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Scale 2827: "Runian"

Scale 2827: Runian, 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




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


Forte Number

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 2587


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

3 (trihemitonic)


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

1 (uncohemitonic)


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.



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.


Prime Form

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

prime: 187


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

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


Interval Structure

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

[1, 2, 5, 1, 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.

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

Interval Spectrum

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


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

Spectra Variation

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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(29, 7, 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
Major TriadsG♯{8,0,3}210.67
Minor Triadsg♯m{8,11,3}121
Diminished Triads{9,0,3}121

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

Parsimonious Voice Leading Between Common Triads of Scale 2827. Created by Ian Ring ©2019 g#m g#m G# G# g#m->G# G#->a°

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.

Central VerticesG♯
Peripheral Verticesg♯m, a°


Modes are the rotational transformation of this scale. Scale 2827 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3461
Scale 3461: Vodian, Ian Ring Music TheoryVodian
3rd mode:
Scale 1889
Scale 1889: Loqian, Ian Ring Music TheoryLoqian
4th mode:
Scale 187
Scale 187: Bedian, Ian Ring Music TheoryBedianThis is the prime mode
5th mode:
Scale 2141
Scale 2141: Nanian, Ian Ring Music TheoryNanian
6th mode:
Scale 1559
Scale 1559: Jowian, Ian Ring Music TheoryJowian


The prime form of this scale is Scale 187

Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian


The hexatonic modal family [2827, 3461, 1889, 187, 2141, 1559] (Forte: 6-Z10) is the complement of the hexatonic modal family [317, 977, 1103, 2599, 3347, 3721] (Forte: 6-Z39)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2827 is 2587

Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian


Only scales that are chiral will have an enantiomorph. Scale 2827 is chiral, and its enantiomorph is scale 2587

Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian


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> 2827       T0I <11,0> 2587
T1 <1,1> 1559      T1I <11,1> 1079
T2 <1,2> 3118      T2I <11,2> 2158
T3 <1,3> 2141      T3I <11,3> 221
T4 <1,4> 187      T4I <11,4> 442
T5 <1,5> 374      T5I <11,5> 884
T6 <1,6> 748      T6I <11,6> 1768
T7 <1,7> 1496      T7I <11,7> 3536
T8 <1,8> 2992      T8I <11,8> 2977
T9 <1,9> 1889      T9I <11,9> 1859
T10 <1,10> 3778      T10I <11,10> 3718
T11 <1,11> 3461      T11I <11,11> 3341
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 697      T0MI <7,0> 937
T1M <5,1> 1394      T1MI <7,1> 1874
T2M <5,2> 2788      T2MI <7,2> 3748
T3M <5,3> 1481      T3MI <7,3> 3401
T4M <5,4> 2962      T4MI <7,4> 2707
T5M <5,5> 1829      T5MI <7,5> 1319
T6M <5,6> 3658      T6MI <7,6> 2638
T7M <5,7> 3221      T7MI <7,7> 1181
T8M <5,8> 2347      T8MI <7,8> 2362
T9M <5,9> 599      T9MI <7,9> 629
T10M <5,10> 1198      T10MI <7,10> 1258
T11M <5,11> 2396      T11MI <7,11> 2516

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 2825Scale 2825: Rumian, Ian Ring Music TheoryRumian
Scale 2829Scale 2829: Rupian, Ian Ring Music TheoryRupian
Scale 2831Scale 2831: Ruqian, Ian Ring Music TheoryRuqian
Scale 2819Scale 2819: Rujian, Ian Ring Music TheoryRujian
Scale 2823Scale 2823: Rulian, Ian Ring Music TheoryRulian
Scale 2835Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic
Scale 2843Scale 2843: Sorian, Ian Ring Music TheorySorian
Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
Scale 2891Scale 2891: Phrogian, Ian Ring Music TheoryPhrogian
Scale 2955Scale 2955: Thorian, Ian Ring Music TheoryThorian
Scale 2571Scale 2571: Pukian, Ian Ring Music TheoryPukian
Scale 2699Scale 2699: Sythimic, Ian Ring Music TheorySythimic
Scale 2315Scale 2315: Orkian, Ian Ring Music TheoryOrkian
Scale 3339Scale 3339: Smuian, Ian Ring Music TheorySmuian
Scale 3851Scale 3851: Yilian, Ian Ring Music TheoryYilian
Scale 779Scale 779: Etrian, Ian Ring Music TheoryEtrian
Scale 1803Scale 1803: Lapian, Ian Ring Music TheoryLapian

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 ( 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.