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Scale 927: "Gaptyllic"

Scale 927: Gaptyllic, 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
Gaptyllic
Dozenal
Fuhian

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,1,2,3,4,7,8,9}

Forte Number

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

8-8

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.

[2]

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.

6 (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.

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.

7

Prime Form

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

yes

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, 1, 1, 1, 3, 1, 1, 3]

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, 4, 4, 5, 6, 3>

Interval Spectrum

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

p6m5n4s4d6t3

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

Spectra Variation

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

2

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.

2.5

Polygon Perimeter

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

5.934

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.

[4]

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.

(44, 16, 93)

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{0,4,7}331.67
G♯{8,0,3}331.67
A{9,1,4}242
Minor Triadscm{0,3,7}242
c♯m{1,4,8}331.67
am{9,0,4}331.67
Augmented TriadsC+{0,4,8}421.33
Diminished Triadsc♯°{1,4,7}242
{9,0,3}242

The following pitch classes are not present in any of the common triads: {2}

Parsimonious Voice Leading Between Common Triads of Scale 927. Created by Ian Ring ©2019 cm cm C C cm->C G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° c#m c#m C+->c#m C+->G# am am C+->am c#°->c#m A A c#m->A G#->a° a°->am am->A

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+
Peripheral Verticescm, c♯°, a°, A

Modes

Modes are the rotational transformation of this scale. Scale 927 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2511
Scale 2511: Aeroptyllic, Ian Ring Music TheoryAeroptyllic
3rd mode:
Scale 3303
Scale 3303: Mylyllic, Ian Ring Music TheoryMylyllic
4th mode:
Scale 3699
Scale 3699: Galyllic, Ian Ring Music TheoryGalyllic
5th mode:
Scale 3897
Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic
6th mode:
Scale 999
Scale 999: Ionodyllic, Ian Ring Music TheoryIonodyllic
7th mode:
Scale 2547
Scale 2547: Raga Ramkali, Ian Ring Music TheoryRaga Ramkali
8th mode:
Scale 3321
Scale 3321: Epagyllic, Ian Ring Music TheoryEpagyllic

Prime

This is the prime form of this scale.

Complement

The octatonic modal family [927, 2511, 3303, 3699, 3897, 999, 2547, 3321] (Forte: 8-8) is the complement of the tetratonic modal family [99, 387, 2097, 2241] (Forte: 4-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 927 is 3897

Scale 3897Scale 3897: Kalyllic, Ian Ring Music TheoryKalyllic

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> 927       T0I <11,0> 3897
T1 <1,1> 1854      T1I <11,1> 3699
T2 <1,2> 3708      T2I <11,2> 3303
T3 <1,3> 3321      T3I <11,3> 2511
T4 <1,4> 2547      T4I <11,4> 927
T5 <1,5> 999      T5I <11,5> 1854
T6 <1,6> 1998      T6I <11,6> 3708
T7 <1,7> 3996      T7I <11,7> 3321
T8 <1,8> 3897      T8I <11,8> 2547
T9 <1,9> 3699      T9I <11,9> 999
T10 <1,10> 3303      T10I <11,10> 1998
T11 <1,11> 2511      T11I <11,11> 3996
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3897      T0MI <7,0> 927
T1M <5,1> 3699      T1MI <7,1> 1854
T2M <5,2> 3303      T2MI <7,2> 3708
T3M <5,3> 2511      T3MI <7,3> 3321
T4M <5,4> 927       T4MI <7,4> 2547
T5M <5,5> 1854      T5MI <7,5> 999
T6M <5,6> 3708      T6MI <7,6> 1998
T7M <5,7> 3321      T7MI <7,7> 3996
T8M <5,8> 2547      T8MI <7,8> 3897
T9M <5,9> 999      T9MI <7,9> 3699
T10M <5,10> 1998      T10MI <7,10> 3303
T11M <5,11> 3996      T11MI <7,11> 2511

The transformations that map this set to itself are: T0, T4I, T4M, T0MI

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 925Scale 925: Chromatic Hypodorian, Ian Ring Music TheoryChromatic Hypodorian
Scale 923Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian
Scale 919Scale 919: Chromatic Phrygian Inverse, Ian Ring Music TheoryChromatic Phrygian Inverse
Scale 911Scale 911: Radian, Ian Ring Music TheoryRadian
Scale 943Scale 943: Aerygyllic, Ian Ring Music TheoryAerygyllic
Scale 959Scale 959: Katylygic, Ian Ring Music TheoryKatylygic
Scale 991Scale 991: Aeolygic, Ian Ring Music TheoryAeolygic
Scale 799Scale 799: Lolian, Ian Ring Music TheoryLolian
Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic
Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian
Scale 415Scale 415: Aeoladian, Ian Ring Music TheoryAeoladian
Scale 1439Scale 1439: Rolyllic, Ian Ring Music TheoryRolyllic
Scale 1951Scale 1951: Marygic, Ian Ring Music TheoryMarygic
Scale 2975Scale 2975: Aeroptygic, Ian Ring Music TheoryAeroptygic

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.