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Scale 2975: "Gaptygic" 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).

Zeitler
Gaptygic
Dozenal
SOZian

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

{0,1,2,3,4,7,8,9,11}

Forte Number

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

9-4

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: 3899

Hemitonia

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

7 (multihemitonic)

Cohemitonia

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

5 (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.

8

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 959

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, an indicator of maximum hierarchization.

no

Interval Structure

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

[1, 1, 1, 1, 3, 1, 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.

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

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.5, 0, 0, 0.333, 0.5, 0>

Interval Spectrum

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

p7m7n6s6d7t3

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

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.

2.683

Polygon Perimeter

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

6.038

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(41, 103, 190)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.688

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.34

Generator

This scale has no generator.

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

E{4,8,11}332
G{7,11,2}263
G♯{8,0,3}431.87
A{9,1,4}263
c♯m{1,4,8}352.4
em{4,7,11}342.2
g♯m{8,11,3}442.07
am{9,0,4}352.4
D♯+{3,7,11}452.27
g♯°{8,11,2}252.8
{9,0,3}242.47

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.

Diameter 6 3 no E, G♯ G, A

Modes

Modes are the rotational transformation of this scale. Scale 2975 can be rotated to make 8 other scales. The 1st mode is itself.

 2nd mode:Scale 3535 Aeroptygic 3rd mode:Scale 3815 Mylygic 4th mode:Scale 3955 Galygic 5th mode:Scale 4025 Kalygic 6th mode:Scale 1015 Ionodygic 7th mode:Scale 2555 Bythygic 8th mode:Scale 3325 Epygic 9th mode:Scale 1855 Marygic

Prime

The prime form of this scale is Scale 959

 Scale 959 Katylygic

Complement

The enneatonic modal family [2975, 3535, 3815, 3955, 4025, 1015, 2555, 3325, 1855] (Forte: 9-4) is the complement of the tritonic modal family [35, 385, 2065] (Forte: 3-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2975 is 3899

 Scale 3899 Katorygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2975 is chiral, and its enantiomorph is scale 3899

 Scale 3899 Katorygic

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> 2975       T0I <11,0> 3899
T1 <1,1> 1855      T1I <11,1> 3703
T2 <1,2> 3710      T2I <11,2> 3311
T3 <1,3> 3325      T3I <11,3> 2527
T4 <1,4> 2555      T4I <11,4> 959
T5 <1,5> 1015      T5I <11,5> 1918
T6 <1,6> 2030      T6I <11,6> 3836
T7 <1,7> 4060      T7I <11,7> 3577
T8 <1,8> 4025      T8I <11,8> 3059
T9 <1,9> 3955      T9I <11,9> 2023
T10 <1,10> 3815      T10I <11,10> 4046
T11 <1,11> 3535      T11I <11,11> 3997
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4025      T0MI <7,0> 959
T1M <5,1> 3955      T1MI <7,1> 1918
T2M <5,2> 3815      T2MI <7,2> 3836
T3M <5,3> 3535      T3MI <7,3> 3577
T4M <5,4> 2975       T4MI <7,4> 3059
T5M <5,5> 1855      T5MI <7,5> 2023
T6M <5,6> 3710      T6MI <7,6> 4046
T7M <5,7> 3325      T7MI <7,7> 3997
T8M <5,8> 2555      T8MI <7,8> 3899
T9M <5,9> 1015      T9MI <7,9> 3703
T10M <5,10> 2030      T10MI <7,10> 3311
T11M <5,11> 4060      T11MI <7,11> 2527

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

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 2973 Panyllic Scale 2971 Aeolynyllic Scale 2967 Madyllic Scale 2959 Dygyllic Scale 2991 Zanygic Scale 3007 Zyryllian Scale 3039 Godyllian Scale 2847 Phracryllic Scale 2911 Katygic Scale 2719 Zocryllic Scale 2463 Ionathyllic Scale 3487 Byptygic Scale 3999 Decatonic Chromatic 6 Scale 927 Koptyllic Scale 1951 Gonygic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.