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Scale 695: "Sarian"

Scale 695: Sarian, 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
Sarian
Dozenal
ERFian

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

 {0,1,2,4,5,7,9}

Forte Number

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

 7-27

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: 3497

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.

 2

Modes

Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 7

Prime Form

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

 yes

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, 2, 1, 2, 2, 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, 4, 4, 4, 5, 1>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.25, 0.5, 0.5, 0.333, 0.75, 0>

Interval Spectrum

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

 p5m4n4s4d3t

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> = {4,5,6,7}
<4> = {5,6,7,8}
<5> = {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.

 2.286

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.

2.549

Polygon Perimeter

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

5.967

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.

(9, 35, 98)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.686

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.222

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.

Generator

This scale has no generator.

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}241.86
F{5,9,0}231.57
A{9,1,4}321.29
Minor Triadsdm{2,5,9}142.14
am{9,0,4}331.43
Augmented TriadsC♯+{1,5,9}331.43
Diminished Triadsc♯°{1,4,7}231.71
Parsimonious Voice Leading Between Common Triads of Scale 695. Created by Ian Ring ©2019 C C c#° c#° C->c#° am am C->am A A c#°->A C#+ C#+ dm dm C#+->dm F F C#+->F C#+->A F->am am->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
Radius2
Self-Centeredno
Central VerticesA
Peripheral VerticesC, dm

Modes

Modes are the rotational transformation of this scale. Scale 695 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2395		Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
3rd mode:
Scale 3245		Scale 3245: Mela Varunapriya, Ian Ring Music TheoryMela Varunapriya
4th mode:
Scale 1835		Scale 1835: Byptian, Ian Ring Music TheoryByptian
5th mode:
Scale 2965		Scale 2965: Darian, Ian Ring Music TheoryDarian
6th mode:
Scale 1765		Scale 1765: Lonian, Ian Ring Music TheoryLonian
7th mode:
Scale 1465		Scale 1465: Mela Ragavardhani, Ian Ring Music TheoryMela Ragavardhani

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [695, 2395, 3245, 1835, 2965, 1765, 1465] (Forte: 7-27) is the complement of the pentatonic modal family [299, 689, 1417, 1573, 2197] (Forte: 5-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 695 is 3497

Scale 3497Scale 3497: Phrolian, Ian Ring Music TheoryPhrolian

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
11111011010100695k = 1h = 1
21111011010100695k = 2h = 1
31111011010100695k = 3h = 1
4211(10)1(10)(10)(10)0695k = 4h = 2
5211(10)1(10)(10)(10)0695k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 695 is chiral, and its enantiomorph is scale 3497

Scale 3497Scale 3497: Phrolian, Ian Ring Music TheoryPhrolian

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 695       T0I <11,0> 3497
T1 <1,1> 1390      T1I <11,1> 2899
T2 <1,2> 2780      T2I <11,2> 1703
T3 <1,3> 1465      T3I <11,3> 3406
T4 <1,4> 2930      T4I <11,4> 2717
T5 <1,5> 1765      T5I <11,5> 1339
T6 <1,6> 3530      T6I <11,6> 2678
T7 <1,7> 2965      T7I <11,7> 1261
T8 <1,8> 1835      T8I <11,8> 2522
T9 <1,9> 3670      T9I <11,9> 949
T10 <1,10> 3245      T10I <11,10> 1898
T11 <1,11> 2395      T11I <11,11> 3796
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3875      T0MI <7,0> 2207
T1M <5,1> 3655      T1MI <7,1> 319
T2M <5,2> 3215      T2MI <7,2> 638
T3M <5,3> 2335      T3MI <7,3> 1276
T4M <5,4> 575      T4MI <7,4> 2552
T5M <5,5> 1150      T5MI <7,5> 1009
T6M <5,6> 2300      T6MI <7,6> 2018
T7M <5,7> 505      T7MI <7,7> 4036
T8M <5,8> 1010      T8MI <7,8> 3977
T9M <5,9> 2020      T9MI <7,9> 3859
T10M <5,10> 4040      T10MI <7,10> 3623
T11M <5,11> 3985      T11MI <7,11> 3151

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 693Scale 693: Arezzo Major Diatonic Hexachord, Ian Ring Music TheoryArezzo Major Diatonic Hexachord
Scale 691Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga Kalavati
Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian
Scale 703Scale 703: Aerocryllic, Ian Ring Music TheoryAerocryllic
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 687Scale 687: Aeolythian, Ian Ring Music TheoryAeolythian
Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic
Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
Scale 631Scale 631: Zygian, Ian Ring Music TheoryZygian
Scale 823Scale 823: Stodian, Ian Ring Music TheoryStodian
Scale 951Scale 951: Thogyllic, Ian Ring Music TheoryThogyllic
Scale 183Scale 183: BEBian, Ian Ring Music TheoryBEBian
Scale 439Scale 439: Bythian, Ian Ring Music TheoryBythian
Scale 1207Scale 1207: Aeoloptian, Ian Ring Music TheoryAeoloptian
Scale 1719Scale 1719: Lyryllic, Ian Ring Music TheoryLyryllic
Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.