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Scale 1375: "Bothyllic"

Scale 1375: Bothyllic, 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
Bothyllic

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,1,2,3,4,6,8,10}

Forte Number

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

8-21

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.

[2]

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.

7

Prime Form

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

yes

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.

[4, 7, 4, 6, 4, 3]

Interval Spectrum

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

p4m6n4s7d4t3

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

Spectra Variation

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

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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.

[4]

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
Major TriadsF♯{6,10,1}242
G♯{8,0,3}242
Minor Triadsc♯m{1,4,8}242
d♯m{3,6,10}242
Augmented TriadsC+{0,4,8}242
D+{2,6,10}242
Diminished Triads{0,3,6}242
a♯°{10,1,4}242
Parsimonious Voice Leading Between Common Triads of Scale 1375. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ c#m c#m C+->c#m C+->G# a#° a#° c#m->a#° D+ D+ D+->d#m F# F# D+->F# F#->a#°

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1375 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2735
Scale 2735: Gynyllic, Ian Ring Music TheoryGynyllic
3rd mode:
Scale 3415
Scale 3415: Ionaptyllic, Ian Ring Music TheoryIonaptyllic
4th mode:
Scale 3755
Scale 3755: Phryryllic, Ian Ring Music TheoryPhryryllic
5th mode:
Scale 3925
Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic
6th mode:
Scale 2005
Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
7th mode:
Scale 1525
Scale 1525: Sodyllic, Ian Ring Music TheorySodyllic
8th mode:
Scale 1405
Scale 1405: Goryllic, Ian Ring Music TheoryGoryllic

Prime

This is the prime form of this scale.

Complement

The octatonic modal family [1375, 2735, 3415, 3755, 3925, 2005, 1525, 1405] (Forte: 8-21) is the complement of the tetratonic modal family [85, 1045, 1285, 1345] (Forte: 4-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1375 is 3925

Scale 3925Scale 3925: Thyryllic, Ian Ring Music TheoryThyryllic

Transformations:

T0 1375  T0I 3925
T1 2750  T1I 3755
T2 1405  T2I 3415
T3 2810  T3I 2735
T4 1525  T4I 1375
T5 3050  T5I 2750
T6 2005  T6I 1405
T7 4010  T7I 2810
T8 3925  T8I 1525
T9 3755  T9I 3050
T10 3415  T10I 2005
T11 2735  T11I 4010

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 1373Scale 1373: Storian, Ian Ring Music TheoryStorian
Scale 1371Scale 1371: Superlocrian, Ian Ring Music TheorySuperlocrian
Scale 1367Scale 1367: Leading Whole-Tone Inverse, Ian Ring Music TheoryLeading Whole-Tone Inverse
Scale 1359Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic
Scale 1407Scale 1407: Tharygic, Ian Ring Music TheoryTharygic
Scale 1311Scale 1311: Bynian, Ian Ring Music TheoryBynian
Scale 1343Scale 1343: Zalyllic, Ian Ring Music TheoryZalyllic
Scale 1439Scale 1439: Rolyllic, Ian Ring Music TheoryRolyllic
Scale 1503Scale 1503: Padygic, Ian Ring Music TheoryPadygic
Scale 1119Scale 1119: Rarian, Ian Ring Music TheoryRarian
Scale 1247Scale 1247: Aeodyllic, Ian Ring Music TheoryAeodyllic
Scale 1631Scale 1631: Rynyllic, Ian Ring Music TheoryRynyllic
Scale 1887Scale 1887: Aerocrygic, Ian Ring Music TheoryAerocrygic
Scale 351Scale 351: Epanian, Ian Ring Music TheoryEpanian
Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic
Scale 2399Scale 2399: Zanyllic, Ian Ring Music TheoryZanyllic
Scale 3423Scale 3423: Lothygic, Ian Ring Music TheoryLothygic

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.