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Scale 2887: "Gaptian"

Scale 2887: Gaptian, 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
Gaptian

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

{0,1,2,6,8,9,11}

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-Z36

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: 3163

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

3

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: 367

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, 1, 2, 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.

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

Interval Spectrum

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

p4m3n4s4d4t2

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

Spectra Variation

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

3.143

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.

2.299

Polygon Perimeter

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

5.803

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

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}221.2
Minor Triadsf♯m{6,9,1}231.4
bm{11,2,6}231.4
Diminished Triadsf♯°{6,9,0}142
g♯°{8,11,2}142
Parsimonious Voice Leading Between Common Triads of Scale 2887. Created by Ian Ring ©2019 D D f#m f#m D->f#m bm bm D->bm f#° f#° f#°->f#m g#° g#° g#°->bm

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 VerticesD
Peripheral Verticesf♯°, g♯°

Modes

Modes are the rotational transformation of this scale. Scale 2887 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3491
Scale 3491: Tharian, Ian Ring Music TheoryTharian
3rd mode:
Scale 3793
Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
4th mode:
Scale 493
Scale 493: Rygian, Ian Ring Music TheoryRygian
5th mode:
Scale 1147
Scale 1147: Epynian, Ian Ring Music TheoryEpynian
6th mode:
Scale 2621
Scale 2621: Ionogian, Ian Ring Music TheoryIonogian
7th mode:
Scale 1679
Scale 1679: Kydian, Ian Ring Music TheoryKydian

Prime

The prime form of this scale is Scale 367

Scale 367Scale 367: Aerodian, Ian Ring Music TheoryAerodian

Complement

The heptatonic modal family [2887, 3491, 3793, 493, 1147, 2621, 1679] (Forte: 7-Z36) is the complement of the pentatonic modal family [151, 737, 1801, 2123, 3109] (Forte: 5-Z36)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2887 is 3163

Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2887 is chiral, and its enantiomorph is scale 3163

Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian

Transformations:

T0 2887  T0I 3163
T1 1679  T1I 2231
T2 3358  T2I 367
T3 2621  T3I 734
T4 1147  T4I 1468
T5 2294  T5I 2936
T6 493  T6I 1777
T7 986  T7I 3554
T8 1972  T8I 3013
T9 3944  T9I 1931
T10 3793  T10I 3862
T11 3491  T11I 3629

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 2885Scale 2885: Byrimic, Ian Ring Music TheoryByrimic
Scale 2883Scale 2883, Ian Ring Music Theory
Scale 2891Scale 2891: Phrogian, Ian Ring Music TheoryPhrogian
Scale 2895Scale 2895: Aeoryllic, Ian Ring Music TheoryAeoryllic
Scale 2903Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic
Scale 2919Scale 2919: Molyllic, Ian Ring Music TheoryMolyllic
Scale 2823Scale 2823, Ian Ring Music Theory
Scale 2855Scale 2855: Epocrain, Ian Ring Music TheoryEpocrain
Scale 2951Scale 2951, Ian Ring Music Theory
Scale 3015Scale 3015: Laptyllic, Ian Ring Music TheoryLaptyllic
Scale 2631Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
Scale 2759Scale 2759: Mela Pavani, Ian Ring Music TheoryMela Pavani
Scale 2375Scale 2375: Aeolaptimic, Ian Ring Music TheoryAeolaptimic
Scale 3399Scale 3399: Zonian, Ian Ring Music TheoryZonian
Scale 3911Scale 3911: Katyryllic, Ian Ring Music TheoryKatyryllic
Scale 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 1863Scale 1863: Pycrian, Ian Ring Music TheoryPycrian

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.