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Scale 2267: "Padian"

Scale 2267: Padian, 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

Zeitler
Padian
Dozenal
Nunian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

{0,1,3,4,6,7,11}

Forte Number

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

7-Z38

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: 2915

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.

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.

6

Prime Form

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

no
prime: 439

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

Interval Spectrum

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

p4m4n4s3d4t2

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

Spectra Variation

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

2.571

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.

2.299

Polygon Perimeter

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

5.803

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.

(27, 24, 90)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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
Major TriadsC{0,4,7}331.43
B{11,3,6}241.86
Minor Triadscm{0,3,7}321.29
em{4,7,11}231.57
Augmented TriadsD♯+{3,7,11}331.43
Diminished Triads{0,3,6}231.71
c♯°{1,4,7}142.14
Parsimonious Voice Leading Between Common Triads of Scale 2267. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ c#° c#° C->c#° em em C->em D#+->em D#+->B

view full size

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.

Diameter4
Radius2
Self-Centeredno
Central Verticescm
Peripheral Verticesc♯°, B

Modes

Modes are the rotational transformation of this scale. Scale 2267 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3181
Scale 3181: Rolian, Ian Ring Music TheoryRolian
3rd mode:
Scale 1819
Scale 1819: Pydian, Ian Ring Music TheoryPydian
4th mode:
Scale 2957
Scale 2957: Thygian, Ian Ring Music TheoryThygian
5th mode:
Scale 1763
Scale 1763: Katalian, Ian Ring Music TheoryKatalian
6th mode:
Scale 2929
Scale 2929: Aeolathian, Ian Ring Music TheoryAeolathian
7th mode:
Scale 439
Scale 439: Bythian, Ian Ring Music TheoryBythianThis is the prime mode

Prime

The prime form of this scale is Scale 439

Scale 439Scale 439: Bythian, Ian Ring Music TheoryBythian

Complement

The heptatonic modal family [2267, 3181, 1819, 2957, 1763, 2929, 439] (Forte: 7-Z38) is the complement of the pentatonic modal family [295, 625, 905, 2195, 3145] (Forte: 5-Z38)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2267 is 2915

Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2267 is chiral, and its enantiomorph is scale 2915

Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian

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> 2267       T0I <11,0> 2915
T1 <1,1> 439      T1I <11,1> 1735
T2 <1,2> 878      T2I <11,2> 3470
T3 <1,3> 1756      T3I <11,3> 2845
T4 <1,4> 3512      T4I <11,4> 1595
T5 <1,5> 2929      T5I <11,5> 3190
T6 <1,6> 1763      T6I <11,6> 2285
T7 <1,7> 3526      T7I <11,7> 475
T8 <1,8> 2957      T8I <11,8> 950
T9 <1,9> 1819      T9I <11,9> 1900
T10 <1,10> 3638      T10I <11,10> 3800
T11 <1,11> 3181      T11I <11,11> 3505
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2537      T0MI <7,0> 755
T1M <5,1> 979      T1MI <7,1> 1510
T2M <5,2> 1958      T2MI <7,2> 3020
T3M <5,3> 3916      T3MI <7,3> 1945
T4M <5,4> 3737      T4MI <7,4> 3890
T5M <5,5> 3379      T5MI <7,5> 3685
T6M <5,6> 2663      T6MI <7,6> 3275
T7M <5,7> 1231      T7MI <7,7> 2455
T8M <5,8> 2462      T8MI <7,8> 815
T9M <5,9> 829      T9MI <7,9> 1630
T10M <5,10> 1658      T10MI <7,10> 3260
T11M <5,11> 3316      T11MI <7,11> 2425

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 2265Scale 2265: Raga Rasamanjari, Ian Ring Music TheoryRaga Rasamanjari
Scale 2269Scale 2269: Pygian, Ian Ring Music TheoryPygian
Scale 2271Scale 2271: Poptyllic, Ian Ring Music TheoryPoptyllic
Scale 2259Scale 2259: Raga Mandari, Ian Ring Music TheoryRaga Mandari
Scale 2263Scale 2263: Lycrian, Ian Ring Music TheoryLycrian
Scale 2251Scale 2251: Zodimic, Ian Ring Music TheoryZodimic
Scale 2283Scale 2283: Aeolyptian, Ian Ring Music TheoryAeolyptian
Scale 2299Scale 2299: Phraptyllic, Ian Ring Music TheoryPhraptyllic
Scale 2203Scale 2203: Dorimic, Ian Ring Music TheoryDorimic
Scale 2235Scale 2235: Bathian, Ian Ring Music TheoryBathian
Scale 2139Scale 2139: Namian, Ian Ring Music TheoryNamian
Scale 2395Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
Scale 2523Scale 2523: Mirage Scale, Ian Ring Music TheoryMirage Scale
Scale 2779Scale 2779: Shostakovich, Ian Ring Music TheoryShostakovich
Scale 3291Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian

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.