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Scale 3631: "Gydyllic"

Scale 3631: Gydyllic, 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
Gydyllic

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

{0,1,2,3,5,9,10,11}

Forte Number

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

8-2

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: 3727

Hemitonia

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

6 (multihemitonic)

Cohemitonia

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

5 (multicohemitonic)

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.

no
prime: 383

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, 1, 1, 2, 4, 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.

<6, 6, 5, 5, 4, 2>

Interval Spectrum

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

p4m5n5s6d6t2

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

Spectra Variation

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

3.25

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.

2.366

Polygon Perimeter

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

5.838

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.

(90, 48, 126)

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 TriadsF{5,9,0}241.86
A♯{10,2,5}341.71
Minor Triadsdm{2,5,9}231.57
a♯m{10,1,5}231.57
Augmented TriadsC♯+{1,5,9}331.43
Diminished Triads{9,0,3}152.57
{11,2,5}152.43
Parsimonious Voice Leading Between Common Triads of Scale 3631. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F a#m a#m C#+->a#m A# A# dm->A# F->a° a#m->A# 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 VerticesC♯+, dm, a♯m
Peripheral Verticesa°, b°

Modes

Modes are the rotational transformation of this scale. Scale 3631 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3863
Scale 3863: Eparyllic, Ian Ring Music TheoryEparyllic
3rd mode:
Scale 3979
Scale 3979: Dynyllic, Ian Ring Music TheoryDynyllic
4th mode:
Scale 4037
Scale 4037: Ionyllic, Ian Ring Music TheoryIonyllic
5th mode:
Scale 2033
Scale 2033: Stolyllic, Ian Ring Music TheoryStolyllic
6th mode:
Scale 383
Scale 383: Logyllic, Ian Ring Music TheoryLogyllicThis is the prime mode
7th mode:
Scale 2239
Scale 2239: Dacryllic, Ian Ring Music TheoryDacryllic
8th mode:
Scale 3167
Scale 3167: Thynyllic, Ian Ring Music TheoryThynyllic

Prime

The prime form of this scale is Scale 383

Scale 383Scale 383: Logyllic, Ian Ring Music TheoryLogyllic

Complement

The octatonic modal family [3631, 3863, 3979, 4037, 2033, 383, 2239, 3167] (Forte: 8-2) is the complement of the tetratonic modal family [23, 1793, 2059, 3077] (Forte: 4-2)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3631 is 3727

Scale 3727Scale 3727: Tholyllic, Ian Ring Music TheoryTholyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3631 is chiral, and its enantiomorph is scale 3727

Scale 3727Scale 3727: Tholyllic, Ian Ring Music TheoryTholyllic

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> 3631       T0I <11,0> 3727
T1 <1,1> 3167      T1I <11,1> 3359
T2 <1,2> 2239      T2I <11,2> 2623
T3 <1,3> 383      T3I <11,3> 1151
T4 <1,4> 766      T4I <11,4> 2302
T5 <1,5> 1532      T5I <11,5> 509
T6 <1,6> 3064      T6I <11,6> 1018
T7 <1,7> 2033      T7I <11,7> 2036
T8 <1,8> 4066      T8I <11,8> 4072
T9 <1,9> 4037      T9I <11,9> 4049
T10 <1,10> 3979      T10I <11,10> 4003
T11 <1,11> 3863      T11I <11,11> 3911
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1711      T0MI <7,0> 3757
T1M <5,1> 3422      T1MI <7,1> 3419
T2M <5,2> 2749      T2MI <7,2> 2743
T3M <5,3> 1403      T3MI <7,3> 1391
T4M <5,4> 2806      T4MI <7,4> 2782
T5M <5,5> 1517      T5MI <7,5> 1469
T6M <5,6> 3034      T6MI <7,6> 2938
T7M <5,7> 1973      T7MI <7,7> 1781
T8M <5,8> 3946      T8MI <7,8> 3562
T9M <5,9> 3797      T9MI <7,9> 3029
T10M <5,10> 3499      T10MI <7,10> 1963
T11M <5,11> 2903      T11MI <7,11> 3926

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 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 3627Scale 3627: Kalian, Ian Ring Music TheoryKalian
Scale 3623Scale 3623: Aerocrian, Ian Ring Music TheoryAerocrian
Scale 3639Scale 3639: Paptyllic, Ian Ring Music TheoryPaptyllic
Scale 3647Scale 3647: Nonatonic Chromatic 4, Ian Ring Music TheoryNonatonic Chromatic 4
Scale 3599Scale 3599: Heptatonic Chromatic 4, Ian Ring Music TheoryHeptatonic Chromatic 4
Scale 3615Scale 3615: Octatonic Chromatic 4, Ian Ring Music TheoryOctatonic Chromatic 4
Scale 3663Scale 3663: Sonyllic, Ian Ring Music TheorySonyllic
Scale 3695Scale 3695: Kodygic, Ian Ring Music TheoryKodygic
Scale 3759Scale 3759: Darygic, Ian Ring Music TheoryDarygic
Scale 3887Scale 3887: Phrathygic, Ian Ring Music TheoryPhrathygic
Scale 3119Scale 3119: Tikian, Ian Ring Music TheoryTikian
Scale 3375Scale 3375: Vecian, Ian Ring Music TheoryVecian
Scale 2607Scale 2607: Aerolian, Ian Ring Music TheoryAerolian
Scale 1583Scale 1583: Salian, Ian Ring Music TheorySalian

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.