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Scale 695: "Sarian"

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
Centre of Gravity Distance When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.	0.177045
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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Major Triads	C	{0,4,7}	2	4	1.86
	F	{5,9,0}	2	3	1.57
	A	{9,1,4}	3	2	1.29
Minor Triads	dm	{2,5,9}	1	4	2.14
Minor Triads	am	{9,0,4}	3	3	1.43
Augmented Triads	C♯+	{1,5,9}	3	3	1.43
Diminished Triads	c♯°	{1,4,7}	2	3	1.71

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.

Diameter	4
Radius	2
Self-Centered	no
Central Vertices	A
Peripheral Vertices	C, 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				Zoptian
3rd mode: Scale 3245				Mela Varunapriya
4th mode: Scale 1835				Byptian
5th mode: Scale 2965				Darian
6th mode: Scale 1765				Lonian
7th mode: Scale 1465				Mela 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 3497

Phrolian

Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

Contradictions (9)

Ambiguities(35)

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.

k	Hierarchizability	Breakdown Pattern
1	1	`111011010100`
2	1	`111011010100`
3	1	`111011010100`
4	2	`11(10)1(10)(10)(10)0`
5	2	`11(10)1(10)(10)(10)0`

Enantiomorph

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

Scale 3497

Phrolian

Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center	(0.176007, -0.019139)
Distance from Center	0.177045
Angle in degrees measured clockwise starting from the root.	83.794
Angle in cents 100 cents = 1 semitone.	279.313333

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 693				Arezzo Major Diatonic Hexachord
Scale 691				Raga Kalavati
Scale 699				Aerothian
Scale 703				Aerocryllic
Scale 679				Lanimic
Scale 687				Aeolythian
Scale 663				Phrynimic
Scale 727				Phradian
Scale 759				Katalyllic
Scale 567				Aeoladimic
Scale 631				Zygian
Scale 823				Stodian
Scale 951				Thogyllic
Scale 183				BEBian
Scale 439				Bythian
Scale 1207				Aeoloptian
Scale 1719				Lyryllic
Scale 2743				Staptyllic

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. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.