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Scale 2445: "Zadimic"

Scale 2445: Zadimic, 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
Zadimic
Dozenal
Pakian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,2,3,7,8,11}

Forte Number

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

6-Z19

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: 1587

Hemitonia

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

3 (trihemitonic)

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.

5

Prime Form

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

no
prime: 411

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.

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

<3, 1, 3, 4, 3, 1>

Interval Spectrum

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

p3m4n3sd3t

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

Spectra Variation

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

2.333

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.

2.116

Polygon Perimeter

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

5.699

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

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.

(8, 18, 62)

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 TriadsG{7,11,2}231.5
G♯{8,0,3}231.5
Minor Triadscm{0,3,7}231.5
g♯m{8,11,3}321.17
Augmented TriadsD♯+{3,7,11}321.17
Diminished Triadsg♯°{8,11,2}231.5
Parsimonious Voice Leading Between Common Triads of Scale 2445. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# Parsimonious Voice Leading Between Common Triads of Scale 2445. Created by Ian Ring ©2019 G D#+->G g#m g#m D#+->g#m g#° g#° G->g#° g#°->g#m g#m->G#

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.

Diameter3
Radius2
Self-Centeredno
Central VerticesD♯+, g♯m
Peripheral Verticescm, G, g♯°, G♯

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There are 2 ways that this hexatonic scale can be split into two common triads.


Minor: {0, 3, 7}
Diminished: {8, 11, 2}

Major: {7, 11, 2}
Major: {8, 0, 3}

Modes

Modes are the rotational transformation of this scale. Scale 2445 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1635
Scale 1635: Sygimic, Ian Ring Music TheorySygimic
3rd mode:
Scale 2865
Scale 2865: Solimic, Ian Ring Music TheorySolimic
4th mode:
Scale 435
Scale 435: Raga Purna Pancama, Ian Ring Music TheoryRaga Purna Pancama
5th mode:
Scale 2265
Scale 2265: Raga Rasamanjari, Ian Ring Music TheoryRaga Rasamanjari
6th mode:
Scale 795
Scale 795: Aeologimic, Ian Ring Music TheoryAeologimic

Prime

The prime form of this scale is Scale 411

Scale 411Scale 411: Lygimic, Ian Ring Music TheoryLygimic

Complement

The hexatonic modal family [2445, 1635, 2865, 435, 2265, 795] (Forte: 6-Z19) is the complement of the hexatonic modal family [615, 825, 915, 2355, 2505, 3225] (Forte: 6-Z44)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2445 is 1587

Scale 1587Scale 1587: Raga Rudra Pancama, Ian Ring Music TheoryRaga Rudra Pancama

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2445 is chiral, and its enantiomorph is scale 1587

Scale 1587Scale 1587: Raga Rudra Pancama, Ian Ring Music TheoryRaga Rudra Pancama

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> 2445       T0I <11,0> 1587
T1 <1,1> 795      T1I <11,1> 3174
T2 <1,2> 1590      T2I <11,2> 2253
T3 <1,3> 3180      T3I <11,3> 411
T4 <1,4> 2265      T4I <11,4> 822
T5 <1,5> 435      T5I <11,5> 1644
T6 <1,6> 870      T6I <11,6> 3288
T7 <1,7> 1740      T7I <11,7> 2481
T8 <1,8> 3480      T8I <11,8> 867
T9 <1,9> 2865      T9I <11,9> 1734
T10 <1,10> 1635      T10I <11,10> 3468
T11 <1,11> 3270      T11I <11,11> 2841
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3225      T0MI <7,0> 807
T1M <5,1> 2355      T1MI <7,1> 1614
T2M <5,2> 615      T2MI <7,2> 3228
T3M <5,3> 1230      T3MI <7,3> 2361
T4M <5,4> 2460      T4MI <7,4> 627
T5M <5,5> 825      T5MI <7,5> 1254
T6M <5,6> 1650      T6MI <7,6> 2508
T7M <5,7> 3300      T7MI <7,7> 921
T8M <5,8> 2505      T8MI <7,8> 1842
T9M <5,9> 915      T9MI <7,9> 3684
T10M <5,10> 1830      T10MI <7,10> 3273
T11M <5,11> 3660      T11MI <7,11> 2451

The transformations that map this set to itself are: T0

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 2447Scale 2447: Thagian, Ian Ring Music TheoryThagian
Scale 2441Scale 2441: Kyritonic, Ian Ring Music TheoryKyritonic
Scale 2443Scale 2443: Panimic, Ian Ring Music TheoryPanimic
Scale 2437Scale 2437: Pafian, Ian Ring Music TheoryPafian
Scale 2453Scale 2453: Raga Latika, Ian Ring Music TheoryRaga Latika
Scale 2461Scale 2461: Sagian, Ian Ring Music TheorySagian
Scale 2477Scale 2477: Harmonic Minor, Ian Ring Music TheoryHarmonic Minor
Scale 2509Scale 2509: Double Harmonic Minor, Ian Ring Music TheoryDouble Harmonic Minor
Scale 2317Scale 2317: Odoian, Ian Ring Music TheoryOdoian
Scale 2381Scale 2381: Takemitsu Linea Mode 1, Ian Ring Music TheoryTakemitsu Linea Mode 1
Scale 2189Scale 2189: Zagitonic, Ian Ring Music TheoryZagitonic
Scale 2701Scale 2701: Hawaiian, Ian Ring Music TheoryHawaiian
Scale 2957Scale 2957: Thygian, Ian Ring Music TheoryThygian
Scale 3469Scale 3469: Monian, Ian Ring Music TheoryMonian
Scale 397Scale 397: Aeolian Pentatonic, Ian Ring Music TheoryAeolian Pentatonic
Scale 1421Scale 1421: Raga Trimurti, Ian Ring Music TheoryRaga Trimurti

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.