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Scale 357: "Banitonic"

Scale 357: Banitonic, 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

41161837294116105072918310504116183
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
Banitonic

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

{0,2,5,6,8}

Forte Number

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

5-28

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: 1233

Hemitonia

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

1 (unhemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

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.

4

Prime Form

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

no
prime: 333

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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.

[1, 2, 2, 2, 1, 2]

Interval Spectrum

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

pm2n2s2dt2

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

Spectra Variation

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

2.4

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.

2.049

Polygon Perimeter

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

5.664

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

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
Minor Triadsfm{5,8,0}110.5
Diminished Triads{2,5,8}110.5

The following pitch classes are not present in any of the common triads: {6}

Parsimonious Voice Leading Between Common Triads of Scale 357. Created by Ian Ring ©2019 fm fm d°->fm

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 357 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1113
Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2
3rd mode:
Scale 651
Scale 651: Golitonic, Ian Ring Music TheoryGolitonic
4th mode:
Scale 2373
Scale 2373: Dyptitonic, Ian Ring Music TheoryDyptitonic
5th mode:
Scale 1617
Scale 1617: Phronitonic, Ian Ring Music TheoryPhronitonic

Prime

The prime form of this scale is Scale 333

Scale 333Scale 333: Bogitonic, Ian Ring Music TheoryBogitonic

Complement

The pentatonic modal family [357, 1113, 651, 2373, 1617] (Forte: 5-28) is the complement of the heptatonic modal family [747, 1431, 1629, 1881, 2421, 2763, 3429] (Forte: 7-28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 357 is 1233

Scale 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 357 is chiral, and its enantiomorph is scale 1233

Scale 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic

Transformations:

T0 357  T0I 1233
T1 714  T1I 2466
T2 1428  T2I 837
T3 2856  T3I 1674
T4 1617  T4I 3348
T5 3234  T5I 2601
T6 2373  T6I 1107
T7 651  T7I 2214
T8 1302  T8I 333
T9 2604  T9I 666
T10 1113  T10I 1332
T11 2226  T11I 2664

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 359Scale 359: Bothimic, Ian Ring Music TheoryBothimic
Scale 353Scale 353, Ian Ring Music Theory
Scale 355Scale 355: Aeoloritonic, Ian Ring Music TheoryAeoloritonic
Scale 361Scale 361: Bocritonic, Ian Ring Music TheoryBocritonic
Scale 365Scale 365: Marimic, Ian Ring Music TheoryMarimic
Scale 373Scale 373: Epagimic, Ian Ring Music TheoryEpagimic
Scale 325Scale 325: Messiaen Truncated Mode 6, Ian Ring Music TheoryMessiaen Truncated Mode 6
Scale 341Scale 341: Bothitonic, Ian Ring Music TheoryBothitonic
Scale 293Scale 293: Raga Haripriya, Ian Ring Music TheoryRaga Haripriya
Scale 421Scale 421: Han-kumoi, Ian Ring Music TheoryHan-kumoi
Scale 485Scale 485: Stoptimic, Ian Ring Music TheoryStoptimic
Scale 101Scale 101, Ian Ring Music Theory
Scale 229Scale 229, Ian Ring Music Theory
Scale 613Scale 613: Phralitonic, Ian Ring Music TheoryPhralitonic
Scale 869Scale 869: Kothimic, Ian Ring Music TheoryKothimic
Scale 1381Scale 1381: Padimic, Ian Ring Music TheoryPadimic
Scale 2405Scale 2405: Katalimic, Ian Ring Music TheoryKatalimic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.