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Scale 2575: "Pumian"

Scale 2575: Pumian, 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
Pumian

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

{0,1,2,3,9,11}

Forte Number

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

6-2

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: 3595

Hemitonia

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

4 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

3 (tricohemitonic)

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.

5

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: 95

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.

[1, 1, 1, 6, 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.

<4, 4, 3, 2, 1, 1>

Interval Spectrum

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

pm2n3s4d4t

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

Spectra Variation

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

4.667

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.

1.433

Polygon Perimeter

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

5.071

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.

(34, 13, 55)

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
Diminished Triads{9,0,3}000

The following pitch classes are not present in any of the common triads: {1,2,11}

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 2575 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3335
Scale 3335: Vadian, Ian Ring Music TheoryVadian
3rd mode:
Scale 3715
Scale 3715: Xician, Ian Ring Music TheoryXician
4th mode:
Scale 3905
Scale 3905: Yusian, Ian Ring Music TheoryYusian
5th mode:
Scale 125
Scale 125: Atwian, Ian Ring Music TheoryAtwian
6th mode:
Scale 1055
Scale 1055: Gihian, Ian Ring Music TheoryGihian

Prime

The prime form of this scale is Scale 95

Scale 95Scale 95: Arkian, Ian Ring Music TheoryArkian

Complement

The hexatonic modal family [2575, 3335, 3715, 3905, 125, 1055] (Forte: 6-2) is the complement of the hexatonic modal family [95, 1985, 2095, 3095, 3595, 3845] (Forte: 6-2)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2575 is 3595

Scale 3595Scale 3595: Wihian, Ian Ring Music TheoryWihian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2575 is chiral, and its enantiomorph is scale 3595

Scale 3595Scale 3595: Wihian, Ian Ring Music TheoryWihian

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> 2575       T0I <11,0> 3595
T1 <1,1> 1055      T1I <11,1> 3095
T2 <1,2> 2110      T2I <11,2> 2095
T3 <1,3> 125      T3I <11,3> 95
T4 <1,4> 250      T4I <11,4> 190
T5 <1,5> 500      T5I <11,5> 380
T6 <1,6> 1000      T6I <11,6> 760
T7 <1,7> 2000      T7I <11,7> 1520
T8 <1,8> 4000      T8I <11,8> 3040
T9 <1,9> 3905      T9I <11,9> 1985
T10 <1,10> 3715      T10I <11,10> 3970
T11 <1,11> 3335      T11I <11,11> 3845
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1705      T0MI <7,0> 685
T1M <5,1> 3410      T1MI <7,1> 1370
T2M <5,2> 2725      T2MI <7,2> 2740
T3M <5,3> 1355      T3MI <7,3> 1385
T4M <5,4> 2710      T4MI <7,4> 2770
T5M <5,5> 1325      T5MI <7,5> 1445
T6M <5,6> 2650      T6MI <7,6> 2890
T7M <5,7> 1205      T7MI <7,7> 1685
T8M <5,8> 2410      T8MI <7,8> 3370
T9M <5,9> 725      T9MI <7,9> 2645
T10M <5,10> 1450      T10MI <7,10> 1195
T11M <5,11> 2900      T11MI <7,11> 2390

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 2573Scale 2573: Pulian, Ian Ring Music TheoryPulian
Scale 2571Scale 2571: Pukian, Ian Ring Music TheoryPukian
Scale 2567Scale 2567: Puhian, Ian Ring Music TheoryPuhian
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 2591Scale 2591: Puwian, Ian Ring Music TheoryPuwian
Scale 2607Scale 2607: Aerolian, Ian Ring Music TheoryAerolian
Scale 2639Scale 2639: Dothian, Ian Ring Music TheoryDothian
Scale 2703Scale 2703: Galian, Ian Ring Music TheoryGalian
Scale 2831Scale 2831: Ruqian, Ian Ring Music TheoryRuqian
Scale 2063Scale 2063: Pentatonic Chromatic 2, Ian Ring Music TheoryPentatonic Chromatic 2
Scale 2319Scale 2319: Oduian, Ian Ring Music TheoryOduian
Scale 3087Scale 3087: Hexatonic Chromatic 3, Ian Ring Music TheoryHexatonic Chromatic 3
Scale 3599Scale 3599: Heptatonic Chromatic 4, Ian Ring Music TheoryHeptatonic Chromatic 4
Scale 527Scale 527: Dedian, Ian Ring Music TheoryDedian
Scale 1551Scale 1551: Jorian, Ian Ring Music TheoryJorian

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.