The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 171: "Pruian"

Scale 171: Pruian, 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
Pruian

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

{0,1,3,5,7}

Forte Number

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

5-24

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: 2721

Hemitonia

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

1 (unhemitonic)

Cohemitonia

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

0 (ancohemitonic)

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.

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.

no

Interval Structure

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

[1, 2, 2, 2, 5]

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.

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

Interval Spectrum

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

p2m2ns3dt

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

Spectra Variation

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

3.2

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.

1.799

Polygon Perimeter

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

5.449

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.

(9, 3, 32)

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 Triadscm{0,3,7}000

The following pitch classes are not present in any of the common triads: {1,5}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 171 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2133
Scale 2133: Raga Kumurdaki, Ian Ring Music TheoryRaga Kumurdaki
3rd mode:
Scale 1557
Scale 1557: Jovian, Ian Ring Music TheoryJovian
4th mode:
Scale 1413
Scale 1413: Iruian, Ian Ring Music TheoryIruian
5th mode:
Scale 1377
Scale 1377: Insian, Ian Ring Music TheoryInsian

Prime

This is the prime form of this scale.

Complement

The pentatonic modal family [171, 2133, 1557, 1413, 1377] (Forte: 5-24) is the complement of the heptatonic modal family [687, 1401, 1509, 1941, 2391, 3243, 3669] (Forte: 7-24)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 171 is 2721

Scale 2721Scale 2721: Raga Puruhutika, Ian Ring Music TheoryRaga Puruhutika

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 171 is chiral, and its enantiomorph is scale 2721

Scale 2721Scale 2721: Raga Puruhutika, Ian Ring Music TheoryRaga Puruhutika

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> 171       T0I <11,0> 2721
T1 <1,1> 342      T1I <11,1> 1347
T2 <1,2> 684      T2I <11,2> 2694
T3 <1,3> 1368      T3I <11,3> 1293
T4 <1,4> 2736      T4I <11,4> 2586
T5 <1,5> 1377      T5I <11,5> 1077
T6 <1,6> 2754      T6I <11,6> 2154
T7 <1,7> 1413      T7I <11,7> 213
T8 <1,8> 2826      T8I <11,8> 426
T9 <1,9> 1557      T9I <11,9> 852
T10 <1,10> 3114      T10I <11,10> 1704
T11 <1,11> 2133      T11I <11,11> 3408
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2091      T0MI <7,0> 2691
T1M <5,1> 87      T1MI <7,1> 1287
T2M <5,2> 174      T2MI <7,2> 2574
T3M <5,3> 348      T3MI <7,3> 1053
T4M <5,4> 696      T4MI <7,4> 2106
T5M <5,5> 1392      T5MI <7,5> 117
T6M <5,6> 2784      T6MI <7,6> 234
T7M <5,7> 1473      T7MI <7,7> 468
T8M <5,8> 2946      T8MI <7,8> 936
T9M <5,9> 1797      T9MI <7,9> 1872
T10M <5,10> 3594      T10MI <7,10> 3744
T11M <5,11> 3093      T11MI <7,11> 3393

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 169Scale 169: Vietnamese Tetratonic, Ian Ring Music TheoryVietnamese Tetratonic
Scale 173Scale 173: Raga Purnalalita, Ian Ring Music TheoryRaga Purnalalita
Scale 175Scale 175: Bewian, Ian Ring Music TheoryBewian
Scale 163Scale 163: Bapian, Ian Ring Music TheoryBapian
Scale 167Scale 167: Barian, Ian Ring Music TheoryBarian
Scale 179Scale 179: Beyian, Ian Ring Music TheoryBeyian
Scale 187Scale 187: Bedian, Ian Ring Music TheoryBedian
Scale 139Scale 139: Ayoian, Ian Ring Music TheoryAyoian
Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian
Scale 203Scale 203: Ichian, Ian Ring Music TheoryIchian
Scale 235Scale 235: Bihian, Ian Ring Music TheoryBihian
Scale 43Scale 43: Alfian, Ian Ring Music TheoryAlfian
Scale 107Scale 107: Ansian, Ian Ring Music TheoryAnsian
Scale 299Scale 299: Raga Chitthakarshini, Ian Ring Music TheoryRaga Chitthakarshini
Scale 427Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha Simantini
Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic
Scale 1195Scale 1195: Raga Gandharavam, Ian Ring Music TheoryRaga Gandharavam
Scale 2219Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic

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.