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Scale 3735: "Xupian"

Scale 3735: Xupian, 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

Dozenal
Xupian

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

{0,1,2,4,7,9,10,11}

Forte Number

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

8-10

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.

[5.5]

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.

5 (multihemitonic)

Cohemitonia

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

4 (multicohemitonic)

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.

3

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.

no
prime: 765

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.

[1, 1, 2, 3, 2, 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.

<5, 6, 6, 4, 5, 2>

Interval Spectrum

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

p5m4n6s6d5t2

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

Spectra Variation

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

2.75

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.

2.616

Polygon Perimeter

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

6.002

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.

[11]

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.

(65, 66, 144)

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 TriadsC{0,4,7}341.9
G{7,11,2}242.1
A{9,1,4}342.1
Minor Triadsem{4,7,11}341.9
gm{7,10,2}342.1
am{9,0,4}242.1
Diminished Triadsc♯°{1,4,7}242.1
{4,7,10}242.1
{7,10,1}242.1
a♯°{10,1,4}242.1
Parsimonious Voice Leading Between Common Triads of Scale 3735. Created by Ian Ring ©2019 C C c#° c#° C->c#° em em C->em am am C->am A A c#°->A e°->em gm gm e°->gm Parsimonious Voice Leading Between Common Triads of Scale 3735. Created by Ian Ring ©2019 G em->G g°->gm a#° a#° g°->a#° gm->G am->A A->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3735 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3915
Scale 3915: Yuyian, Ian Ring Music TheoryYuyian
3rd mode:
Scale 4005
Scale 4005: Zibian, Ian Ring Music TheoryZibian
4th mode:
Scale 2025
Scale 2025: Mivian, Ian Ring Music TheoryMivian
5th mode:
Scale 765
Scale 765: Erkian, Ian Ring Music TheoryErkianThis is the prime mode
6th mode:
Scale 1215
Scale 1215: Hibian, Ian Ring Music TheoryHibian
7th mode:
Scale 2655
Scale 2655: Qojian, Ian Ring Music TheoryQojian
8th mode:
Scale 3375
Scale 3375: Vecian, Ian Ring Music TheoryVecian

Prime

The prime form of this scale is Scale 765

Scale 765Scale 765: Erkian, Ian Ring Music TheoryErkian

Complement

The octatonic modal family [3735, 3915, 4005, 2025, 765, 1215, 2655, 3375] (Forte: 8-10) is the complement of the tetratonic modal family [45, 1035, 1665, 2565] (Forte: 4-10)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3735 is 3375

Scale 3375Scale 3375: Vecian, Ian Ring Music TheoryVecian

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> 3735       T0I <11,0> 3375
T1 <1,1> 3375      T1I <11,1> 2655
T2 <1,2> 2655      T2I <11,2> 1215
T3 <1,3> 1215      T3I <11,3> 2430
T4 <1,4> 2430      T4I <11,4> 765
T5 <1,5> 765      T5I <11,5> 1530
T6 <1,6> 1530      T6I <11,6> 3060
T7 <1,7> 3060      T7I <11,7> 2025
T8 <1,8> 2025      T8I <11,8> 4050
T9 <1,9> 4050      T9I <11,9> 4005
T10 <1,10> 4005      T10I <11,10> 3915
T11 <1,11> 3915      T11I <11,11> 3735
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4005      T0MI <7,0> 1215
T1M <5,1> 3915      T1MI <7,1> 2430
T2M <5,2> 3735       T2MI <7,2> 765
T3M <5,3> 3375      T3MI <7,3> 1530
T4M <5,4> 2655      T4MI <7,4> 3060
T5M <5,5> 1215      T5MI <7,5> 2025
T6M <5,6> 2430      T6MI <7,6> 4050
T7M <5,7> 765      T7MI <7,7> 4005
T8M <5,8> 1530      T8MI <7,8> 3915
T9M <5,9> 3060      T9MI <7,9> 3735
T10M <5,10> 2025      T10MI <7,10> 3375
T11M <5,11> 4050      T11MI <7,11> 2655

The transformations that map this set to itself are: T0, T11I, T2M, T9MI

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 3733Scale 3733: Gycrian, Ian Ring Music TheoryGycrian
Scale 3731Scale 3731: Aeryrian, Ian Ring Music TheoryAeryrian
Scale 3739Scale 3739: Epanyllic, Ian Ring Music TheoryEpanyllic
Scale 3743Scale 3743: Thadygic, Ian Ring Music TheoryThadygic
Scale 3719Scale 3719: Xofian, Ian Ring Music TheoryXofian
Scale 3727Scale 3727: Tholyllic, Ian Ring Music TheoryTholyllic
Scale 3751Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
Scale 3767Scale 3767: Chromatic Bebop, Ian Ring Music TheoryChromatic Bebop
Scale 3799Scale 3799: Aeralygic, Ian Ring Music TheoryAeralygic
Scale 3607Scale 3607: Wopian, Ian Ring Music TheoryWopian
Scale 3671Scale 3671: Aeonyllic, Ian Ring Music TheoryAeonyllic
Scale 3863Scale 3863: Eparyllic, Ian Ring Music TheoryEparyllic
Scale 3991Scale 3991: Badygic, Ian Ring Music TheoryBadygic
Scale 3223Scale 3223: Thyphian, Ian Ring Music TheoryThyphian
Scale 3479Scale 3479: Rothyllic, Ian Ring Music TheoryRothyllic
Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian
Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian

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.