The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 881: "Aerothimic"

Scale 881: Aerothimic, 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
Aerothimic
Dozenal
Droian

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

{0,4,5,6,8,9}

Forte Number

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

6-15

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: 473

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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: 311

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.

[4, 1, 1, 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.

<3, 2, 3, 4, 2, 1>

Interval Spectrum

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

p2m4n3s2d3t

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> = {1,2,3,4}
<2> = {2,3,4,5,7}
<3> = {4,6,8}
<4> = {5,7,8,9,10}
<5> = {8,9,10,11}

Spectra Variation

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

3.333

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.

2.116

Polygon Perimeter

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

5.699

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.

(21, 18, 64)

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}321
Minor Triadsfm{5,8,0}221.2
am{9,0,4}221.2
Augmented TriadsC+{0,4,8}231.4
Diminished Triadsf♯°{6,9,0}131.6
Parsimonious Voice Leading Between Common Triads of Scale 881. Created by Ian Ring ©2019 C+ C+ fm fm C+->fm am am C+->am F F fm->F f#° f#° F->f#° F->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
Radius2
Self-Centeredno
Central Verticesfm, F, am
Peripheral VerticesC+, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 881 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 311
Scale 311: Stagimic, Ian Ring Music TheoryStagimicThis is the prime mode
3rd mode:
Scale 2203
Scale 2203: Dorimic, Ian Ring Music TheoryDorimic
4th mode:
Scale 3149
Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
5th mode:
Scale 1811
Scale 1811: Kyptimic, Ian Ring Music TheoryKyptimic
6th mode:
Scale 2953
Scale 2953: Ionylimic, Ian Ring Music TheoryIonylimic

Prime

The prime form of this scale is Scale 311

Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic

Complement

The hexatonic modal family [881, 311, 2203, 3149, 1811, 2953] (Forte: 6-15) is the complement of the hexatonic modal family [311, 881, 1811, 2203, 2953, 3149] (Forte: 6-15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 881 is 473

Scale 473Scale 473: Aeralimic, Ian Ring Music TheoryAeralimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 881 is chiral, and its enantiomorph is scale 473

Scale 473Scale 473: Aeralimic, Ian Ring Music TheoryAeralimic

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> 881       T0I <11,0> 473
T1 <1,1> 1762      T1I <11,1> 946
T2 <1,2> 3524      T2I <11,2> 1892
T3 <1,3> 2953      T3I <11,3> 3784
T4 <1,4> 1811      T4I <11,4> 3473
T5 <1,5> 3622      T5I <11,5> 2851
T6 <1,6> 3149      T6I <11,6> 1607
T7 <1,7> 2203      T7I <11,7> 3214
T8 <1,8> 311      T8I <11,8> 2333
T9 <1,9> 622      T9I <11,9> 571
T10 <1,10> 1244      T10I <11,10> 1142
T11 <1,11> 2488      T11I <11,11> 2284
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 851      T0MI <7,0> 2393
T1M <5,1> 1702      T1MI <7,1> 691
T2M <5,2> 3404      T2MI <7,2> 1382
T3M <5,3> 2713      T3MI <7,3> 2764
T4M <5,4> 1331      T4MI <7,4> 1433
T5M <5,5> 2662      T5MI <7,5> 2866
T6M <5,6> 1229      T6MI <7,6> 1637
T7M <5,7> 2458      T7MI <7,7> 3274
T8M <5,8> 821      T8MI <7,8> 2453
T9M <5,9> 1642      T9MI <7,9> 811
T10M <5,10> 3284      T10MI <7,10> 1622
T11M <5,11> 2473      T11MI <7,11> 3244

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 883Scale 883: Ralian, Ian Ring Music TheoryRalian
Scale 885Scale 885: Sathian, Ian Ring Music TheorySathian
Scale 889Scale 889: Borian, Ian Ring Music TheoryBorian
Scale 865Scale 865: Jahian, Ian Ring Music TheoryJahian
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 849Scale 849: Aerynitonic, Ian Ring Music TheoryAerynitonic
Scale 817Scale 817: Zothitonic, Ian Ring Music TheoryZothitonic
Scale 945Scale 945: Raga Saravati, Ian Ring Music TheoryRaga Saravati
Scale 1009Scale 1009: Katyptian, Ian Ring Music TheoryKatyptian
Scale 625Scale 625: Ionyptitonic, Ian Ring Music TheoryIonyptitonic
Scale 753Scale 753: Aeronimic, Ian Ring Music TheoryAeronimic
Scale 369Scale 369: Laditonic, Ian Ring Music TheoryLaditonic
Scale 1393Scale 1393: Mycrimic, Ian Ring Music TheoryMycrimic
Scale 1905Scale 1905: Katacrian, Ian Ring Music TheoryKatacrian
Scale 2929Scale 2929: Aeolathian, Ian Ring Music TheoryAeolathian

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.