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Scale 2577: "Punian"

Scale 2577: Punian, 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
Punian

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

{0,4,9,11}

Forte Number

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

4-14

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: 267

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.

2

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: 141

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

Interval Spectrum

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

p2mnsd

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> = {3,5,7,9}
<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, 2, 18)

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: {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 2577 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 417
Scale 417: Copian, Ian Ring Music TheoryCopian
3rd mode:
Scale 141
Scale 141: Babian, Ian Ring Music TheoryBabianThis is the prime mode
4th mode:
Scale 1059
Scale 1059: Gikian, Ian Ring Music TheoryGikian

Prime

The prime form of this scale is Scale 141

Scale 141Scale 141: Babian, Ian Ring Music TheoryBabian

Complement

The tetratonic modal family [2577, 417, 141, 1059] (Forte: 4-14) is the complement of the octatonic modal family [759, 1839, 1977, 2427, 2967, 3261, 3531, 3813] (Forte: 8-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2577 is 267

Scale 267Scale 267: Bobian, Ian Ring Music TheoryBobian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2577 is chiral, and its enantiomorph is scale 267

Scale 267Scale 267: Bobian, Ian Ring Music TheoryBobian

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> 2577       T0I <11,0> 267
T1 <1,1> 1059      T1I <11,1> 534
T2 <1,2> 2118      T2I <11,2> 1068
T3 <1,3> 141      T3I <11,3> 2136
T4 <1,4> 282      T4I <11,4> 177
T5 <1,5> 564      T5I <11,5> 354
T6 <1,6> 1128      T6I <11,6> 708
T7 <1,7> 2256      T7I <11,7> 1416
T8 <1,8> 417      T8I <11,8> 2832
T9 <1,9> 834      T9I <11,9> 1569
T10 <1,10> 1668      T10I <11,10> 3138
T11 <1,11> 3336      T11I <11,11> 2181
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 897      T0MI <7,0> 57
T1M <5,1> 1794      T1MI <7,1> 114
T2M <5,2> 3588      T2MI <7,2> 228
T3M <5,3> 3081      T3MI <7,3> 456
T4M <5,4> 2067      T4MI <7,4> 912
T5M <5,5> 39      T5MI <7,5> 1824
T6M <5,6> 78      T6MI <7,6> 3648
T7M <5,7> 156      T7MI <7,7> 3201
T8M <5,8> 312      T8MI <7,8> 2307
T9M <5,9> 624      T9MI <7,9> 519
T10M <5,10> 1248      T10MI <7,10> 1038
T11M <5,11> 2496      T11MI <7,11> 2076

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 2579Scale 2579: Pupian, Ian Ring Music TheoryPupian
Scale 2581Scale 2581: Raga Neroshta, Ian Ring Music TheoryRaga Neroshta
Scale 2585Scale 2585: Otlian, Ian Ring Music TheoryOtlian
Scale 2561Scale 2561: Podian, Ian Ring Music TheoryPodian
Scale 2569Scale 2569: Pujian, Ian Ring Music TheoryPujian
Scale 2593Scale 2593: Puxian, Ian Ring Music TheoryPuxian
Scale 2609Scale 2609: Raga Bhinna Shadja, Ian Ring Music TheoryRaga Bhinna Shadja
Scale 2641Scale 2641: Raga Hindol, Ian Ring Music TheoryRaga Hindol
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 2833Scale 2833: Dolitonic, Ian Ring Music TheoryDolitonic
Scale 2065Scale 2065: Motian, Ian Ring Music TheoryMotian
Scale 2321Scale 2321: Zyphic, Ian Ring Music TheoryZyphic
Scale 3089Scale 3089: Tirian, Ian Ring Music TheoryTirian
Scale 3601Scale 3601: Wilian, Ian Ring Music TheoryWilian
Scale 529Scale 529: Raga Bilwadala, Ian Ring Music TheoryRaga Bilwadala
Scale 1553Scale 1553: Josian, Ian Ring Music TheoryJosian

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.