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Scale 3825: "Pynyllic"

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: Pynyllic

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	{0,4,5,6,7,9,10,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	8-6
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.	[2]
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.	no
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	6 (multihemitonic)
Cohemitonia A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.	4 (multicohemitonic)
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 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: 495
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.	[4, 1, 1, 1, 2, 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.	<6, 5, 4, 4, 6, 3>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p⁶m⁴n⁴s⁵d⁶t³
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> = {1,2,4} <2> = {2,3,5} <3> = {3,4,6} <4> = {5,7} <5> = {6,8,9} <6> = {7,9,10} <7> = {8,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	2.5
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.	2.366
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.838
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.	[4]
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.	(60, 47, 124)

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	3	1.5
Major Triads	F	{5,9,0}	2	4	1.83
Minor Triads	em	{4,7,11}	2	4	1.83
Minor Triads	am	{9,0,4}	2	3	1.5
Diminished Triads	e°	{4,7,10}	1	5	2.5
Diminished Triads	f♯°	{6,9,0}	1	5	2.5

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	5
Radius	3
Self-Centered	no
Central Vertices	C, am
Peripheral Vertices	e°, f♯°

Modes

Modes are the rotational transformation of this scale. Scale 3825 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode: Scale 495	Bocryllic	This is the prime mode
3rd mode: Scale 2295	Kogyllic
4th mode: Scale 3195	Raryllic
5th mode: Scale 3645	Zycryllic
6th mode: Scale 1935	Mycryllic
7th mode: Scale 3015	Laptyllic
8th mode: Scale 3555	Pylyllic

Prime

The prime form of this scale is Scale 495

Scale 495

Bocryllic

Complement

The octatonic modal family [3825, 495, 2295, 3195, 3645, 1935, 3015, 3555] (Forte: 8-6) is the complement of the tetratonic modal family [135, 225, 2115, 3105] (Forte: 4-6)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3825 is 495

Scale 495

Bocryllic

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>	3825	T₀I	<11,0>	495
T₁	<1,1>	3555	T₁I	<11,1>	990
T₂	<1,2>	3015	T₂I	<11,2>	1980
T₃	<1,3>	1935	T₃I	<11,3>	3960
T₄	<1,4>	3870	T₄I	<11,4>	3825
T₅	<1,5>	3645	T₅I	<11,5>	3555
T₆	<1,6>	3195	T₆I	<11,6>	3015
T₇	<1,7>	2295	T₇I	<11,7>	1935
T₈	<1,8>	495	T₈I	<11,8>	3870
T₉	<1,9>	990	T₉I	<11,9>	3645
T₁₀	<1,10>	1980	T₁₀I	<11,10>	3195
T₁₁	<1,11>	3960	T₁₁I	<11,11>	2295
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	3015	T₀MI	<7,0>	3195
T₁M	<5,1>	1935	T₁MI	<7,1>	2295
T₂M	<5,2>	3870	T₂MI	<7,2>	495
T₃M	<5,3>	3645	T₃MI	<7,3>	990
T₄M	<5,4>	3195	T₄MI	<7,4>	1980
T₅M	<5,5>	2295	T₅MI	<7,5>	3960
T₆M	<5,6>	495	T₆MI	<7,6>	3825
T₇M	<5,7>	990	T₇MI	<7,7>	3555
T₈M	<5,8>	1980	T₈MI	<7,8>	3015
T₉M	<5,9>	3960	T₉MI	<7,9>	1935
T₁₀M	<5,10>	3825	T₁₀MI	<7,10>	3870
T₁₁M	<5,11>	3555	T₁₁MI	<7,11>	3645

The transformations that map this set to itself are: T₀, T₄I, T₁₀M, T₆MI

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 3827				Bodygic
Scale 3829				Taishikicho
Scale 3833				Dycrygic
Scale 3809				Yelian
Scale 3817				Zoryllic
Scale 3793				Aeopian
Scale 3761				Raga Madhuri
Scale 3697				Ionarian
Scale 3953				Thagyllic
Scale 4081				Nonatonic Chromatic Descending
Scale 3313				Aeolacrian
Scale 3569				Aeoladyllic
Scale 2801				Zogian
Scale 1777				Saptian

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.