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# Scale 1557: "Jovian" ### 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).

Dozenal
Jovian

## 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

{0,2,4,9,10}

#### Forte Number

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

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: 1293

#### 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 Starr/Rahn algorithm.

no
prime: 171

#### 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.

[2, 2, 5, 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.

<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 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, 3, 32)

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

The following pitch classes are not present in any of the common triads: {2,10}

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 1557 can be rotated to make 4 other scales. The 1st mode is itself.

 2nd mode:Scale 1413 Iruian 3rd mode:Scale 1377 Insian 4th mode:Scale 171 Pruian This is the prime mode 5th mode:Scale 2133 Raga Kumurdaki

## Prime

The prime form of this scale is Scale 171

 Scale 171 Pruian

## Complement

The pentatonic modal family [1557, 1413, 1377, 171, 2133] (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 1557 is 1293

 Scale 1293 Huxian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1557 is chiral, and its enantiomorph is scale 1293

 Scale 1293 Huxian

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

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 1559 Jowian Scale 1553 Josian Scale 1555 Jotian Scale 1561 Joxian Scale 1565 Jozian Scale 1541 Jilian Scale 1549 Joqian Scale 1573 Raga Guhamanohari Scale 1589 Raga Rageshri Scale 1621 Scriabin's Prometheus Scale 1685 Zeracrimic Scale 1813 Katothimic Scale 1045 Gibian Scale 1301 Koditonic Scale 533 Dehian Scale 2581 Raga Neroshta Scale 3605 Olkian

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.