MI <7,5> 1913
T6M <5,6> 2542      T6MI <7,6> 3826
T7M <5,7> 989      T7MI <7,7> 3557
T8M <5,8> 1978      T8MI <7,8> 3019
T9M <5,9> 3956      T9MI <7,9> 1943
T10M <5,10> 3817      T10MI <7,10> 3886
T11M <5,11> 3539      T11MI <7,11> 3677

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 3761Scale 3761: Raga Madhuri, Ian Ring Music TheoryRaga Madhuri
Scale 3765Scale 3765: Dominant Bebop, Ian Ring Music TheoryDominant Bebop
Scale 3767Scale 3767: Chromatic Bebop, Ian Ring Music TheoryChromatic Bebop
Scale 3771Scale 3771: Stophygic, Ian Ring Music TheoryStophygic
Scale 3747Scale 3747: Myrian, Ian Ring Music TheoryMyrian
Scale 3755Scale 3755: Phryryllic, Ian Ring Music TheoryPhryryllic
Scale 3731Scale 3731: Aeryrian, Ian Ring Music TheoryAeryrian
Scale 3795Scale 3795: Epothyllic, Ian Ring Music TheoryEpothyllic
Scale 3827Scale 3827: Dorygic, Ian Ring Music TheoryDorygic
Scale 3635Scale 3635: Katygian, Ian Ring Music TheoryKatygian
Scale 3699Scale 3699: Aeolylyllic, Ian Ring Music TheoryAeolylyllic
Scale 3891Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
Scale 4019Scale 4019: Lonygic, Ian Ring Music TheoryLonygic
Scale 3251Scale 3251: Mela Hatakambari, Ian Ring Music TheoryMela Hatakambari
Scale 3507Scale 3507: Ponyllic, Ian Ring Music TheoryPonyllic
Scale 2739Scale 2739: Mela Suryakanta, Ian Ring Music TheoryMela Suryakanta
Scale 1715Scale 1715: Harmonic Minor Inverse, Ian Ring Music TheoryHarmonic Minor Inverse

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.

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