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Scale 1001: "Badian"

Scale 1001: Badian, 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
Badian
Dozenal
Gezian

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

{0,3,5,6,7,8,9}

Forte Number

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

7-10

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: 761

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.

6

Prime Form

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

no
prime: 607

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

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, 5, 3, 3, 2>

Interval Spectrum

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

p3m3n5s4d4t2

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

Spectra Variation

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

3.143

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.

2.433

Polygon Perimeter

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

5.899

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.

(46, 40, 104)

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{5,9,0}331.63
G♯{8,0,3}331.63
Minor Triadscm{0,3,7}231.75
fm{5,8,0}231.75
Diminished Triads{0,3,6}231.88
d♯°{3,6,9}231.88
f♯°{6,9,0}231.75
{9,0,3}231.75
Parsimonious Voice Leading Between Common Triads of Scale 1001. Created by Ian Ring ©2019 cm cm c°->cm d#° d#° c°->d#° G# G# cm->G# f#° f#° d#°->f#° fm fm F F fm->F fm->G# F->f#° F->a° G#->a°

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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1001 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 637
Scale 637: Debussy's Heptatonic, Ian Ring Music TheoryDebussy's Heptatonic
3rd mode:
Scale 1183
Scale 1183: Sadian, Ian Ring Music TheorySadian
4th mode:
Scale 2639
Scale 2639: Dothian, Ian Ring Music TheoryDothian
5th mode:
Scale 3367
Scale 3367: Moptian, Ian Ring Music TheoryMoptian
6th mode:
Scale 3731
Scale 3731: Aeryrian, Ian Ring Music TheoryAeryrian
7th mode:
Scale 3913
Scale 3913: Bonian, Ian Ring Music TheoryBonian

Prime

The prime form of this scale is Scale 607

Scale 607Scale 607: Kadian, Ian Ring Music TheoryKadian

Complement

The heptatonic modal family [1001, 637, 1183, 2639, 3367, 3731, 3913] (Forte: 7-10) is the complement of the pentatonic modal family [91, 1547, 1729, 2093, 2821] (Forte: 5-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1001 is 761

Scale 761Scale 761: Ponian, Ian Ring Music TheoryPonian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1001 is chiral, and its enantiomorph is scale 761

Scale 761Scale 761: Ponian, Ian Ring Music TheoryPonian

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> 1001       T0I <11,0> 761
T1 <1,1> 2002      T1I <11,1> 1522
T2 <1,2> 4004      T2I <11,2> 3044
T3 <1,3> 3913      T3I <11,3> 1993
T4 <1,4> 3731      T4I <11,4> 3986
T5 <1,5> 3367      T5I <11,5> 3877
T6 <1,6> 2639      T6I <11,6> 3659
T7 <1,7> 1183      T7I <11,7> 3223
T8 <1,8> 2366      T8I <11,8> 2351
T9 <1,9> 637      T9I <11,9> 607
T10 <1,10> 1274      T10I <11,10> 1214
T11 <1,11> 2548      T11I <11,11> 2428
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2651      T0MI <7,0> 2891
T1M <5,1> 1207      T1MI <7,1> 1687
T2M <5,2> 2414      T2MI <7,2> 3374
T3M <5,3> 733      T3MI <7,3> 2653
T4M <5,4> 1466      T4MI <7,4> 1211
T5M <5,5> 2932      T5MI <7,5> 2422
T6M <5,6> 1769      T6MI <7,6> 749
T7M <5,7> 3538      T7MI <7,7> 1498
T8M <5,8> 2981      T8MI <7,8> 2996
T9M <5,9> 1867      T9MI <7,9> 1897
T10M <5,10> 3734      T10MI <7,10> 3794
T11M <5,11> 3373      T11MI <7,11> 3493

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 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 1005Scale 1005: Radyllic, Ian Ring Music TheoryRadyllic
Scale 993Scale 993: Gavian, Ian Ring Music TheoryGavian
Scale 997Scale 997: Rycrian, Ian Ring Music TheoryRycrian
Scale 1009Scale 1009: Katyptian, Ian Ring Music TheoryKatyptian
Scale 1017Scale 1017: Dythyllic, Ian Ring Music TheoryDythyllic
Scale 969Scale 969: Ionogimic, Ian Ring Music TheoryIonogimic
Scale 985Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
Scale 937Scale 937: Stothimic, Ian Ring Music TheoryStothimic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 745Scale 745: Kolimic, Ian Ring Music TheoryKolimic
Scale 489Scale 489: Phrathimic, Ian Ring Music TheoryPhrathimic
Scale 1513Scale 1513: Stathian, Ian Ring Music TheoryStathian
Scale 2025Scale 2025: Mivian, Ian Ring Music TheoryMivian
Scale 3049Scale 3049: Phrydyllic, Ian Ring Music TheoryPhrydyllic

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.