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Scale 3701: "Bagyllic"

Scale 3701: Bagyllic, 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
Bagyllic
Dozenal
Xetian

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

{0,2,4,5,6,9,10,11}

Forte Number

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

8-16

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: 1487

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: 943

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, 2, 1, 1, 3, 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, 4, 5, 6, 3>

Interval Spectrum

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

p6m5n4s5d5t3

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}
<4> = {5,6,7}
<5> = {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.

1.75

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.

(12, 44, 123)

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 TriadsD{2,6,9}331.67
F{5,9,0}341.89
A♯{10,2,5}331.67
Minor Triadsdm{2,5,9}331.56
am{9,0,4}152.67
bm{11,2,6}252.33
Augmented TriadsD+{2,6,10}341.78
Diminished Triadsf♯°{6,9,0}242
{11,2,5}242.22
Parsimonious Voice Leading Between Common Triads of Scale 3701. Created by Ian Ring ©2019 dm dm D D dm->D F F dm->F A# A# dm->A# D+ D+ D->D+ f#° f#° D->f#° D+->A# bm bm D+->bm F->f#° am am F->am A#->b° b°->bm

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 Verticesdm, D, A♯
Peripheral Verticesam, bm

Modes

Modes are the rotational transformation of this scale. Scale 3701 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1949
Scale 1949: Mathyllic, Ian Ring Music TheoryMathyllic
3rd mode:
Scale 1511
Scale 1511: Styptyllic, Ian Ring Music TheoryStyptyllic
4th mode:
Scale 2803
Scale 2803: Raga Bhatiyar, Ian Ring Music TheoryRaga Bhatiyar
5th mode:
Scale 3449
Scale 3449: Bacryllic, Ian Ring Music TheoryBacryllic
6th mode:
Scale 943
Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllicThis is the prime mode
7th mode:
Scale 2519
Scale 2519: Dathyllic, Ian Ring Music TheoryDathyllic
8th mode:
Scale 3307
Scale 3307: Boptyllic, Ian Ring Music TheoryBoptyllic

Prime

The prime form of this scale is Scale 943

Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic

Complement

The octatonic modal family [3701, 1949, 1511, 2803, 3449, 943, 2519, 3307] (Forte: 8-16) is the complement of the tetratonic modal family [163, 389, 1121, 2129] (Forte: 4-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3701 is 1487

Scale 1487Scale 1487: Mothyllic, Ian Ring Music TheoryMothyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3701 is chiral, and its enantiomorph is scale 1487

Scale 1487Scale 1487: Mothyllic, Ian Ring Music TheoryMothyllic

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> 3701       T0I <11,0> 1487
T1 <1,1> 3307      T1I <11,1> 2974
T2 <1,2> 2519      T2I <11,2> 1853
T3 <1,3> 943      T3I <11,3> 3706
T4 <1,4> 1886      T4I <11,4> 3317
T5 <1,5> 3772      T5I <11,5> 2539
T6 <1,6> 3449      T6I <11,6> 983
T7 <1,7> 2803      T7I <11,7> 1966
T8 <1,8> 1511      T8I <11,8> 3932
T9 <1,9> 3022      T9I <11,9> 3769
T10 <1,10> 1949      T10I <11,10> 3443
T11 <1,11> 3898      T11I <11,11> 2791
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1991      T0MI <7,0> 3197
T1M <5,1> 3982      T1MI <7,1> 2299
T2M <5,2> 3869      T2MI <7,2> 503
T3M <5,3> 3643      T3MI <7,3> 1006
T4M <5,4> 3191      T4MI <7,4> 2012
T5M <5,5> 2287      T5MI <7,5> 4024
T6M <5,6> 479      T6MI <7,6> 3953
T7M <5,7> 958      T7MI <7,7> 3811
T8M <5,8> 1916      T8MI <7,8> 3527
T9M <5,9> 3832      T9MI <7,9> 2959
T10M <5,10> 3569      T10MI <7,10> 1823
T11M <5,11> 3043      T11MI <7,11> 3646

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 3703Scale 3703: Katalygic, Ian Ring Music TheoryKatalygic
Scale 3697Scale 3697: Ionarian, Ian Ring Music TheoryIonarian
Scale 3699Scale 3699: Galyllic, Ian Ring Music TheoryGalyllic
Scale 3705Scale 3705: Messiaen Mode 4 Inverse, Ian Ring Music TheoryMessiaen Mode 4 Inverse
Scale 3709Scale 3709: Katynygic, Ian Ring Music TheoryKatynygic
Scale 3685Scale 3685: Kodian, Ian Ring Music TheoryKodian
Scale 3693Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3637Scale 3637: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
Scale 3765Scale 3765: Dominant Bebop, Ian Ring Music TheoryDominant Bebop
Scale 3829Scale 3829: Taishikicho, Ian Ring Music TheoryTaishikicho
Scale 3957Scale 3957: Porygic, Ian Ring Music TheoryPorygic
Scale 3189Scale 3189: Aeolonian, Ian Ring Music TheoryAeolonian
Scale 3445Scale 3445: Messiaen Mode 6 Inverse, Ian Ring Music TheoryMessiaen Mode 6 Inverse
Scale 2677Scale 2677: Thodian, Ian Ring Music TheoryThodian
Scale 1653Scale 1653: Minor Romani Inverse, Ian Ring Music TheoryMinor Romani 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, 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.