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Scale 855: "Porian"

Scale 855: Porian, 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
Porian

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

{0,1,2,4,6,8,9}

Forte Number

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

7-30

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: 3417

Hemitonia

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

3 (trihemitonic)

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.

yes

Deep Scale

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

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 2, 2, 2, 1, 3]

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.

<3, 4, 3, 5, 4, 2>

Interval Spectrum

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

p4m5n3s4d3t2

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}
<3> = {4,5,6}
<4> = {6,7,8}
<5> = {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.

1.714

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.

2.549

Polygon Perimeter

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

5.967

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

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 TriadsD{2,6,9}142.14
A{9,1,4}321.29
Minor Triadsc♯m{1,4,8}231.71
f♯m{6,9,1}331.43
am{9,0,4}331.43
Augmented TriadsC+{0,4,8}241.86
Diminished Triadsf♯°{6,9,0}231.57
Parsimonious Voice Leading Between Common Triads of Scale 855. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m am am C+->am A A c#m->A D D f#m f#m D->f#m f#° f#° f#°->f#m f#°->am f#m->A am->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 VerticesA
Peripheral VerticesC+, D

Modes

Modes are the rotational transformation of this scale. Scale 855 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2475
Scale 2475: Neapolitan Minor, Ian Ring Music TheoryNeapolitan Minor
3rd mode:
Scale 3285
Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
4th mode:
Scale 1845
Scale 1845: Lagian, Ian Ring Music TheoryLagian
5th mode:
Scale 1485
Scale 1485: Minor Romani, Ian Ring Music TheoryMinor Romani
6th mode:
Scale 1395
Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
7th mode:
Scale 2745
Scale 2745: Mela Sulini, Ian Ring Music TheoryMela Sulini

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [855, 2475, 3285, 1845, 1485, 1395, 2745] (Forte: 7-30) is the complement of the pentatonic modal family [339, 789, 1221, 1329, 2217] (Forte: 5-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 855 is 3417

Scale 3417Scale 3417: Golian, Ian Ring Music TheoryGolian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 855 is chiral, and its enantiomorph is scale 3417

Scale 3417Scale 3417: Golian, Ian Ring Music TheoryGolian

Transformations:

T0 855  T0I 3417
T1 1710  T1I 2739
T2 3420  T2I 1383
T3 2745  T3I 2766
T4 1395  T4I 1437
T5 2790  T5I 2874
T6 1485  T6I 1653
T7 2970  T7I 3306
T8 1845  T8I 2517
T9 3690  T9I 939
T10 3285  T10I 1878
T11 2475  T11I 3756

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 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic
Scale 851Scale 851: Raga Hejjajji, Ian Ring Music TheoryRaga Hejjajji
Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic
Scale 839Scale 839: Ionathimic, Ian Ring Music TheoryIonathimic
Scale 847Scale 847: Ganian, Ian Ring Music TheoryGanian
Scale 871Scale 871: Locrian Double-flat 3 Double-flat 7, Ian Ring Music TheoryLocrian Double-flat 3 Double-flat 7
Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic
Scale 791Scale 791: Aeoloptimic, Ian Ring Music TheoryAeoloptimic
Scale 823Scale 823: Stodian, Ian Ring Music TheoryStodian
Scale 919Scale 919: Chromatic Phrygian Inverse, Ian Ring Music TheoryChromatic Phrygian Inverse
Scale 983Scale 983: Thocryllic, Ian Ring Music TheoryThocryllic
Scale 599Scale 599: Thyrimic, Ian Ring Music TheoryThyrimic
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 343Scale 343: Ionorimic, Ian Ring Music TheoryIonorimic
Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse
Scale 1879Scale 1879: Mixoryllic, Ian Ring Music TheoryMixoryllic
Scale 2903Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic

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. 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.