The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 857: "Aeolydimic"

Scale 857: Aeolydimic, 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
Aeolydimic
Dozenal
Feqian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,3,4,6,8,9}

Forte Number

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

6-Z28

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.

[0]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

yes

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

no

Hemitonia

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

2 (dihemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

5

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 619

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.

[3, 1, 2, 2, 1, 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.

<2, 2, 4, 3, 2, 2>

Interval Spectrum

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

p2m3n4s2d2t2

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

Spectra Variation

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

2

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.

5.864

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.

[0]

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.

(4, 13, 58)

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 TriadsG♯{8,0,3}331.43
Minor Triadsam{9,0,4}331.43
Augmented TriadsC+{0,4,8}231.57
Diminished Triads{0,3,6}231.57
d♯°{3,6,9}231.71
f♯°{6,9,0}231.57
{9,0,3}231.57
Parsimonious Voice Leading Between Common Triads of Scale 857. Created by Ian Ring ©2019 d#° d#° c°->d#° G# G# c°->G# C+ C+ C+->G# am am C+->am f#° f#° d#°->f#° f#°->am G#->a° a°->am

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.

Diameter3
Radius3
Self-Centeredyes

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Augmented: {0, 4, 8}
Diminished: {3, 6, 9}

Modes

Modes are the rotational transformation of this scale. Scale 857 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 619
Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian HexatonicThis is the prime mode
3rd mode:
Scale 2357
Scale 2357: Raga Sarasanana, Ian Ring Music TheoryRaga Sarasanana
4th mode:
Scale 1613
Scale 1613: Thylimic, Ian Ring Music TheoryThylimic
5th mode:
Scale 1427
Scale 1427: Lolimic, Ian Ring Music TheoryLolimic
6th mode:
Scale 2761
Scale 2761: Dagimic, Ian Ring Music TheoryDagimic

Prime

The prime form of this scale is Scale 619

Scale 619Scale 619: Double-Phrygian Hexatonic, Ian Ring Music TheoryDouble-Phrygian Hexatonic

Complement

The hexatonic modal family [857, 619, 2357, 1613, 1427, 2761] (Forte: 6-Z28) is the complement of the hexatonic modal family [667, 869, 1241, 1619, 2381, 2857] (Forte: 6-Z49)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 857 is itself, because it is a palindromic scale!

Scale 857Scale 857: Aeolydimic, Ian Ring Music TheoryAeolydimic

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> 857       T0I <11,0> 857
T1 <1,1> 1714      T1I <11,1> 1714
T2 <1,2> 3428      T2I <11,2> 3428
T3 <1,3> 2761      T3I <11,3> 2761
T4 <1,4> 1427      T4I <11,4> 1427
T5 <1,5> 2854      T5I <11,5> 2854
T6 <1,6> 1613      T6I <11,6> 1613
T7 <1,7> 3226      T7I <11,7> 3226
T8 <1,8> 2357      T8I <11,8> 2357
T9 <1,9> 619      T9I <11,9> 619
T10 <1,10> 1238      T10I <11,10> 1238
T11 <1,11> 2476      T11I <11,11> 2476
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 857       T0MI <7,0> 857
T1M <5,1> 1714      T1MI <7,1> 1714
T2M <5,2> 3428      T2MI <7,2> 3428
T3M <5,3> 2761      T3MI <7,3> 2761
T4M <5,4> 1427      T4MI <7,4> 1427
T5M <5,5> 2854      T5MI <7,5> 2854
T6M <5,6> 1613      T6MI <7,6> 1613
T7M <5,7> 3226      T7MI <7,7> 3226
T8M <5,8> 2357      T8MI <7,8> 2357
T9M <5,9> 619      T9MI <7,9> 619
T10M <5,10> 1238      T10MI <7,10> 1238
T11M <5,11> 2476      T11MI <7,11> 2476

The transformations that map this set to itself are: T0, T0I, T0M, T0MI

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 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 861Scale 861: Rylian, Ian Ring Music TheoryRylian
Scale 849Scale 849: Aerynitonic, Ian Ring Music TheoryAerynitonic
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 841Scale 841: Phrothitonic, Ian Ring Music TheoryPhrothitonic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 889Scale 889: Borian, Ian Ring Music TheoryBorian
Scale 793Scale 793: Mocritonic, Ian Ring Music TheoryMocritonic
Scale 825Scale 825: Thyptimic, Ian Ring Music TheoryThyptimic
Scale 921Scale 921: Bogimic, Ian Ring Music TheoryBogimic
Scale 985Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
Scale 601Scale 601: Bycritonic, Ian Ring Music TheoryBycritonic
Scale 729Scale 729: Stygimic, Ian Ring Music TheoryStygimic
Scale 345Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic
Scale 1369Scale 1369: Boptimic, Ian Ring Music TheoryBoptimic
Scale 1881Scale 1881: Katorian, Ian Ring Music TheoryKatorian
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1

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.