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Scale 2715: "Kynian"

Scale 2715: Kynian, 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
Kynian
Dozenal
Rewian

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

{0,1,3,4,7,9,11}

Forte Number

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

7-26

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: 2859

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.

4

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: 699

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

Interval Spectrum

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

p3m5n4s4d3t2

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

Spectra Variation

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

2.286

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

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.

(13, 34, 98)

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}421.25
A{9,1,4}242
Minor Triadscm{0,3,7}331.5
em{4,7,11}231.75
am{9,0,4}331.5
Augmented TriadsD♯+{3,7,11}242
Diminished Triadsc♯°{1,4,7}231.75
{9,0,3}231.75
Parsimonious Voice Leading Between Common Triads of Scale 2715. Created by Ian Ring ©2019 cm cm C C cm->C D#+ D#+ cm->D#+ cm->a° c#° c#° C->c#° em em C->em am am C->am A A c#°->A D#+->em a°->am 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 VerticesC
Peripheral VerticesD♯+, A

Modes

Modes are the rotational transformation of this scale. Scale 2715 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3405
Scale 3405: Stynian, Ian Ring Music TheoryStynian
3rd mode:
Scale 1875
Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale
4th mode:
Scale 2985
Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian
5th mode:
Scale 885
Scale 885: Sathian, Ian Ring Music TheorySathian
6th mode:
Scale 1245
Scale 1245: Lathian, Ian Ring Music TheoryLathian
7th mode:
Scale 1335
Scale 1335: Elephant Scale, Ian Ring Music TheoryElephant Scale

Prime

The prime form of this scale is Scale 699

Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian

Complement

The heptatonic modal family [2715, 3405, 1875, 2985, 885, 1245, 1335] (Forte: 7-26) is the complement of the pentatonic modal family [309, 849, 1101, 1299, 2697] (Forte: 5-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2715 is 2859

Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2715 is chiral, and its enantiomorph is scale 2859

Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian

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> 2715       T0I <11,0> 2859
T1 <1,1> 1335      T1I <11,1> 1623
T2 <1,2> 2670      T2I <11,2> 3246
T3 <1,3> 1245      T3I <11,3> 2397
T4 <1,4> 2490      T4I <11,4> 699
T5 <1,5> 885      T5I <11,5> 1398
T6 <1,6> 1770      T6I <11,6> 2796
T7 <1,7> 3540      T7I <11,7> 1497
T8 <1,8> 2985      T8I <11,8> 2994
T9 <1,9> 1875      T9I <11,9> 1893
T10 <1,10> 3750      T10I <11,10> 3786
T11 <1,11> 3405      T11I <11,11> 3477
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2985      T0MI <7,0> 699
T1M <5,1> 1875      T1MI <7,1> 1398
T2M <5,2> 3750      T2MI <7,2> 2796
T3M <5,3> 3405      T3MI <7,3> 1497
T4M <5,4> 2715       T4MI <7,4> 2994
T5M <5,5> 1335      T5MI <7,5> 1893
T6M <5,6> 2670      T6MI <7,6> 3786
T7M <5,7> 1245      T7MI <7,7> 3477
T8M <5,8> 2490      T8MI <7,8> 2859
T9M <5,9> 885      T9MI <7,9> 1623
T10M <5,10> 1770      T10MI <7,10> 3246
T11M <5,11> 3540      T11MI <7,11> 2397

The transformations that map this set to itself are: T0, T4M

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 2713Scale 2713: Porimic, Ian Ring Music TheoryPorimic
Scale 2717Scale 2717: Epygian, Ian Ring Music TheoryEpygian
Scale 2719Scale 2719: Zocryllic, Ian Ring Music TheoryZocryllic
Scale 2707Scale 2707: Banimic, Ian Ring Music TheoryBanimic
Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian
Scale 2699Scale 2699: Sythimic, Ian Ring Music TheorySythimic
Scale 2731Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major
Scale 2747Scale 2747: Stythyllic, Ian Ring Music TheoryStythyllic
Scale 2779Scale 2779: Shostakovich, Ian Ring Music TheoryShostakovich
Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian
Scale 2651Scale 2651: Panian, Ian Ring Music TheoryPanian
Scale 2843Scale 2843: Sorian, Ian Ring Music TheorySorian
Scale 2971Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic
Scale 2203Scale 2203: Dorimic, Ian Ring Music TheoryDorimic
Scale 2459Scale 2459: Ionocrian, Ian Ring Music TheoryIonocrian
Scale 3227Scale 3227: Aeolocrian, Ian Ring Music TheoryAeolocrian
Scale 3739Scale 3739: Epanyllic, Ian Ring Music TheoryEpanyllic
Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic
Scale 1691Scale 1691: Kathian, Ian Ring Music TheoryKathian

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.