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Scale 1831: "Pothian"

Scale 1831: Pothian, 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
Pothian
Dozenal
LEGIAN

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

{0,1,2,5,8,9,10}

Forte Number

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

7-Z17

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]

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.

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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: 631

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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, 5, 4, 1]

Interval Spectrum

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

p4m5n4s3d4t

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

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.

2.433

Polygon Perimeter

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

5.899

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.

[10]

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

Harmonic Chords

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♯{1,5,8}331.5
F{5,9,0}231.75
A♯{10,2,5}242
Minor Triadsdm{2,5,9}331.5
fm{5,8,0}242
a♯m{10,1,5}231.75
Augmented TriadsC♯+{1,5,9}421.25
Diminished Triads{2,5,8}231.75
Parsimonious Voice Leading Between Common Triads of Scale 1831. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ C#->d° fm fm C#->fm dm dm C#+->dm F F C#+->F a#m a#m C#+->a#m d°->dm A# A# dm->A# fm->F a#m->A#

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 VerticesC♯+
Peripheral Verticesfm, A♯

Modes

Modes are the rotational transformation of this scale. Scale 1831 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2963
Scale 2963: Bygian, Ian Ring Music TheoryBygian
3rd mode:
Scale 3529
Scale 3529: Stalian, Ian Ring Music TheoryStalian
4th mode:
Scale 953
Scale 953: Mela Yagapriya, Ian Ring Music TheoryMela Yagapriya
5th mode:
Scale 631
Scale 631: Zygian, Ian Ring Music TheoryZygianThis is the prime mode
6th mode:
Scale 2363
Scale 2363: Kataptian, Ian Ring Music TheoryKataptian
7th mode:
Scale 3229
Scale 3229: Aeolaptian, Ian Ring Music TheoryAeolaptian

Prime

The prime form of this scale is Scale 631

Scale 631Scale 631: Zygian, Ian Ring Music TheoryZygian

Complement

The heptatonic modal family [1831, 2963, 3529, 953, 631, 2363, 3229] (Forte: 7-Z17) is the complement of the pentatonic modal family [283, 433, 1571, 2189, 2833] (Forte: 5-Z17)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1831 is 3229

Scale 3229Scale 3229: Aeolaptian, Ian Ring Music TheoryAeolaptian

Transformations:

T0 1831  T0I 3229
T1 3662  T1I 2363
T2 3229  T2I 631
T3 2363  T3I 1262
T4 631  T4I 2524
T5 1262  T5I 953
T6 2524  T6I 1906
T7 953  T7I 3812
T8 1906  T8I 3529
T9 3812  T9I 2963
T10 3529  T10I 1831
T11 2963  T11I 3662

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 1829Scale 1829: Pathimic, Ian Ring Music TheoryPathimic
Scale 1827Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic
Scale 1835Scale 1835: Byptian, Ian Ring Music TheoryByptian
Scale 1839Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
Scale 1847Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic
Scale 1799Scale 1799: LAMIAN, Ian Ring Music TheoryLAMIAN
Scale 1815Scale 1815: Godian, Ian Ring Music TheoryGodian
Scale 1863Scale 1863: Pycrian, Ian Ring Music TheoryPycrian
Scale 1895Scale 1895: Salyllic, Ian Ring Music TheorySalyllic
Scale 1959Scale 1959: Katolyllic, Ian Ring Music TheoryKatolyllic
Scale 1575Scale 1575: Zycrimic, Ian Ring Music TheoryZycrimic
Scale 1703Scale 1703: Mela Vanaspati, Ian Ring Music TheoryMela Vanaspati
Scale 1319Scale 1319: Phronimic, Ian Ring Music TheoryPhronimic
Scale 807Scale 807: Raga Suddha Mukhari, Ian Ring Music TheoryRaga Suddha Mukhari
Scale 2855Scale 2855: Epocrian, Ian Ring Music TheoryEpocrian
Scale 3879Scale 3879: Pathyllic, Ian Ring Music TheoryPathyllic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.