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Scale 2939: "Goptygic"

Scale 2939: Goptygic, 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

41161837294116105072918310504116183
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
Goptygic

Analysis

Cardinality

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

9 (enneatonic)

Pitch Class Set

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

{0,1,3,4,5,6,8,9,11}

Forte Number

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

9-11

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: 3035

Hemitonia

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

6 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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.

2

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.

8

Prime Form

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

no
prime: 1775

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, 2, 1, 1, 1, 2, 1, 2, 1] 9

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.

<6, 6, 7, 7, 7, 3>

Interval Spectrum

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

p7m7n7s6d6t3

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

Spectra Variation

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

1.111

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.

2.799

Polygon Perimeter

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

6.106

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".

Proper

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}342.44
E{4,8,11}342.39
F{5,9,0}442.22
G♯{8,0,3}442.28
A{9,1,4}342.44
B{11,3,6}342.67
Minor Triadsc♯m{1,4,8}342.39
fm{5,8,0}442.17
f♯m{6,9,1}342.56
g♯m{8,11,3}342.5
am{9,0,4}442.17
Augmented TriadsC+{0,4,8}542
C♯+{1,5,9}442.33
Diminished Triads{0,3,6}242.72
d♯°{3,6,9}242.72
{5,8,11}242.67
f♯°{6,9,0}242.67
{9,0,3}242.56
Parsimonious Voice Leading Between Common Triads of Scale 2939. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ c#m c#m C+->c#m E E C+->E fm fm C+->fm C+->G# am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F f#m f#m C#+->f#m C#+->A d#° d#° d#°->f#m d#°->B E->f° g#m g#m E->g#m f°->fm fm->F f#° f#° F->f#° F->am f#°->f#m g#m->G# g#m->B G#->a° a°->am am->A

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2939 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3517
Scale 3517: Epocrygic, Ian Ring Music TheoryEpocrygic
3rd mode:
Scale 1903
Scale 1903: Rocrygic, Ian Ring Music TheoryRocrygic
4th mode:
Scale 2999
Scale 2999: Diminishing Nonamode, Ian Ring Music TheoryDiminishing Nonamode
5th mode:
Scale 3547
Scale 3547: Sadygic, Ian Ring Music TheorySadygic
6th mode:
Scale 3821
Scale 3821: Epyrygic, Ian Ring Music TheoryEpyrygic
7th mode:
Scale 1979
Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic
8th mode:
Scale 3037
Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale
9th mode:
Scale 1783
Scale 1783: Youlan Scale, Ian Ring Music TheoryYoulan Scale

Prime

The prime form of this scale is Scale 1775

Scale 1775Scale 1775: Lyrygic, Ian Ring Music TheoryLyrygic

Complement

The enneatonic modal family [2939, 3517, 1903, 2999, 3547, 3821, 1979, 3037, 1783] (Forte: 9-11) is the complement of the tritonic modal family [137, 289, 529] (Forte: 3-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2939 is 3035

Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2939 is chiral, and its enantiomorph is scale 3035

Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic

Transformations:

T0 2939  T0I 3035
T1 1783  T1I 1975
T2 3566  T2I 3950
T3 3037  T3I 3805
T4 1979  T4I 3515
T5 3958  T5I 2935
T6 3821  T6I 1775
T7 3547  T7I 3550
T8 2999  T8I 3005
T9 1903  T9I 1915
T10 3806  T10I 3830
T11 3517  T11I 3565

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 2937Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic
Scale 2941Scale 2941: Laptygic, Ian Ring Music TheoryLaptygic
Scale 2943Scale 2943: Dathyllian, Ian Ring Music TheoryDathyllian
Scale 2931Scale 2931: Zathyllic, Ian Ring Music TheoryZathyllic
Scale 2935Scale 2935: Modygic, Ian Ring Music TheoryModygic
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2907Scale 2907: Magen Abot 2, Ian Ring Music TheoryMagen Abot 2
Scale 2875Scale 2875: Ganyllic, Ian Ring Music TheoryGanyllic
Scale 3003Scale 3003: Genus Chromaticum, Ian Ring Music TheoryGenus Chromaticum
Scale 3067Scale 3067: Goptyllian, Ian Ring Music TheoryGoptyllian
Scale 2683Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
Scale 2811Scale 2811: Barygic, Ian Ring Music TheoryBarygic
Scale 2427Scale 2427: Katoryllic, Ian Ring Music TheoryKatoryllic
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 3963Scale 3963: Aeoryllian, Ian Ring Music TheoryAeoryllian
Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic
Scale 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic

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.