The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3677: "Xafian"

Scale 3677: Xafian, 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
Xafian

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

{0,2,3,4,6,9,10,11}

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-Z29

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: 1871

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.

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.

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: 751

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.

[2, 1, 1, 2, 3, 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, 5, 5, 5, 5, 3>

Interval Spectrum

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

p5m5n5s5d5t3

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

Spectra Variation

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

2.25

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.

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.

(30, 60, 141)

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 TriadsD{2,6,9}341.9
B{11,3,6}341.9
Minor Triadsd♯m{3,6,10}341.9
am{9,0,4}242.3
bm{11,2,6}242.1
Augmented TriadsD+{2,6,10}341.9
Diminished Triads{0,3,6}242.1
d♯°{3,6,9}242.1
f♯°{6,9,0}242.1
{9,0,3}242.3
Parsimonious Voice Leading Between Common Triads of Scale 3677. Created by Ian Ring ©2019 c°->a° B B c°->B D D D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° d#m d#m D+->d#m bm bm D+->bm d#°->d#m d#m->B am am f#°->am a°->am bm->B

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 3677 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1943
Scale 1943: Luxian, Ian Ring Music TheoryLuxian
3rd mode:
Scale 3019
Scale 3019: Subian, Ian Ring Music TheorySubian
4th mode:
Scale 3557
Scale 3557: Wekian, Ian Ring Music TheoryWekian
5th mode:
Scale 1913
Scale 1913: Lofian, Ian Ring Music TheoryLofian
6th mode:
Scale 751
Scale 751: Epoian, Ian Ring Music TheoryEpoianThis is the prime mode
7th mode:
Scale 2423
Scale 2423: Otuian, Ian Ring Music TheoryOtuian
8th mode:
Scale 3259
Scale 3259: Ulian, Ian Ring Music TheoryUlian

Prime

The prime form of this scale is Scale 751

Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian

Complement

The octatonic modal family [3677, 1943, 3019, 3557, 1913, 751, 2423, 3259] (Forte: 8-Z29) is the complement of the tetratonic modal family [139, 353, 1553, 2117] (Forte: 4-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3677 is 1871

Scale 1871Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3677 is chiral, and its enantiomorph is scale 1871

Scale 1871Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic

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> 3677       T0I <11,0> 1871
T1 <1,1> 3259      T1I <11,1> 3742
T2 <1,2> 2423      T2I <11,2> 3389
T3 <1,3> 751      T3I <11,3> 2683
T4 <1,4> 1502      T4I <11,4> 1271
T5 <1,5> 3004      T5I <11,5> 2542
T6 <1,6> 1913      T6I <11,6> 989
T7 <1,7> 3826      T7I <11,7> 1978
T8 <1,8> 3557      T8I <11,8> 3956
T9 <1,9> 3019      T9I <11,9> 3817
T10 <1,10> 1943      T10I <11,10> 3539
T11 <1,11> 3886      T11I <11,11> 2983
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1997      T0MI <7,0> 1661
T1M <5,1> 3994      T1MI <7,1> 3322
T2M <5,2> 3893      T2MI <7,2> 2549
T3M <5,3> 3691      T3MI <7,3> 1003
T4M <5,4> 3287      T4MI <7,4> 2006
T5M <5,5> 2479      T5MI <7,5> 4012
T6M <5,6> 863      T6MI <7,6> 3929
T7M <5,7> 1726      T7MI <7,7> 3763
T8M <5,8> 3452      T8MI <7,8> 3431
T9M <5,9> 2809      T9MI <7,9> 2767
T10M <5,10> 1523      T10MI <7,10> 1439
T11M <5,11> 3046      T11MI <7,11> 2878

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 3679Scale 3679: Rycrygic, Ian Ring Music TheoryRycrygic
Scale 3673Scale 3673: Ranian, Ian Ring Music TheoryRanian
Scale 3675Scale 3675: Monyllic, Ian Ring Music TheoryMonyllic
Scale 3669Scale 3669: Mothian, Ian Ring Music TheoryMothian
Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian
Scale 3693Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic
Scale 3709Scale 3709: Katynygic, Ian Ring Music TheoryKatynygic
Scale 3613Scale 3613: Wosian, Ian Ring Music TheoryWosian
Scale 3645Scale 3645: Zycryllic, Ian Ring Music TheoryZycryllic
Scale 3741Scale 3741: Zydyllic, Ian Ring Music TheoryZydyllic
Scale 3805Scale 3805: Moptygic, Ian Ring Music TheoryMoptygic
Scale 3933Scale 3933: Ionidygic, Ian Ring Music TheoryIonidygic
Scale 3165Scale 3165: Mylian, Ian Ring Music TheoryMylian
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 2653Scale 2653: Sygian, Ian Ring Music TheorySygian
Scale 1629Scale 1629: Synian, Ian Ring Music TheorySynian

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.