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Scale 2615: "Thoptian"

Scale 2615: Thoptian, 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
Thoptian
Dozenal
Qakian

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

{0,1,2,4,5,9,11}

Forte Number

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

7-11

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: 3467

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: 379

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, 1, 2, 1, 4, 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.

<4, 4, 4, 4, 4, 1>

Interval Spectrum

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

p4m4n4s4d4t

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

Spectra Variation

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

3.143

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.

2.299

Polygon Perimeter

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

5.803

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.

(43, 27, 92)

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 TriadsF{5,9,0}231.5
A{9,1,4}231.5
Minor Triadsdm{2,5,9}231.5
am{9,0,4}241.83
Augmented TriadsC♯+{1,5,9}321.17
Diminished Triads{11,2,5}142.17
Parsimonious Voice Leading Between Common Triads of Scale 2615. Created by Ian Ring ©2019 C#+ C#+ dm dm C#+->dm F F C#+->F A A C#+->A dm->b° am am F->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 Verticesam, b°

Modes

Modes are the rotational transformation of this scale. Scale 2615 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3355
Scale 3355: Bagian, Ian Ring Music TheoryBagian
3rd mode:
Scale 3725
Scale 3725: Kyrian, Ian Ring Music TheoryKyrian
4th mode:
Scale 1955
Scale 1955: Sonian, Ian Ring Music TheorySonian
5th mode:
Scale 3025
Scale 3025: Epycrian, Ian Ring Music TheoryEpycrian
6th mode:
Scale 445
Scale 445: Gocrian, Ian Ring Music TheoryGocrian
7th mode:
Scale 1135
Scale 1135: Katolian, Ian Ring Music TheoryKatolian

Prime

The prime form of this scale is Scale 379

Scale 379Scale 379: Aeragian, Ian Ring Music TheoryAeragian

Complement

The heptatonic modal family [2615, 3355, 3725, 1955, 3025, 445, 1135] (Forte: 7-11) is the complement of the pentatonic modal family [157, 929, 1063, 2579, 3337] (Forte: 5-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2615 is 3467

Scale 3467Scale 3467: Katonian, Ian Ring Music TheoryKatonian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2615 is chiral, and its enantiomorph is scale 3467

Scale 3467Scale 3467: Katonian, Ian Ring Music TheoryKatonian

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> 2615       T0I <11,0> 3467
T1 <1,1> 1135      T1I <11,1> 2839
T2 <1,2> 2270      T2I <11,2> 1583
T3 <1,3> 445      T3I <11,3> 3166
T4 <1,4> 890      T4I <11,4> 2237
T5 <1,5> 1780      T5I <11,5> 379
T6 <1,6> 3560      T6I <11,6> 758
T7 <1,7> 3025      T7I <11,7> 1516
T8 <1,8> 1955      T8I <11,8> 3032
T9 <1,9> 3910      T9I <11,9> 1969
T10 <1,10> 3725      T10I <11,10> 3938
T11 <1,11> 3355      T11I <11,11> 3781
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1955      T0MI <7,0> 2237
T1M <5,1> 3910      T1MI <7,1> 379
T2M <5,2> 3725      T2MI <7,2> 758
T3M <5,3> 3355      T3MI <7,3> 1516
T4M <5,4> 2615       T4MI <7,4> 3032
T5M <5,5> 1135      T5MI <7,5> 1969
T6M <5,6> 2270      T6MI <7,6> 3938
T7M <5,7> 445      T7MI <7,7> 3781
T8M <5,8> 890      T8MI <7,8> 3467
T9M <5,9> 1780      T9MI <7,9> 2839
T10M <5,10> 3560      T10MI <7,10> 1583
T11M <5,11> 3025      T11MI <7,11> 3166

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 2613Scale 2613: Raga Hamsa Vinodini, Ian Ring Music TheoryRaga Hamsa Vinodini
Scale 2611Scale 2611: Raga Vasanta, Ian Ring Music TheoryRaga Vasanta
Scale 2619Scale 2619: Ionyrian, Ian Ring Music TheoryIonyrian
Scale 2623Scale 2623: Aerylyllic, Ian Ring Music TheoryAerylyllic
Scale 2599Scale 2599: Malimic, Ian Ring Music TheoryMalimic
Scale 2607Scale 2607: Aerolian, Ian Ring Music TheoryAerolian
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 2647Scale 2647: Dadian, Ian Ring Music TheoryDadian
Scale 2679Scale 2679: Rathyllic, Ian Ring Music TheoryRathyllic
Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic
Scale 2871Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic
Scale 2103Scale 2103: Murian, Ian Ring Music TheoryMurian
Scale 2359Scale 2359: Gadian, Ian Ring Music TheoryGadian
Scale 3127Scale 3127: Topian, Ian Ring Music TheoryTopian
Scale 3639Scale 3639: Paptyllic, Ian Ring Music TheoryPaptyllic
Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
Scale 1591Scale 1591: Rodian, Ian Ring Music TheoryRodian

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.