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Scale 2199: "Dyptimic"

Scale 2199: Dyptimic, 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
Dyptimic
Dozenal
Nixian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,1,2,4,7,11}

Forte Number

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

6-Z40

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: 3363

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.

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.

5

Prime Form

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

no
prime: 303

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, 3, 4, 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, 3, 3, 2, 3, 1>

Interval Spectrum

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

p3m2n3s3d3t

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

Spectra Variation

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

3.667

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.

2.116

Polygon Perimeter

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

5.699

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.

(29, 16, 64)

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}221
G{7,11,2}131.5
Minor Triadsem{4,7,11}221
Diminished Triadsc♯°{1,4,7}131.5
Parsimonious Voice Leading Between Common Triads of Scale 2199. Created by Ian Ring ©2019 C C c#° c#° C->c#° em em C->em Parsimonious Voice Leading Between Common Triads of Scale 2199. Created by Ian Ring ©2019 G em->G

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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC, em
Peripheral Verticesc♯°, G

Modes

Modes are the rotational transformation of this scale. Scale 2199 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3147
Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic
3rd mode:
Scale 3621
Scale 3621: Gylimic, Ian Ring Music TheoryGylimic
4th mode:
Scale 1929
Scale 1929: Aeolycrimic, Ian Ring Music TheoryAeolycrimic
5th mode:
Scale 753
Scale 753: Aeronimic, Ian Ring Music TheoryAeronimic
6th mode:
Scale 303
Scale 303: Golimic, Ian Ring Music TheoryGolimicThis is the prime mode

Prime

The prime form of this scale is Scale 303

Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic

Complement

The hexatonic modal family [2199, 3147, 3621, 1929, 753, 303] (Forte: 6-Z40) is the complement of the hexatonic modal family [183, 1761, 1803, 2139, 2949, 3117] (Forte: 6-Z11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2199 is 3363

Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2199 is chiral, and its enantiomorph is scale 3363

Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic

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> 2199       T0I <11,0> 3363
T1 <1,1> 303      T1I <11,1> 2631
T2 <1,2> 606      T2I <11,2> 1167
T3 <1,3> 1212      T3I <11,3> 2334
T4 <1,4> 2424      T4I <11,4> 573
T5 <1,5> 753      T5I <11,5> 1146
T6 <1,6> 1506      T6I <11,6> 2292
T7 <1,7> 3012      T7I <11,7> 489
T8 <1,8> 1929      T8I <11,8> 978
T9 <1,9> 3858      T9I <11,9> 1956
T10 <1,10> 3621      T10I <11,10> 3912
T11 <1,11> 3147      T11I <11,11> 3729
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3489      T0MI <7,0> 183
T1M <5,1> 2883      T1MI <7,1> 366
T2M <5,2> 1671      T2MI <7,2> 732
T3M <5,3> 3342      T3MI <7,3> 1464
T4M <5,4> 2589      T4MI <7,4> 2928
T5M <5,5> 1083      T5MI <7,5> 1761
T6M <5,6> 2166      T6MI <7,6> 3522
T7M <5,7> 237      T7MI <7,7> 2949
T8M <5,8> 474      T8MI <7,8> 1803
T9M <5,9> 948      T9MI <7,9> 3606
T10M <5,10> 1896      T10MI <7,10> 3117
T11M <5,11> 3792      T11MI <7,11> 2139

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 2197Scale 2197: Raga Hamsadhvani, Ian Ring Music TheoryRaga Hamsadhvani
Scale 2195Scale 2195: Zalitonic, Ian Ring Music TheoryZalitonic
Scale 2203Scale 2203: Dorimic, Ian Ring Music TheoryDorimic
Scale 2207Scale 2207: Mygian, Ian Ring Music TheoryMygian
Scale 2183Scale 2183: Nenian, Ian Ring Music TheoryNenian
Scale 2191Scale 2191: Thydimic, Ian Ring Music TheoryThydimic
Scale 2215Scale 2215: Ranimic, Ian Ring Music TheoryRanimic
Scale 2231Scale 2231: Macrian, Ian Ring Music TheoryMacrian
Scale 2263Scale 2263: Lycrian, Ian Ring Music TheoryLycrian
Scale 2071Scale 2071: Moxian, Ian Ring Music TheoryMoxian
Scale 2135Scale 2135: Nakian, Ian Ring Music TheoryNakian
Scale 2327Scale 2327: Epalimic, Ian Ring Music TheoryEpalimic
Scale 2455Scale 2455: Bothian, Ian Ring Music TheoryBothian
Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian
Scale 3223Scale 3223: Thyphian, Ian Ring Music TheoryThyphian
Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian
Scale 1175Scale 1175: Epycrimic, Ian Ring Music TheoryEpycrimic

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.