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Scale 3145: "Stolitonic"

Scale 3145: Stolitonic, 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
Stolitonic
Dozenal
Tozian

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

{0,3,6,10,11}

Forte Number

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

5-Z38

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: 583

Hemitonia

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

2 (dihemitonic)

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.

4

Prime Form

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

no
prime: 295

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.

[3, 3, 4, 1, 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.

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

Interval Spectrum

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

p2m2n2sd2t

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

Spectra Variation

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

3.2

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.

1.933

Polygon Perimeter

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

5.596

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.

(9, 5, 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 TriadsB{11,3,6}210.67
Minor Triadsd♯m{3,6,10}121
Diminished Triads{0,3,6}121
Parsimonious Voice Leading Between Common Triads of Scale 3145. Created by Ian Ring ©2019 B B c°->B d#m d#m d#m->B

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 VerticesB
Peripheral Verticesc°, d♯m

Modes

Modes are the rotational transformation of this scale. Scale 3145 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 905
Scale 905: Bylitonic, Ian Ring Music TheoryBylitonic
3rd mode:
Scale 625
Scale 625: Ionyptitonic, Ian Ring Music TheoryIonyptitonic
4th mode:
Scale 295
Scale 295: Gyritonic, Ian Ring Music TheoryGyritonicThis is the prime mode
5th mode:
Scale 2195
Scale 2195: Zalitonic, Ian Ring Music TheoryZalitonic

Prime

The prime form of this scale is Scale 295

Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic

Complement

The pentatonic modal family [3145, 905, 625, 295, 2195] (Forte: 5-Z38) is the complement of the heptatonic modal family [439, 1763, 1819, 2267, 2929, 2957, 3181] (Forte: 7-Z38)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3145 is 583

Scale 583Scale 583: Aeritonic, Ian Ring Music TheoryAeritonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3145 is chiral, and its enantiomorph is scale 583

Scale 583Scale 583: Aeritonic, Ian Ring Music TheoryAeritonic

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> 3145       T0I <11,0> 583
T1 <1,1> 2195      T1I <11,1> 1166
T2 <1,2> 295      T2I <11,2> 2332
T3 <1,3> 590      T3I <11,3> 569
T4 <1,4> 1180      T4I <11,4> 1138
T5 <1,5> 2360      T5I <11,5> 2276
T6 <1,6> 625      T6I <11,6> 457
T7 <1,7> 1250      T7I <11,7> 914
T8 <1,8> 2500      T8I <11,8> 1828
T9 <1,9> 905      T9I <11,9> 3656
T10 <1,10> 1810      T10I <11,10> 3217
T11 <1,11> 3620      T11I <11,11> 2339
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 205      T0MI <7,0> 1633
T1M <5,1> 410      T1MI <7,1> 3266
T2M <5,2> 820      T2MI <7,2> 2437
T3M <5,3> 1640      T3MI <7,3> 779
T4M <5,4> 3280      T4MI <7,4> 1558
T5M <5,5> 2465      T5MI <7,5> 3116
T6M <5,6> 835      T6MI <7,6> 2137
T7M <5,7> 1670      T7MI <7,7> 179
T8M <5,8> 3340      T8MI <7,8> 358
T9M <5,9> 2585      T9MI <7,9> 716
T10M <5,10> 1075      T10MI <7,10> 1432
T11M <5,11> 2150      T11MI <7,11> 2864

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 3147Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic
Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
Scale 3137Scale 3137, Ian Ring Music Theory
Scale 3141Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
Scale 3153Scale 3153: Zathitonic, Ian Ring Music TheoryZathitonic
Scale 3161Scale 3161: Kodimic, Ian Ring Music TheoryKodimic
Scale 3177Scale 3177: Rothimic, Ian Ring Music TheoryRothimic
Scale 3081Scale 3081: Temian, Ian Ring Music TheoryTemian
Scale 3113Scale 3113: Tigian, Ian Ring Music TheoryTigian
Scale 3209Scale 3209: Aeraphitonic, Ian Ring Music TheoryAeraphitonic
Scale 3273Scale 3273: Raga Jivantini, Ian Ring Music TheoryRaga Jivantini
Scale 3401Scale 3401: Palimic, Ian Ring Music TheoryPalimic
Scale 3657Scale 3657: Epynimic, Ian Ring Music TheoryEpynimic
Scale 2121Scale 2121: Nabian, Ian Ring Music TheoryNabian
Scale 2633Scale 2633: Bartók Beta Chord, Ian Ring Music TheoryBartók Beta Chord
Scale 1097Scale 1097: Aeraphic, Ian Ring Music TheoryAeraphic

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.