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Scale 3217: "Molitonic"

Scale 3217: Molitonic, 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
Molitonic
Dozenal
Udoian

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

{0,4,7,10,11}

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: 295

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.

[4, 3, 3, 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 TriadsC{0,4,7}121
Minor Triadsem{4,7,11}210.67
Diminished Triads{4,7,10}121
Parsimonious Voice Leading Between Common Triads of Scale 3217. Created by Ian Ring ©2019 C C em em C->em e°->em

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 Verticesem
Peripheral VerticesC, e°

Modes

Modes are the rotational transformation of this scale. Scale 3217 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 457
Scale 457: Staptitonic, Ian Ring Music TheoryStaptitonic
3rd mode:
Scale 569
Scale 569: Mothitonic, Ian Ring Music TheoryMothitonic
4th mode:
Scale 583
Scale 583: Aeritonic, Ian Ring Music TheoryAeritonic
5th mode:
Scale 2339
Scale 2339: Raga Kshanika, Ian Ring Music TheoryRaga Kshanika

Prime

The prime form of this scale is Scale 295

Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic

Complement

The pentatonic modal family [3217, 457, 569, 583, 2339] (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 3217 is 295

Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3217 is chiral, and its enantiomorph is scale 295

Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic

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

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 3219Scale 3219: Ionaphimic, Ian Ring Music TheoryIonaphimic
Scale 3221Scale 3221: Bycrimic, Ian Ring Music TheoryBycrimic
Scale 3225Scale 3225: Ionalimic, Ian Ring Music TheoryIonalimic
Scale 3201Scale 3201: Urtian, Ian Ring Music TheoryUrtian
Scale 3209Scale 3209: Aeraphitonic, Ian Ring Music TheoryAeraphitonic
Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian
Scale 3249Scale 3249: Raga Tilang, Ian Ring Music TheoryRaga Tilang
Scale 3281Scale 3281: Raga Vijayavasanta, Ian Ring Music TheoryRaga Vijayavasanta
Scale 3089Scale 3089: Tirian, Ian Ring Music TheoryTirian
Scale 3153Scale 3153: Zathitonic, Ian Ring Music TheoryZathitonic
Scale 3345Scale 3345: Zylitonic, Ian Ring Music TheoryZylitonic
Scale 3473Scale 3473: Lathimic, Ian Ring Music TheoryLathimic
Scale 3729Scale 3729: Starimic, Ian Ring Music TheoryStarimic
Scale 2193Scale 2193: Major Seventh, Ian Ring Music TheoryMajor Seventh
Scale 2705Scale 2705: Raga Mamata, Ian Ring Music TheoryRaga Mamata
Scale 1169Scale 1169: Raga Mahathi, Ian Ring Music TheoryRaga Mahathi

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.