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Scale 1315: "Pyritonic"

Scale 1315: Pyritonic, 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
Pyritonic
Dozenal
Ibsian

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,1,5,8,10}

Forte Number

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

5-27

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: 2197

Hemitonia

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

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

2

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.

4

Prime Form

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

no
prime: 299

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, 4, 3, 2, 2]

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.

<1, 2, 2, 2, 3, 0>

Interval Spectrum

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

p3m2n2s2d

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

Spectra Variation

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

2.8

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.

2.049

Polygon Perimeter

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

5.664

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.

(3, 8, 36)

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}210.67
Minor Triadsfm{5,8,0}121
a♯m{10,1,5}121
Parsimonious Voice Leading Between Common Triads of Scale 1315. Created by Ian Ring ©2019 C# C# fm fm C#->fm a#m a#m C#->a#m

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesC♯
Peripheral Verticesfm, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 1315 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2705
Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
3rd mode:
Scale 425
Scale 425: Raga Kokil Pancham, Ian Ring Music TheoryRaga Kokil Pancham
4th mode:
Scale 565
Scale 565: Aeolyphritonic, Ian Ring Music TheoryAeolyphritonic
5th mode:
Scale 1165
Scale 1165: Gycritonic, Ian Ring Music TheoryGycritonic

Prime

The prime form of this scale is Scale 299

Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini

Complement

The pentatonic modal family [1315, 2705, 425, 565, 1165] (Forte: 5-27) is the complement of the heptatonic modal family [695, 1465, 1765, 1835, 2395, 2965, 3245] (Forte: 7-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1315 is 2197

Scale 2197Scale 2197: Raga Hamsadhvani, Ian Ring Music TheoryRaga Hamsadhvani

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1315 is chiral, and its enantiomorph is scale 2197

Scale 2197Scale 2197: Raga Hamsadhvani, Ian Ring Music TheoryRaga Hamsadhvani

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> 1315       T0I <11,0> 2197
T1 <1,1> 2630      T1I <11,1> 299
T2 <1,2> 1165      T2I <11,2> 598
T3 <1,3> 2330      T3I <11,3> 1196
T4 <1,4> 565      T4I <11,4> 2392
T5 <1,5> 1130      T5I <11,5> 689
T6 <1,6> 2260      T6I <11,6> 1378
T7 <1,7> 425      T7I <11,7> 2756
T8 <1,8> 850      T8I <11,8> 1417
T9 <1,9> 1700      T9I <11,9> 2834
T10 <1,10> 3400      T10I <11,10> 1573
T11 <1,11> 2705      T11I <11,11> 3146
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 55      T0MI <7,0> 3457
T1M <5,1> 110      T1MI <7,1> 2819
T2M <5,2> 220      T2MI <7,2> 1543
T3M <5,3> 440      T3MI <7,3> 3086
T4M <5,4> 880      T4MI <7,4> 2077
T5M <5,5> 1760      T5MI <7,5> 59
T6M <5,6> 3520      T6MI <7,6> 118
T7M <5,7> 2945      T7MI <7,7> 236
T8M <5,8> 1795      T8MI <7,8> 472
T9M <5,9> 3590      T9MI <7,9> 944
T10M <5,10> 3085      T10MI <7,10> 1888
T11M <5,11> 2075      T11MI <7,11> 3776

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 1313Scale 1313: Iplian, Ian Ring Music TheoryIplian
Scale 1317Scale 1317: Chaio, Ian Ring Music TheoryChaio
Scale 1319Scale 1319: Phronimic, Ian Ring Music TheoryPhronimic
Scale 1323Scale 1323: Ritsu, Ian Ring Music TheoryRitsu
Scale 1331Scale 1331: Raga Vasantabhairavi, Ian Ring Music TheoryRaga Vasantabhairavi
Scale 1283Scale 1283: Hurian, Ian Ring Music TheoryHurian
Scale 1299Scale 1299: Aerophitonic, Ian Ring Music TheoryAerophitonic
Scale 1347Scale 1347: Igoian, Ian Ring Music TheoryIgoian
Scale 1379Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic
Scale 1443Scale 1443: Raga Phenadyuti, Ian Ring Music TheoryRaga Phenadyuti
Scale 1059Scale 1059: Gikian, Ian Ring Music TheoryGikian
Scale 1187Scale 1187: Kokin-joshi, Ian Ring Music TheoryKokin-joshi
Scale 1571Scale 1571: Lagitonic, Ian Ring Music TheoryLagitonic
Scale 1827Scale 1827: Katygimic, Ian Ring Music TheoryKatygimic
Scale 291Scale 291: Raga Lavangi, Ian Ring Music TheoryRaga Lavangi
Scale 803Scale 803: Loritonic, Ian Ring Music TheoryLoritonic
Scale 2339Scale 2339: Raga Kshanika, Ian Ring Music TheoryRaga Kshanika
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic

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.