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Scale 789: "Zogitonic"

Scale 789: Zogitonic, 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
Zogitonic

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,2,4,8,9}

Forte Number

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

5-30

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: 1305

Hemitonia

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

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

4

Prime Form

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

no
prime: 339

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.

[2, 2, 4, 1, 3] 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.

<1, 2, 1, 3, 2, 1>

Interval Spectrum

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

p2m3ns2dt

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> = {1,2,3,4}
<2> = {4,5,6}
<3> = {6,7,8}
<4> = {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.

2.049

Polygon Perimeter

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

5.664

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
Minor Triadsam{9,0,4}110.5
Augmented TriadsC+{0,4,8}110.5

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

Parsimonious Voice Leading Between Common Triads of Scale 789. Created by Ian Ring ©2019 C+ C+ am am C+->am

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 789 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1221
Scale 1221: Epyritonic, Ian Ring Music TheoryEpyritonic
3rd mode:
Scale 1329
Scale 1329: Epygitonic, Ian Ring Music TheoryEpygitonic
4th mode:
Scale 339
Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonicThis is the prime mode
5th mode:
Scale 2217
Scale 2217: Kagitonic, Ian Ring Music TheoryKagitonic

Prime

The prime form of this scale is Scale 339

Scale 339Scale 339: Zaptitonic, Ian Ring Music TheoryZaptitonic

Complement

The pentatonic modal family [789, 1221, 1329, 339, 2217] (Forte: 5-30) is the complement of the heptatonic modal family [855, 1395, 1485, 1845, 2475, 2745, 3285] (Forte: 7-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 789 is 1305

Scale 1305Scale 1305: Dynitonic, Ian Ring Music TheoryDynitonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 789 is chiral, and its enantiomorph is scale 1305

Scale 1305Scale 1305: Dynitonic, Ian Ring Music TheoryDynitonic

Transformations:

T0 789  T0I 1305
T1 1578  T1I 2610
T2 3156  T2I 1125
T3 2217  T3I 2250
T4 339  T4I 405
T5 678  T5I 810
T6 1356  T6I 1620
T7 2712  T7I 3240
T8 1329  T8I 2385
T9 2658  T9I 675
T10 1221  T10I 1350
T11 2442  T11I 2700

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 791Scale 791: Aeoloptimic, Ian Ring Music TheoryAeoloptimic
Scale 785Scale 785: Aeoloric, Ian Ring Music TheoryAeoloric
Scale 787Scale 787: Aeolapritonic, Ian Ring Music TheoryAeolapritonic
Scale 793Scale 793: Mocritonic, Ian Ring Music TheoryMocritonic
Scale 797Scale 797: Katocrimic, Ian Ring Music TheoryKatocrimic
Scale 773Scale 773, Ian Ring Music Theory
Scale 781Scale 781, Ian Ring Music Theory
Scale 805Scale 805: Rothitonic, Ian Ring Music TheoryRothitonic
Scale 821Scale 821: Aeranimic, Ian Ring Music TheoryAeranimic
Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 917Scale 917: Dygimic, Ian Ring Music TheoryDygimic
Scale 533Scale 533, Ian Ring Music Theory
Scale 661Scale 661: Major Pentatonic, Ian Ring Music TheoryMajor Pentatonic
Scale 277Scale 277: Mixolyric, Ian Ring Music TheoryMixolyric
Scale 1301Scale 1301: Koditonic, Ian Ring Music TheoryKoditonic
Scale 1813Scale 1813: Katothimic, Ian Ring Music TheoryKatothimic
Scale 2837Scale 2837: Aelothimic, Ian Ring Music TheoryAelothimic

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.