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Scale 3857: "Ponimic"

Scale 3857: Ponimic, 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
Ponimic
Dozenal
Yopian

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

{0,4,8,9,10,11}

Forte Number

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

6-Z37

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.

[4]

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.

no

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.

3 (tricohemitonic)

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.

4

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: 287

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

Interval Spectrum

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

p2m3n2s3d4t

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

Spectra Variation

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

4

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.

1.866

Polygon Perimeter

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

5.535

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.

[8]

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.

(41, 4, 50)

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 TriadsE{4,8,11}121
Minor Triadsam{9,0,4}121
Augmented TriadsC+{0,4,8}210.67

The following pitch classes are not present in any of the common triads: {10}

Parsimonious Voice Leading Between Common Triads of Scale 3857. Created by Ian Ring ©2019 C+ C+ E E C+->E am am C+->am

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 VerticesC+
Peripheral VerticesE, am

Modes

Modes are the rotational transformation of this scale. Scale 3857 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 497
Scale 497: Kadimic, Ian Ring Music TheoryKadimic
3rd mode:
Scale 287
Scale 287: Gynimic, Ian Ring Music TheoryGynimicThis is the prime mode
4th mode:
Scale 2191
Scale 2191: Thydimic, Ian Ring Music TheoryThydimic
5th mode:
Scale 3143
Scale 3143: Polimic, Ian Ring Music TheoryPolimic
6th mode:
Scale 3619
Scale 3619: Thanimic, Ian Ring Music TheoryThanimic

Prime

The prime form of this scale is Scale 287

Scale 287Scale 287: Gynimic, Ian Ring Music TheoryGynimic

Complement

The hexatonic modal family [3857, 497, 287, 2191, 3143, 3619] (Forte: 6-Z37) is the complement of the hexatonic modal family [119, 1799, 2107, 2947, 3101, 3521] (Forte: 6-Z4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3857 is 287

Scale 287Scale 287: Gynimic, Ian Ring Music TheoryGynimic

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> 3857       T0I <11,0> 287
T1 <1,1> 3619      T1I <11,1> 574
T2 <1,2> 3143      T2I <11,2> 1148
T3 <1,3> 2191      T3I <11,3> 2296
T4 <1,4> 287      T4I <11,4> 497
T5 <1,5> 574      T5I <11,5> 994
T6 <1,6> 1148      T6I <11,6> 1988
T7 <1,7> 2296      T7I <11,7> 3976
T8 <1,8> 497      T8I <11,8> 3857
T9 <1,9> 994      T9I <11,9> 3619
T10 <1,10> 1988      T10I <11,10> 3143
T11 <1,11> 3976      T11I <11,11> 2191
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 917      T0MI <7,0> 1337
T1M <5,1> 1834      T1MI <7,1> 2674
T2M <5,2> 3668      T2MI <7,2> 1253
T3M <5,3> 3241      T3MI <7,3> 2506
T4M <5,4> 2387      T4MI <7,4> 917
T5M <5,5> 679      T5MI <7,5> 1834
T6M <5,6> 1358      T6MI <7,6> 3668
T7M <5,7> 2716      T7MI <7,7> 3241
T8M <5,8> 1337      T8MI <7,8> 2387
T9M <5,9> 2674      T9MI <7,9> 679
T10M <5,10> 1253      T10MI <7,10> 1358
T11M <5,11> 2506      T11MI <7,11> 2716

The transformations that map this set to itself are: T0, T8I

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 3859Scale 3859: Aeolarian, Ian Ring Music TheoryAeolarian
Scale 3861Scale 3861: Phroptian, Ian Ring Music TheoryPhroptian
Scale 3865Scale 3865: Starian, Ian Ring Music TheoryStarian
Scale 3841Scale 3841: Pentatonic Chromatic Descending, Ian Ring Music TheoryPentatonic Chromatic Descending
Scale 3849Scale 3849: Yikian, Ian Ring Music TheoryYikian
Scale 3873Scale 3873: Yoyian, Ian Ring Music TheoryYoyian
Scale 3889Scale 3889: Parian, Ian Ring Music TheoryParian
Scale 3921Scale 3921: Pythian, Ian Ring Music TheoryPythian
Scale 3985Scale 3985: Thadian, Ian Ring Music TheoryThadian
Scale 3601Scale 3601: Wilian, Ian Ring Music TheoryWilian
Scale 3729Scale 3729: Starimic, Ian Ring Music TheoryStarimic
Scale 3345Scale 3345: Zylitonic, Ian Ring Music TheoryZylitonic
Scale 2833Scale 2833: Dolitonic, Ian Ring Music TheoryDolitonic
Scale 1809Scale 1809: Ranitonic, Ian Ring Music TheoryRanitonic

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.