The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3875: "Aeryptian"

Scale 3875: Aeryptian, 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
Aeryptian
Dozenal
Yozian

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

{0,1,5,8,9,10,11}

Forte Number

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

7-3

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: 2207

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.

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.

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.

6

Prime Form

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

no
prime: 319

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.

[1, 4, 3, 1, 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, 4, 4, 4, 3, 1>

Interval Spectrum

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

p3m4n4s4d5t

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

Spectra Variation

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

3.714

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.

2.183

Polygon Perimeter

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

5.734

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.

(67, 23, 86)

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 TriadsC♯{1,5,8}221.33
F{5,9,0}221.33
Minor Triadsfm{5,8,0}331.33
a♯m{10,1,5}142
Augmented TriadsC♯+{1,5,9}331.33
Diminished Triads{5,8,11}142
Parsimonious Voice Leading Between Common Triads of Scale 3875. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F a#m a#m C#+->a#m f°->fm fm->F

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 VerticesC♯, F
Peripheral Verticesf°, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 3875 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3985
Scale 3985: Thadian, Ian Ring Music TheoryThadian
3rd mode:
Scale 505
Scale 505: Sanian, Ian Ring Music TheorySanian
4th mode:
Scale 575
Scale 575: Ionydian, Ian Ring Music TheoryIonydian
5th mode:
Scale 2335
Scale 2335: Epydian, Ian Ring Music TheoryEpydian
6th mode:
Scale 3215
Scale 3215: Katydian, Ian Ring Music TheoryKatydian
7th mode:
Scale 3655
Scale 3655: Mathian, Ian Ring Music TheoryMathian

Prime

The prime form of this scale is Scale 319

Scale 319Scale 319: Epodian, Ian Ring Music TheoryEpodian

Complement

The heptatonic modal family [3875, 3985, 505, 575, 2335, 3215, 3655] (Forte: 7-3) is the complement of the pentatonic modal family [55, 1795, 2075, 2945, 3085] (Forte: 5-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3875 is 2207

Scale 2207Scale 2207: Mygian, Ian Ring Music TheoryMygian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3875 is chiral, and its enantiomorph is scale 2207

Scale 2207Scale 2207: Mygian, Ian Ring Music TheoryMygian

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> 3875       T0I <11,0> 2207
T1 <1,1> 3655      T1I <11,1> 319
T2 <1,2> 3215      T2I <11,2> 638
T3 <1,3> 2335      T3I <11,3> 1276
T4 <1,4> 575      T4I <11,4> 2552
T5 <1,5> 1150      T5I <11,5> 1009
T6 <1,6> 2300      T6I <11,6> 2018
T7 <1,7> 505      T7I <11,7> 4036
T8 <1,8> 1010      T8I <11,8> 3977
T9 <1,9> 2020      T9I <11,9> 3859
T10 <1,10> 4040      T10I <11,10> 3623
T11 <1,11> 3985      T11I <11,11> 3151
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 695      T0MI <7,0> 3497
T1M <5,1> 1390      T1MI <7,1> 2899
T2M <5,2> 2780      T2MI <7,2> 1703
T3M <5,3> 1465      T3MI <7,3> 3406
T4M <5,4> 2930      T4MI <7,4> 2717
T5M <5,5> 1765      T5MI <7,5> 1339
T6M <5,6> 3530      T6MI <7,6> 2678
T7M <5,7> 2965      T7MI <7,7> 1261
T8M <5,8> 1835      T8MI <7,8> 2522
T9M <5,9> 3670      T9MI <7,9> 949
T10M <5,10> 3245      T10MI <7,10> 1898
T11M <5,11> 2395      T11MI <7,11> 3796

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 3873Scale 3873: Yoyian, Ian Ring Music TheoryYoyian
Scale 3877Scale 3877: Thanian, Ian Ring Music TheoryThanian
Scale 3879Scale 3879: Pathyllic, Ian Ring Music TheoryPathyllic
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 3891Scale 3891: Ryryllic, Ian Ring Music TheoryRyryllic
Scale 3843Scale 3843: Hexatonic Chromatic 5, Ian Ring Music TheoryHexatonic Chromatic 5
Scale 3859Scale 3859: Aeolarian, Ian Ring Music TheoryAeolarian
Scale 3907Scale 3907, Ian Ring Music Theory
Scale 3939Scale 3939: Dogyllic, Ian Ring Music TheoryDogyllic
Scale 4003Scale 4003: Sadyllic, Ian Ring Music TheorySadyllic
Scale 3619Scale 3619: Thanimic, Ian Ring Music TheoryThanimic
Scale 3747Scale 3747: Myrian, Ian Ring Music TheoryMyrian
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic
Scale 2851Scale 2851: Katoptimic, Ian Ring Music TheoryKatoptimic
Scale 1827Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic

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.