The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1877: "Aeroptian"

Scale 1877: Aeroptian, 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
Aeroptian
Dozenal
Lijian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,2,4,6,8,9,10}

Forte Number

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

7-33

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.

[3]

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.

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.

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.

5

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.

6

Prime Form

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

no
prime: 1367

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, 2, 2, 1, 1, 2]

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, 6, 2, 6, 2, 3>

Interval Spectrum

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

p2m6n2s6d2t3

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

Spectra Variation

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

1.429

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.

yes

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.

2.665

Polygon Perimeter

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

6.035

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.

[6]

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".

Proper

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.

(0, 35, 80)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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}231.4
Minor Triadsam{9,0,4}231.4
Augmented TriadsC+{0,4,8}142
D+{2,6,10}142
Diminished Triadsf♯°{6,9,0}221.2
Parsimonious Voice Leading Between Common Triads of Scale 1877. Created by Ian Ring ©2019 C+ C+ am am C+->am D D D+ D+ D->D+ f#° f#° D->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.

Diameter4
Radius2
Self-Centeredno
Central Verticesf♯°
Peripheral VerticesC+, D+

Modes

Modes are the rotational transformation of this scale. Scale 1877 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1493
Scale 1493: Lydian Minor, Ian Ring Music TheoryLydian Minor
3rd mode:
Scale 1397
Scale 1397: Major Locrian, Ian Ring Music TheoryMajor Locrian
4th mode:
Scale 1373
Scale 1373: Storian, Ian Ring Music TheoryStorian
5th mode:
Scale 1367
Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone InverseThis is the prime mode
6th mode:
Scale 2731
Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major
7th mode:
Scale 3413
Scale 3413: Leading Whole-tone, Ian Ring Music TheoryLeading Whole-tone

Prime

The prime form of this scale is Scale 1367

Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse

Complement

The heptatonic modal family [1877, 1493, 1397, 1373, 1367, 2731, 3413] (Forte: 7-33) is the complement of the pentatonic modal family [341, 1109, 1301, 1349, 1361] (Forte: 5-33)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1877 is 1373

Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian

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> 1877       T0I <11,0> 1373
T1 <1,1> 3754      T1I <11,1> 2746
T2 <1,2> 3413      T2I <11,2> 1397
T3 <1,3> 2731      T3I <11,3> 2794
T4 <1,4> 1367      T4I <11,4> 1493
T5 <1,5> 2734      T5I <11,5> 2986
T6 <1,6> 1373      T6I <11,6> 1877
T7 <1,7> 2746      T7I <11,7> 3754
T8 <1,8> 1397      T8I <11,8> 3413
T9 <1,9> 2794      T9I <11,9> 2731
T10 <1,10> 1493      T10I <11,10> 1367
T11 <1,11> 2986      T11I <11,11> 2734
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1877       T0MI <7,0> 1373
T1M <5,1> 3754      T1MI <7,1> 2746
T2M <5,2> 3413      T2MI <7,2> 1397
T3M <5,3> 2731      T3MI <7,3> 2794
T4M <5,4> 1367      T4MI <7,4> 1493
T5M <5,5> 2734      T5MI <7,5> 2986
T6M <5,6> 1373      T6MI <7,6> 1877
T7M <5,7> 2746      T7MI <7,7> 3754
T8M <5,8> 1397      T8MI <7,8> 3413
T9M <5,9> 2794      T9MI <7,9> 2731
T10M <5,10> 1493      T10MI <7,10> 1367
T11M <5,11> 2986      T11MI <7,11> 2734

The transformations that map this set to itself are: T0, T6I, T0M, T6MI

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 1879Scale 1879: Mixoryllic, Ian Ring Music TheoryMixoryllic
Scale 1873Scale 1873: Dathimic, Ian Ring Music TheoryDathimic
Scale 1875Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale
Scale 1881Scale 1881: Katorian, Ian Ring Music TheoryKatorian
Scale 1885Scale 1885: Saptyllic, Ian Ring Music TheorySaptyllic
Scale 1861Scale 1861: Phrygimic, Ian Ring Music TheoryPhrygimic
Scale 1869Scale 1869: Katyrian, Ian Ring Music TheoryKatyrian
Scale 1893Scale 1893: Ionylian, Ian Ring Music TheoryIonylian
Scale 1909Scale 1909: Epicryllic, Ian Ring Music TheoryEpicryllic
Scale 1813Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
Scale 1845Scale 1845: Lagian, Ian Ring Music TheoryLagian
Scale 1941Scale 1941: Aeranian, Ian Ring Music TheoryAeranian
Scale 2005Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
Scale 1621Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus
Scale 1749Scale 1749: Acoustic, Ian Ring Music TheoryAcoustic
Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 2901Scale 2901: Lydian Augmented, Ian Ring Music TheoryLydian Augmented
Scale 3925Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic

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.