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Scale 1565: "Jozian"

Scale 1565: Jozian, 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

Dozenal
Jozian

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

{0,2,3,4,9,10}

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-Z12

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: 1805

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.

1 (uncohemitonic)

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: 215

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, 1, 5, 1, 2]

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, 3, 2, 2, 3, 2>

Interval Spectrum

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

p3m2n2s3d3t2

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

Spectra Variation

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

3.333

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.

1.866

Polygon Perimeter

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

5.485

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.

(22, 14, 61)

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
Diminished Triads{9,0,3}110.5

The following pitch classes are not present in any of the common triads: {2,10}

Parsimonious Voice Leading Between Common Triads of Scale 1565. Created by Ian Ring ©2019 am am a°->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 1565 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1415
Scale 1415: Impian, Ian Ring Music TheoryImpian
3rd mode:
Scale 2755
Scale 2755: Rivian, Ian Ring Music TheoryRivian
4th mode:
Scale 3425
Scale 3425: Vihian, Ian Ring Music TheoryVihian
5th mode:
Scale 235
Scale 235: Bihian, Ian Ring Music TheoryBihian
6th mode:
Scale 2165
Scale 2165: Necian, Ian Ring Music TheoryNecian

Prime

The prime form of this scale is Scale 215

Scale 215Scale 215: Bivian, Ian Ring Music TheoryBivian

Complement

The hexatonic modal family [1565, 1415, 2755, 3425, 235, 2165] (Forte: 6-Z12) is the complement of the hexatonic modal family [335, 965, 1265, 2215, 3155, 3625] (Forte: 6-Z41)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1565 is 1805

Scale 1805Scale 1805: Laqian, Ian Ring Music TheoryLaqian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1565 is chiral, and its enantiomorph is scale 1805

Scale 1805Scale 1805: Laqian, Ian Ring Music TheoryLaqian

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> 1565       T0I <11,0> 1805
T1 <1,1> 3130      T1I <11,1> 3610
T2 <1,2> 2165      T2I <11,2> 3125
T3 <1,3> 235      T3I <11,3> 2155
T4 <1,4> 470      T4I <11,4> 215
T5 <1,5> 940      T5I <11,5> 430
T6 <1,6> 1880      T6I <11,6> 860
T7 <1,7> 3760      T7I <11,7> 1720
T8 <1,8> 3425      T8I <11,8> 3440
T9 <1,9> 2755      T9I <11,9> 2785
T10 <1,10> 1415      T10I <11,10> 1475
T11 <1,11> 2830      T11I <11,11> 2950
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1805      T0MI <7,0> 1565
T1M <5,1> 3610      T1MI <7,1> 3130
T2M <5,2> 3125      T2MI <7,2> 2165
T3M <5,3> 2155      T3MI <7,3> 235
T4M <5,4> 215      T4MI <7,4> 470
T5M <5,5> 430      T5MI <7,5> 940
T6M <5,6> 860      T6MI <7,6> 1880
T7M <5,7> 1720      T7MI <7,7> 3760
T8M <5,8> 3440      T8MI <7,8> 3425
T9M <5,9> 2785      T9MI <7,9> 2755
T10M <5,10> 1475      T10MI <7,10> 1415
T11M <5,11> 2950      T11MI <7,11> 2830

The transformations that map this set to itself are: T0, 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 1567Scale 1567: Jobian, Ian Ring Music TheoryJobian
Scale 1561Scale 1561: Joxian, Ian Ring Music TheoryJoxian
Scale 1563Scale 1563: Joyian, Ian Ring Music TheoryJoyian
Scale 1557Scale 1557: Jovian, Ian Ring Music TheoryJovian
Scale 1549Scale 1549: Joqian, Ian Ring Music TheoryJoqian
Scale 1581Scale 1581: Raga Bagesri, Ian Ring Music TheoryRaga Bagesri
Scale 1597Scale 1597: Aeolodian, Ian Ring Music TheoryAeolodian
Scale 1629Scale 1629: Synian, Ian Ring Music TheorySynian
Scale 1693Scale 1693: Dogian, Ian Ring Music TheoryDogian
Scale 1821Scale 1821: Aeradian, Ian Ring Music TheoryAeradian
Scale 1053Scale 1053: Gigian, Ian Ring Music TheoryGigian
Scale 1309Scale 1309: Pogimic, Ian Ring Music TheoryPogimic
Scale 541Scale 541: Demian, Ian Ring Music TheoryDemian
Scale 2589Scale 2589: Puvian, Ian Ring Music TheoryPuvian
Scale 3613Scale 3613: Wosian, Ian Ring Music TheoryWosian

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.