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Scale 3867: "Storyllic"

Scale 3867: Storyllic, 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

41161837294116105072918310504116183
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
Storyllic

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

{0,1,3,4,8,9,10,11}

Forte Number

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

8-4

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: 2847

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.

4 (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.

3

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: 447

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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, 5, 5, 5, 5, 2]

Interval Spectrum

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

p5m5n5s5d6t2

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

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

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 TriadsE{4,8,11}242
G♯{8,0,3}341.78
A{9,1,4}341.89
Minor Triadsc♯m{1,4,8}231.78
g♯m{8,11,3}252.33
am{9,0,4}331.56
Augmented TriadsC+{0,4,8}431.44
Diminished Triads{9,0,3}231.89
a♯°{10,1,4}152.67
Parsimonious Voice Leading Between Common Triads of Scale 3867. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E G# G# C+->G# am am C+->am A A c#m->A g#m g#m E->g#m g#m->G# G#->a° a°->am am->A a#° a#° A->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.

Diameter5
Radius3
Self-Centeredno
Central VerticesC+, c♯m, a°, am
Peripheral Verticesg♯m, a♯°

Modes

Modes are the rotational transformation of this scale. Scale 3867 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3981
Scale 3981: Phrycryllic, Ian Ring Music TheoryPhrycryllic
3rd mode:
Scale 2019
Scale 2019: Palyllic, Ian Ring Music TheoryPalyllic
4th mode:
Scale 3057
Scale 3057: Phroryllic, Ian Ring Music TheoryPhroryllic
5th mode:
Scale 447
Scale 447: Thyphyllic, Ian Ring Music TheoryThyphyllicThis is the prime mode
6th mode:
Scale 2271
Scale 2271: Poptyllic, Ian Ring Music TheoryPoptyllic
7th mode:
Scale 3183
Scale 3183: Mixonyllic, Ian Ring Music TheoryMixonyllic
8th mode:
Scale 3639
Scale 3639: Paptyllic, Ian Ring Music TheoryPaptyllic

Prime

The prime form of this scale is Scale 447

Scale 447Scale 447: Thyphyllic, Ian Ring Music TheoryThyphyllic

Complement

The octatonic modal family [3867, 3981, 2019, 3057, 447, 2271, 3183, 3639] (Forte: 8-4) is the complement of the tetratonic modal family [39, 897, 2067, 3081] (Forte: 4-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3867 is 2847

Scale 2847Scale 2847: Phracryllic, Ian Ring Music TheoryPhracryllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3867 is chiral, and its enantiomorph is scale 2847

Scale 2847Scale 2847: Phracryllic, Ian Ring Music TheoryPhracryllic

Transformations:

T0 3867  T0I 2847
T1 3639  T1I 1599
T2 3183  T2I 3198
T3 2271  T3I 2301
T4 447  T4I 507
T5 894  T5I 1014
T6 1788  T6I 2028
T7 3576  T7I 4056
T8 3057  T8I 4017
T9 2019  T9I 3939
T10 4038  T10I 3783
T11 3981  T11I 3471

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 3865Scale 3865: Starian, Ian Ring Music TheoryStarian
Scale 3869Scale 3869: Bygyllic, Ian Ring Music TheoryBygyllic
Scale 3871Scale 3871: Aerynygic, Ian Ring Music TheoryAerynygic
Scale 3859Scale 3859: Aeolarian, Ian Ring Music TheoryAeolarian
Scale 3863Scale 3863: Eparyllic, Ian Ring Music TheoryEparyllic
Scale 3851Scale 3851, Ian Ring Music Theory
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 3899Scale 3899: Katorygic, Ian Ring Music TheoryKatorygic
Scale 3931Scale 3931: Aerygic, Ian Ring Music TheoryAerygic
Scale 3995Scale 3995: Ionygic, Ian Ring Music TheoryIonygic
Scale 3611Scale 3611, Ian Ring Music Theory
Scale 3739Scale 3739: Epanyllic, Ian Ring Music TheoryEpanyllic
Scale 3355Scale 3355: Bagian, Ian Ring Music TheoryBagian
Scale 2843Scale 2843: Sorian, Ian Ring Music TheorySorian
Scale 1819Scale 1819: Pydian, Ian Ring Music TheoryPydian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.