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Scale 3179: "Daptian"

Scale 3179: Daptian, 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
Daptian
Dozenal
Tuvian

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

{0,1,3,5,6,10,11}

Forte Number

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

7-14

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: 2759

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.

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.

6

Prime Form

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

no
prime: 431

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, 2, 2, 1, 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.

<4, 4, 3, 3, 5, 2>

Interval Spectrum

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

p5m3n3s4d4t2

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

Spectra Variation

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

2.857

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.

(25, 38, 102)

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♯{6,10,1}231.4
B{11,3,6}231.4
Minor Triadsd♯m{3,6,10}221.2
a♯m{10,1,5}142
Diminished Triads{0,3,6}142
Parsimonious Voice Leading Between Common Triads of Scale 3179. Created by Ian Ring ©2019 B B c°->B d#m d#m F# F# d#m->F# d#m->B a#m a#m F#->a#m

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 Verticesd♯m
Peripheral Verticesc°, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 3179 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3637
Scale 3637: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
3rd mode:
Scale 1933
Scale 1933: Mocrian, Ian Ring Music TheoryMocrian
4th mode:
Scale 1507
Scale 1507: Zynian, Ian Ring Music TheoryZynian
5th mode:
Scale 2801
Scale 2801: Zogian, Ian Ring Music TheoryZogian
6th mode:
Scale 431
Scale 431: Epyrian, Ian Ring Music TheoryEpyrianThis is the prime mode
7th mode:
Scale 2263
Scale 2263: Lycrian, Ian Ring Music TheoryLycrian

Prime

The prime form of this scale is Scale 431

Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian

Complement

The heptatonic modal family [3179, 3637, 1933, 1507, 2801, 431, 2263] (Forte: 7-14) is the complement of the pentatonic modal family [167, 901, 1249, 2131, 3113] (Forte: 5-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3179 is 2759

Scale 2759Scale 2759: Mela Pavani, Ian Ring Music TheoryMela Pavani

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3179 is chiral, and its enantiomorph is scale 2759

Scale 2759Scale 2759: Mela Pavani, Ian Ring Music TheoryMela Pavani

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> 3179       T0I <11,0> 2759
T1 <1,1> 2263      T1I <11,1> 1423
T2 <1,2> 431      T2I <11,2> 2846
T3 <1,3> 862      T3I <11,3> 1597
T4 <1,4> 1724      T4I <11,4> 3194
T5 <1,5> 3448      T5I <11,5> 2293
T6 <1,6> 2801      T6I <11,6> 491
T7 <1,7> 1507      T7I <11,7> 982
T8 <1,8> 3014      T8I <11,8> 1964
T9 <1,9> 1933      T9I <11,9> 3928
T10 <1,10> 3866      T10I <11,10> 3761
T11 <1,11> 3637      T11I <11,11> 3427
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 239      T0MI <7,0> 3809
T1M <5,1> 478      T1MI <7,1> 3523
T2M <5,2> 956      T2MI <7,2> 2951
T3M <5,3> 1912      T3MI <7,3> 1807
T4M <5,4> 3824      T4MI <7,4> 3614
T5M <5,5> 3553      T5MI <7,5> 3133
T6M <5,6> 3011      T6MI <7,6> 2171
T7M <5,7> 1927      T7MI <7,7> 247
T8M <5,8> 3854      T8MI <7,8> 494
T9M <5,9> 3613      T9MI <7,9> 988
T10M <5,10> 3131      T10MI <7,10> 1976
T11M <5,11> 2167      T11MI <7,11> 3952

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 3177Scale 3177: Rothimic, Ian Ring Music TheoryRothimic
Scale 3181Scale 3181: Rolian, Ian Ring Music TheoryRolian
Scale 3183Scale 3183: Mixonyllic, Ian Ring Music TheoryMixonyllic
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 3175Scale 3175: Eponian, Ian Ring Music TheoryEponian
Scale 3187Scale 3187: Koptian, Ian Ring Music TheoryKoptian
Scale 3195Scale 3195: Raryllic, Ian Ring Music TheoryRaryllic
Scale 3147Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic
Scale 3163Scale 3163: Rogian, Ian Ring Music TheoryRogian
Scale 3115Scale 3115: Tihian, Ian Ring Music TheoryTihian
Scale 3243Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
Scale 3307Scale 3307: Boptyllic, Ian Ring Music TheoryBoptyllic
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3691Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic
Scale 2155Scale 2155: Newian, Ian Ring Music TheoryNewian
Scale 2667Scale 2667: Byrian, Ian Ring Music TheoryByrian
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form

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.