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Scale 899: "Foqian"

Scale 899: Foqian, 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
Foqian

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,1,7,8,9}

Forte Number

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

5-6

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: 2105

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

4

Prime Form

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

no
prime: 103

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, 6, 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.

<3, 1, 1, 2, 2, 1>

Interval Spectrum

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

p2m2nsd3t

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> = {1,3,6}
<2> = {2,4,7}
<3> = {5,8,10}
<4> = {6,9,11}

Spectra Variation

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

4

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.

1.25

Polygon Perimeter

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

4.967

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.

(16, 1, 30)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 899 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2497
Scale 2497: Peqian, Ian Ring Music TheoryPeqian
3rd mode:
Scale 103
Scale 103: Apuian, Ian Ring Music TheoryApuianThis is the prime mode
4th mode:
Scale 2099
Scale 2099: Raga Megharanji, Ian Ring Music TheoryRaga Megharanji
5th mode:
Scale 3097
Scale 3097: Tiwian, Ian Ring Music TheoryTiwian

Prime

The prime form of this scale is Scale 103

Scale 103Scale 103: Apuian, Ian Ring Music TheoryApuian

Complement

The pentatonic modal family [899, 2497, 103, 2099, 3097] (Forte: 5-6) is the complement of the heptatonic modal family [415, 995, 2255, 2545, 3175, 3635, 3865] (Forte: 7-6)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 899 is 2105

Scale 2105Scale 2105: Rigian, Ian Ring Music TheoryRigian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 899 is chiral, and its enantiomorph is scale 2105

Scale 2105Scale 2105: Rigian, Ian Ring Music TheoryRigian

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> 899       T0I <11,0> 2105
T1 <1,1> 1798      T1I <11,1> 115
T2 <1,2> 3596      T2I <11,2> 230
T3 <1,3> 3097      T3I <11,3> 460
T4 <1,4> 2099      T4I <11,4> 920
T5 <1,5> 103      T5I <11,5> 1840
T6 <1,6> 206      T6I <11,6> 3680
T7 <1,7> 412      T7I <11,7> 3265
T8 <1,8> 824      T8I <11,8> 2435
T9 <1,9> 1648      T9I <11,9> 775
T10 <1,10> 3296      T10I <11,10> 1550
T11 <1,11> 2497      T11I <11,11> 3100
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2609      T0MI <7,0> 395
T1M <5,1> 1123      T1MI <7,1> 790
T2M <5,2> 2246      T2MI <7,2> 1580
T3M <5,3> 397      T3MI <7,3> 3160
T4M <5,4> 794      T4MI <7,4> 2225
T5M <5,5> 1588      T5MI <7,5> 355
T6M <5,6> 3176      T6MI <7,6> 710
T7M <5,7> 2257      T7MI <7,7> 1420
T8M <5,8> 419      T8MI <7,8> 2840
T9M <5,9> 838      T9MI <7,9> 1585
T10M <5,10> 1676      T10MI <7,10> 3170
T11M <5,11> 3352      T11MI <7,11> 2245

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 897Scale 897: Fopian, Ian Ring Music TheoryFopian
Scale 901Scale 901: Bofian, Ian Ring Music TheoryBofian
Scale 903Scale 903: Fosian, Ian Ring Music TheoryFosian
Scale 907Scale 907: Tholimic, Ian Ring Music TheoryTholimic
Scale 915Scale 915: Raga Kalagada, Ian Ring Music TheoryRaga Kalagada
Scale 931Scale 931: Raga Kalakanthi, Ian Ring Music TheoryRaga Kalakanthi
Scale 963Scale 963: Gacian, Ian Ring Music TheoryGacian
Scale 771Scale 771: Esoian, Ian Ring Music TheoryEsoian
Scale 835Scale 835: Fecian, Ian Ring Music TheoryFecian
Scale 643Scale 643: Duxian, Ian Ring Music TheoryDuxian
Scale 387Scale 387: Ciwian, Ian Ring Music TheoryCiwian
Scale 1411Scale 1411: Iroian, Ian Ring Music TheoryIroian
Scale 1923Scale 1923: Lulian, Ian Ring Music TheoryLulian
Scale 2947Scale 2947: Sijian, Ian Ring Music TheorySijian

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.