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Scale 1671: "Kemian"

Scale 1671: Kemian, 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
Kemian

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

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

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-Z11

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: 3117

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.

5

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 183

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, an indicator of maximum hierarchization.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 5, 2, 1, 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.

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

Interval Spectrum

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

p3m2n3s3d3t

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

Spectra Variation

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

3.667

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.

1.866

Polygon Perimeter

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

5.485

Myhill Property

A scale has Myhill Property if the Distribution Spectra have 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.

(27, 11, 59)

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
Minor Triadsgm{7,10,2}110.5
Diminished Triads{7,10,1}110.5

The following pitch classes are not present in any of the common triads: {0,9}

Parsimonious Voice Leading Between Common Triads of Scale 1671. Created by Ian Ring ©2019 gm gm g°->gm

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1671 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2883
Scale 2883: Savian, Ian Ring Music TheorySavian
3rd mode:
Scale 3489
Scale 3489: Vuvian, Ian Ring Music TheoryVuvian
4th mode:
Scale 237
Scale 237: Bijian, Ian Ring Music TheoryBijian
5th mode:
Scale 1083
Scale 1083: Goyian, Ian Ring Music TheoryGoyian
6th mode:
Scale 2589
Scale 2589: Puvian, Ian Ring Music TheoryPuvian

Prime

The prime form of this scale is Scale 183

Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian

Complement

The hexatonic modal family [1671, 2883, 3489, 237, 1083, 2589] (Forte: 6-Z11) is the complement of the hexatonic modal family [303, 753, 1929, 2199, 3147, 3621] (Forte: 6-Z40)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1671 is 3117

Scale 3117Scale 3117: Tijian, Ian Ring Music TheoryTijian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1671 is chiral, and its enantiomorph is scale 3117

Scale 3117Scale 3117: Tijian, Ian Ring Music TheoryTijian

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> 1671       T0I <11,0> 3117
T1 <1,1> 3342      T1I <11,1> 2139
T2 <1,2> 2589      T2I <11,2> 183
T3 <1,3> 1083      T3I <11,3> 366
T4 <1,4> 2166      T4I <11,4> 732
T5 <1,5> 237      T5I <11,5> 1464
T6 <1,6> 474      T6I <11,6> 2928
T7 <1,7> 948      T7I <11,7> 1761
T8 <1,8> 1896      T8I <11,8> 3522
T9 <1,9> 3792      T9I <11,9> 2949
T10 <1,10> 3489      T10I <11,10> 1803
T11 <1,11> 2883      T11I <11,11> 3606
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3621      T0MI <7,0> 1167
T1M <5,1> 3147      T1MI <7,1> 2334
T2M <5,2> 2199      T2MI <7,2> 573
T3M <5,3> 303      T3MI <7,3> 1146
T4M <5,4> 606      T4MI <7,4> 2292
T5M <5,5> 1212      T5MI <7,5> 489
T6M <5,6> 2424      T6MI <7,6> 978
T7M <5,7> 753      T7MI <7,7> 1956
T8M <5,8> 1506      T8MI <7,8> 3912
T9M <5,9> 3012      T9MI <7,9> 3729
T10M <5,10> 1929      T10MI <7,10> 3363
T11M <5,11> 3858      T11MI <7,11> 2631

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 1669Scale 1669: Raga Matha Kokila, Ian Ring Music TheoryRaga Matha Kokila
Scale 1667Scale 1667: Kekian, Ian Ring Music TheoryKekian
Scale 1675Scale 1675: Raga Salagavarali, Ian Ring Music TheoryRaga Salagavarali
Scale 1679Scale 1679: Kydian, Ian Ring Music TheoryKydian
Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian
Scale 1703Scale 1703: Mela Vanaspati, Ian Ring Music TheoryMela Vanaspati
Scale 1735Scale 1735: Mela Navanitam, Ian Ring Music TheoryMela Navanitam
Scale 1543Scale 1543: Jomian, Ian Ring Music TheoryJomian
Scale 1607Scale 1607: Epytimic, Ian Ring Music TheoryEpytimic
Scale 1799Scale 1799: Lamian, Ian Ring Music TheoryLamian
Scale 1927Scale 1927: Lunian, Ian Ring Music TheoryLunian
Scale 1159Scale 1159: Hasian, Ian Ring Music TheoryHasian
Scale 1415Scale 1415: Impian, Ian Ring Music TheoryImpian
Scale 647Scale 647: Duzian, Ian Ring Music TheoryDuzian
Scale 2695Scale 2695: Rakian, Ian Ring Music TheoryRakian
Scale 3719Scale 3719: Xofian, Ian Ring Music TheoryXofian

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.