The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1369: "Boptimic"

Scale 1369: Boptimic, 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
Boptimic
Dozenal
Iltian

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

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

Forte Number

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

6-34

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: 853

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.

5

Prime Form

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

no
prime: 683

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.

[3, 1, 2, 2, 2, 2]

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, 4, 2, 4, 2, 2>

Interval Spectrum

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

p2m4n2s4dt2

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

Spectra Variation

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

1.667

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.

2.482

Polygon Perimeter

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

5.932

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".

Proper

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.

(0, 4, 45)

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 TriadsG♯{8,0,3}221
Minor Triadsd♯m{3,6,10}131.5
Augmented TriadsC+{0,4,8}131.5
Diminished Triads{0,3,6}221
Parsimonious Voice Leading Between Common Triads of Scale 1369. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ C+->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°, G♯
Peripheral VerticesC+, d♯m

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There is 1 way that this hexatonic scale can be split into two common triads.


Augmented: {0, 4, 8}
Minor: {3, 6, 10}

Modes

Modes are the rotational transformation of this scale. Scale 1369 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 683
Scale 683: Stogimic, Ian Ring Music TheoryStogimicThis is the prime mode
3rd mode:
Scale 2389
Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
4th mode:
Scale 1621
Scale 1621: Scriabin's Prometheus, Ian Ring Music TheoryScriabin's Prometheus
5th mode:
Scale 1429
Scale 1429: Bythimic, Ian Ring Music TheoryBythimic
6th mode:
Scale 1381
Scale 1381: Padimic, Ian Ring Music TheoryPadimic

Prime

The prime form of this scale is Scale 683

Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic

Complement

The hexatonic modal family [1369, 683, 2389, 1621, 1429, 1381] (Forte: 6-34) is the complement of the hexatonic modal family [683, 1369, 1381, 1429, 1621, 2389] (Forte: 6-34)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1369 is 853

Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1369 is chiral, and its enantiomorph is scale 853

Scale 853Scale 853: Epothimic, Ian Ring Music TheoryEpothimic

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> 1369       T0I <11,0> 853
T1 <1,1> 2738      T1I <11,1> 1706
T2 <1,2> 1381      T2I <11,2> 3412
T3 <1,3> 2762      T3I <11,3> 2729
T4 <1,4> 1429      T4I <11,4> 1363
T5 <1,5> 2858      T5I <11,5> 2726
T6 <1,6> 1621      T6I <11,6> 1357
T7 <1,7> 3242      T7I <11,7> 2714
T8 <1,8> 2389      T8I <11,8> 1333
T9 <1,9> 683      T9I <11,9> 2666
T10 <1,10> 1366      T10I <11,10> 1237
T11 <1,11> 2732      T11I <11,11> 2474
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 349      T0MI <7,0> 1873
T1M <5,1> 698      T1MI <7,1> 3746
T2M <5,2> 1396      T2MI <7,2> 3397
T3M <5,3> 2792      T3MI <7,3> 2699
T4M <5,4> 1489      T4MI <7,4> 1303
T5M <5,5> 2978      T5MI <7,5> 2606
T6M <5,6> 1861      T6MI <7,6> 1117
T7M <5,7> 3722      T7MI <7,7> 2234
T8M <5,8> 3349      T8MI <7,8> 373
T9M <5,9> 2603      T9MI <7,9> 746
T10M <5,10> 1111      T10MI <7,10> 1492
T11M <5,11> 2222      T11MI <7,11> 2984

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 1371Scale 1371: Superlocrian, Ian Ring Music TheorySuperlocrian
Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian
Scale 1361Scale 1361: Bolitonic, Ian Ring Music TheoryBolitonic
Scale 1365Scale 1365: Whole Tone, Ian Ring Music TheoryWhole Tone
Scale 1353Scale 1353: Raga Harikauns, Ian Ring Music TheoryRaga Harikauns
Scale 1385Scale 1385: Phracrimic, Ian Ring Music TheoryPhracrimic
Scale 1401Scale 1401: Pagian, Ian Ring Music TheoryPagian
Scale 1305Scale 1305: Dynitonic, Ian Ring Music TheoryDynitonic
Scale 1337Scale 1337: Epogimic, Ian Ring Music TheoryEpogimic
Scale 1433Scale 1433: Dynimic, Ian Ring Music TheoryDynimic
Scale 1497Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini
Scale 1113Scale 1113: Locrian Pentatonic 2, Ian Ring Music TheoryLocrian Pentatonic 2
Scale 1241Scale 1241: Pygimic, Ian Ring Music TheoryPygimic
Scale 1625Scale 1625: Lythimic, Ian Ring Music TheoryLythimic
Scale 1881Scale 1881: Katorian, Ian Ring Music TheoryKatorian
Scale 345Scale 345: Gylitonic, Ian Ring Music TheoryGylitonic
Scale 857Scale 857: Aeolydimic, Ian Ring Music TheoryAeolydimic
Scale 2393Scale 2393: Zathimic, Ian Ring Music TheoryZathimic
Scale 3417Scale 3417: Golian, Ian Ring Music TheoryGolian

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.