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Scale 2449: "Zacritonic"

Scale 2449: Zacritonic, 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
Zacritonic
Dozenal
Pamian

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

{0,4,7,8,11}

Forte Number

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

5-21

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: 307

Hemitonia

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

2 (dihemitonic)

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: 307

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.

[4, 3, 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.

<2, 0, 2, 4, 2, 0>

Interval Spectrum

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

p2m4n2d2

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

Spectra Variation

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

2.4

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.

1.933

Polygon Perimeter

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

5.596

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.

(1, 8, 30)

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}221
E{4,8,11}221
Minor Triadsem{4,7,11}221
Augmented TriadsC+{0,4,8}221
Parsimonious Voice Leading Between Common Triads of Scale 2449. Created by Ian Ring ©2019 C C C+ C+ C->C+ em em C->em E E C+->E em->E

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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2449 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 409
Scale 409: Laritonic, Ian Ring Music TheoryLaritonic
3rd mode:
Scale 563
Scale 563: Thacritonic, Ian Ring Music TheoryThacritonic
4th mode:
Scale 2329
Scale 2329: Styditonic, Ian Ring Music TheoryStyditonic
5th mode:
Scale 803
Scale 803: Loritonic, Ian Ring Music TheoryLoritonic

Prime

The prime form of this scale is Scale 307

Scale 307Scale 307: Raga Megharanjani, Ian Ring Music TheoryRaga Megharanjani

Complement

The pentatonic modal family [2449, 409, 563, 2329, 803] (Forte: 5-21) is the complement of the heptatonic modal family [823, 883, 1843, 2459, 2489, 2969, 3277] (Forte: 7-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2449 is 307

Scale 307Scale 307: Raga Megharanjani, Ian Ring Music TheoryRaga Megharanjani

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2449 is chiral, and its enantiomorph is scale 307

Scale 307Scale 307: Raga Megharanjani, Ian Ring Music TheoryRaga Megharanjani

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> 2449       T0I <11,0> 307
T1 <1,1> 803      T1I <11,1> 614
T2 <1,2> 1606      T2I <11,2> 1228
T3 <1,3> 3212      T3I <11,3> 2456
T4 <1,4> 2329      T4I <11,4> 817
T5 <1,5> 563      T5I <11,5> 1634
T6 <1,6> 1126      T6I <11,6> 3268
T7 <1,7> 2252      T7I <11,7> 2441
T8 <1,8> 409      T8I <11,8> 787
T9 <1,9> 818      T9I <11,9> 1574
T10 <1,10> 1636      T10I <11,10> 3148
T11 <1,11> 3272      T11I <11,11> 2201
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2449       T0MI <7,0> 307
T1M <5,1> 803      T1MI <7,1> 614
T2M <5,2> 1606      T2MI <7,2> 1228
T3M <5,3> 3212      T3MI <7,3> 2456
T4M <5,4> 2329      T4MI <7,4> 817
T5M <5,5> 563      T5MI <7,5> 1634
T6M <5,6> 1126      T6MI <7,6> 3268
T7M <5,7> 2252      T7MI <7,7> 2441
T8M <5,8> 409      T8MI <7,8> 787
T9M <5,9> 818      T9MI <7,9> 1574
T10M <5,10> 1636      T10MI <7,10> 3148
T11M <5,11> 3272      T11MI <7,11> 2201

The transformations that map this set to itself are: T0, T0M

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 2451Scale 2451: Raga Bauli, Ian Ring Music TheoryRaga Bauli
Scale 2453Scale 2453: Raga Latika, Ian Ring Music TheoryRaga Latika
Scale 2457Scale 2457: Augmented, Ian Ring Music TheoryAugmented
Scale 2433Scale 2433: Pacian, Ian Ring Music TheoryPacian
Scale 2441Scale 2441: Kyritonic, Ian Ring Music TheoryKyritonic
Scale 2465Scale 2465: Raga Devaranjani, Ian Ring Music TheoryRaga Devaranjani
Scale 2481Scale 2481: Raga Paraju, Ian Ring Music TheoryRaga Paraju
Scale 2513Scale 2513: Aerycrimic, Ian Ring Music TheoryAerycrimic
Scale 2321Scale 2321: Zyphic, Ian Ring Music TheoryZyphic
Scale 2385Scale 2385: Aeolanitonic, Ian Ring Music TheoryAeolanitonic
Scale 2193Scale 2193: Major Seventh, Ian Ring Music TheoryMajor Seventh
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 401Scale 401: Epogic, Ian Ring Music TheoryEpogic
Scale 1425Scale 1425: Ryphitonic, Ian Ring Music TheoryRyphitonic

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.