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Scale 2565: "Pogian"

Scale 2565: Pogian, 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
Pogian

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

{0,2,9,11}

Forte Number

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

4-10

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.

[5.5]

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.

no

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: 45

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, 7, 2, 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.

<1, 2, 2, 0, 1, 0>

Interval Spectrum

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

pn2s2d

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

Spectra Variation

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

4.5

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.

0.866

Polygon Perimeter

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

4.449

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.

[11]

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.

(5, 0, 14)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 2565 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 1665
Scale 1665: Kejian, Ian Ring Music TheoryKejian
3rd mode:
Scale 45
Scale 45: Aprian, Ian Ring Music TheoryAprianThis is the prime mode
4th mode:
Scale 1035
Scale 1035: Givian, Ian Ring Music TheoryGivian

Prime

The prime form of this scale is Scale 45

Scale 45Scale 45: Aprian, Ian Ring Music TheoryAprian

Complement

The tetratonic modal family [2565, 1665, 45, 1035] (Forte: 4-10) is the complement of the octatonic modal family [765, 1215, 2025, 2655, 3375, 3735, 3915, 4005] (Forte: 8-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2565 is 1035

Scale 1035Scale 1035: Givian, Ian Ring Music TheoryGivian

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> 2565       T0I <11,0> 1035
T1 <1,1> 1035      T1I <11,1> 2070
T2 <1,2> 2070      T2I <11,2> 45
T3 <1,3> 45      T3I <11,3> 90
T4 <1,4> 90      T4I <11,4> 180
T5 <1,5> 180      T5I <11,5> 360
T6 <1,6> 360      T6I <11,6> 720
T7 <1,7> 720      T7I <11,7> 1440
T8 <1,8> 1440      T8I <11,8> 2880
T9 <1,9> 2880      T9I <11,9> 1665
T10 <1,10> 1665      T10I <11,10> 3330
T11 <1,11> 3330      T11I <11,11> 2565
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1665      T0MI <7,0> 45
T1M <5,1> 3330      T1MI <7,1> 90
T2M <5,2> 2565       T2MI <7,2> 180
T3M <5,3> 1035      T3MI <7,3> 360
T4M <5,4> 2070      T4MI <7,4> 720
T5M <5,5> 45      T5MI <7,5> 1440
T6M <5,6> 90      T6MI <7,6> 2880
T7M <5,7> 180      T7MI <7,7> 1665
T8M <5,8> 360      T8MI <7,8> 3330
T9M <5,9> 720      T9MI <7,9> 2565
T10M <5,10> 1440      T10MI <7,10> 1035
T11M <5,11> 2880      T11MI <7,11> 2070

The transformations that map this set to itself are: T0, T11I, T2M, T9MI

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 2567Scale 2567: Puhian, Ian Ring Music TheoryPuhian
Scale 2561Scale 2561: Podian, Ian Ring Music TheoryPodian
Scale 2563Scale 2563: Pofian, Ian Ring Music TheoryPofian
Scale 2569Scale 2569: Pujian, Ian Ring Music TheoryPujian
Scale 2573Scale 2573: Pulian, Ian Ring Music TheoryPulian
Scale 2581Scale 2581: Raga Neroshta, Ian Ring Music TheoryRaga Neroshta
Scale 2597Scale 2597: Raga Rasranjani, Ian Ring Music TheoryRaga Rasranjani
Scale 2629Scale 2629: Raga Shubravarni, Ian Ring Music TheoryRaga Shubravarni
Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian
Scale 2821Scale 2821: Rukian, Ian Ring Music TheoryRukian
Scale 2053Scale 2053: Powian, Ian Ring Music TheoryPowian
Scale 2309Scale 2309: Ocuian, Ian Ring Music TheoryOcuian
Scale 3077Scale 3077: Tekian, Ian Ring Music TheoryTekian
Scale 3589Scale 3589: Widian, Ian Ring Music TheoryWidian
Scale 517Scale 517: Aluian, Ian Ring Music TheoryAluian
Scale 1541Scale 1541: Jilian, Ian Ring Music TheoryJilian

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.