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Scale 1553: "Josian"

Scale 1553: Josian, 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
Josian

Analysis

Cardinality

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

4 (tetratonic)

Pitch Class Set

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

{0,4,9,10}

Forte Number

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

4-Z29

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: 269

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.

3

Prime Form

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

no
prime: 139

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, 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.

<1, 1, 1, 1, 1, 1>

Interval Spectrum

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

pmnsdt

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

Spectra Variation

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

3.5

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.

1.366

Polygon Perimeter

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

5.182

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.

(4, 0, 17)

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}000

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

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 1553 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 353
Scale 353: Cebian, Ian Ring Music TheoryCebian
3rd mode:
Scale 139
Scale 139: Ayoian, Ian Ring Music TheoryAyoianThis is the prime mode
4th mode:
Scale 2117
Scale 2117: Raga Sumukam, Ian Ring Music TheoryRaga Sumukam

Prime

The prime form of this scale is Scale 139

Scale 139Scale 139: Ayoian, Ian Ring Music TheoryAyoian

Complement

The tetratonic modal family [1553, 353, 139, 2117] (Forte: 4-Z29) is the complement of the octatonic modal family [751, 1913, 1943, 2423, 3019, 3259, 3557, 3677] (Forte: 8-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1553 is 269

Scale 269Scale 269: Bocian, Ian Ring Music TheoryBocian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1553 is chiral, and its enantiomorph is scale 269

Scale 269Scale 269: Bocian, Ian Ring Music TheoryBocian

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> 1553       T0I <11,0> 269
T1 <1,1> 3106      T1I <11,1> 538
T2 <1,2> 2117      T2I <11,2> 1076
T3 <1,3> 139      T3I <11,3> 2152
T4 <1,4> 278      T4I <11,4> 209
T5 <1,5> 556      T5I <11,5> 418
T6 <1,6> 1112      T6I <11,6> 836
T7 <1,7> 2224      T7I <11,7> 1672
T8 <1,8> 353      T8I <11,8> 3344
T9 <1,9> 706      T9I <11,9> 2593
T10 <1,10> 1412      T10I <11,10> 1091
T11 <1,11> 2824      T11I <11,11> 2182
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 773      T0MI <7,0> 1049
T1M <5,1> 1546      T1MI <7,1> 2098
T2M <5,2> 3092      T2MI <7,2> 101
T3M <5,3> 2089      T3MI <7,3> 202
T4M <5,4> 83      T4MI <7,4> 404
T5M <5,5> 166      T5MI <7,5> 808
T6M <5,6> 332      T6MI <7,6> 1616
T7M <5,7> 664      T7MI <7,7> 3232
T8M <5,8> 1328      T8MI <7,8> 2369
T9M <5,9> 2656      T9MI <7,9> 643
T10M <5,10> 1217      T10MI <7,10> 1286
T11M <5,11> 2434      T11MI <7,11> 2572

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 1555Scale 1555: Jotian, Ian Ring Music TheoryJotian
Scale 1557Scale 1557: Jovian, Ian Ring Music TheoryJovian
Scale 1561Scale 1561: Joxian, Ian Ring Music TheoryJoxian
Scale 1537Scale 1537: Jijian, Ian Ring Music TheoryJijian
Scale 1545Scale 1545: Jonian, Ian Ring Music TheoryJonian
Scale 1569Scale 1569: Jocian, Ian Ring Music TheoryJocian
Scale 1585Scale 1585: Raga Khamaji Durga, Ian Ring Music TheoryRaga Khamaji Durga
Scale 1617Scale 1617: Phronitonic, Ian Ring Music TheoryPhronitonic
Scale 1681Scale 1681: Raga Valaji, Ian Ring Music TheoryRaga Valaji
Scale 1809Scale 1809: Ranitonic, Ian Ring Music TheoryRanitonic
Scale 1041Scale 1041: Hitian, Ian Ring Music TheoryHitian
Scale 1297Scale 1297: Aeolic, Ian Ring Music TheoryAeolic
Scale 529Scale 529: Raga Bilwadala, Ian Ring Music TheoryRaga Bilwadala
Scale 2577Scale 2577: Punian, Ian Ring Music TheoryPunian
Scale 3601Scale 3601: Wilian, Ian Ring Music TheoryWilian

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.