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Scale 1351: "Aeraptimic"

Scale 1351: Aeraptimic, 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
Aeraptimic

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

{0,1,2,6,8,10}

Forte Number

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

6-22

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: 3157

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.

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: 343

Deep Scale

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

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 4, 2, 2, 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, 4, 1, 4, 2, 2>

Interval Spectrum

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

p2m4ns4d2t2

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

2.232

Polygon Perimeter

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

5.767

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 TriadsF♯{6,10,1}110.5
Augmented TriadsD+{2,6,10}110.5

The following pitch classes are not present in any of the common triads: {0,8}

Parsimonious Voice Leading Between Common Triads of Scale 1351. Created by Ian Ring ©2019 D+ D+ F# F# D+->F#

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1351 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2723
Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
3rd mode:
Scale 3409
Scale 3409: Katanimic, Ian Ring Music TheoryKatanimic
4th mode:
Scale 469
Scale 469: Katyrimic, Ian Ring Music TheoryKatyrimic
5th mode:
Scale 1141
Scale 1141: Rynimic, Ian Ring Music TheoryRynimic
6th mode:
Scale 1309
Scale 1309: Pogimic, Ian Ring Music TheoryPogimic

Prime

The prime form of this scale is Scale 343

Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic

Complement

The hexatonic modal family [1351, 2723, 3409, 469, 1141, 1309] (Forte: 6-22) is the complement of the hexatonic modal family [343, 1393, 1477, 1813, 2219, 3157] (Forte: 6-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1351 is 3157

Scale 3157Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1351 is chiral, and its enantiomorph is scale 3157

Scale 3157Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic

Transformations:

T0 1351  T0I 3157
T1 2702  T1I 2219
T2 1309  T2I 343
T3 2618  T3I 686
T4 1141  T4I 1372
T5 2282  T5I 2744
T6 469  T6I 1393
T7 938  T7I 2786
T8 1876  T8I 1477
T9 3752  T9I 2954
T10 3409  T10I 1813
T11 2723  T11I 3626

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 1349Scale 1349: Tholitonic, Ian Ring Music TheoryTholitonic
Scale 1347Scale 1347, Ian Ring Music Theory
Scale 1355Scale 1355: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 1359Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse
Scale 1383Scale 1383: Pynian, Ian Ring Music TheoryPynian
Scale 1287Scale 1287, Ian Ring Music Theory
Scale 1319Scale 1319: Phronimic, Ian Ring Music TheoryPhronimic
Scale 1415Scale 1415, Ian Ring Music Theory
Scale 1479Scale 1479: Mela Jalarnava, Ian Ring Music TheoryMela Jalarnava
Scale 1095Scale 1095: Phrythitonic, Ian Ring Music TheoryPhrythitonic
Scale 1223Scale 1223: Phryptimic, Ian Ring Music TheoryPhryptimic
Scale 1607Scale 1607: Epytimic, Ian Ring Music TheoryEpytimic
Scale 1863Scale 1863: Pycrian, Ian Ring Music TheoryPycrian
Scale 327Scale 327: Syptitonic, Ian Ring Music TheorySyptitonic
Scale 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 2375Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic
Scale 3399Scale 3399: Zonian, Ian Ring Music TheoryZonian

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. 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.