# A Study of Scales

Where we discuss every possible combination of notes

This exploration of scales is based on work by William Zeitler, as published at http://allthescales.org/. In fact much of the material on this page merely repeats Zeitler's findings, presented here along with PHP code I used to generate the scales. This exploration also owes a debt to Andrew Duncan's work on combinatorial music theory. Most of the code in this treatise is the result of continuous exploration and development of PHPMusicTools, an open-source project at GitHub.

## Assumptions

This exploration of scales is based in a musical universe founded on two assumptions:

• Octave Equivalence
We assume that for the purpose of defining a scale, a pitch is functionally equivalent to another pitch separated by an octave. So it follows that if you're playing a scale in one octave, if you continue the pattern into the next octave you will play pitches with the same name.

• 12 tone equal temperament
We're using the 12 tones of an equally-tempered tuning system, as you'd find on a piano. Equal temperament asserts that the perceptual (or functional) relationship between two pitches is the same as the relationship between two other pitches with the same chromatic interval distance. Other tuning systems have the notion of scales, but they're not being considered in this study.

## Representing a scale

When I began piano lessons as a child, I learned that a scale was made up of whole- and half-steps. The scale was essentially a stack of intervals piled up, that happened to add up to an octave. While this view of the scale is useful for understanding diatonic melodies, it's not a convenient paradigm for analysis. Instead, we should regard a scale as being a set of 12 possibilities, and each one is either on or off.

The major scale, in lights.

What we have in the 12-tone system is a binary "word" made of 12 bits. We can assign one bit to each degree of the chromatic scale, and use the power of binary arithmetic and logic to do some pretty awesome analysis with them. When represented as bits it reads from right to left - the lowest bit is the root, and each bit going from right to left ascends by one semitone.

The total number of possible combinations of on and off bits is called the "power set". The number of sets in a power set of size n is (2^n). Using a word of 12 bits, the power set (2^12) is equal to 4096. The fun thing about binary power sets is that we can produce every possible combination, by merely invoking the integers from 0 (no tones) to 4095 (all 12 tones).

This means that every possible combination of tones in the 12-tone set can be represented by a number between 0 and 4095. We don't need to remember the fancy names like "phrygian", we can just call it scale number 1451. Convenient!

\$allscales = range(0, 4095);
decimalbinary
0 000000000000 no notes in the scale
1 000000000001 just the root tone
1365 010101010101 whole tone scale
2741 101010110101 major scale
4095 111111111111 chromatic scale

An important concept here is that any set of tones can be represented by a number. This number is not "ordinal" - it's not merely describing the position of the set in an indexed scheme; it's also not "cardinal" because it's not describing an amount of something. This is a nominal number because the number *is* the scale. You can do binary arithmetic with it, and you are adding and subtracting scales with no need to convert the scale into some other representation.

## Scale Finder

Binary:
Decimal:

Because scales are cyclical - they repeat and continue beyond a single octave - it is intuitive to represent them in "bracelet" notation. This visual representation is useful because it helps conceptualize modes and rotation (which we'll get into further below).

Here is the major scale, represented as a bracelet. Black beads are on bits, white beads are off bits.

If you imagine the bracelet with clock numbers, the topmost bead - at 12 o'clock - is the root, or 1st degree of the scale. Beware that in some of the math, this is the "0" position, as if we're counting semitones above the root. To play a scale ascending, start from the top and move around it clockwise. To play the scale descending, start at 12 o'clock and go around counterclockwise.

## What is a scale?

Or more importantly, what is *not* a scale?

Now that we have the superset of all possible sets of tones, we can whittle it down to exclude ones that we don't consider to be a legitimate "scale". We can do this with just two rules.

• #### A scale starts on the root tone.

This means any set of notes that doesn't have that first bit turned on is not eligible to be called a scale. This cuts our power set in exactly half, leaving 2048 sets.

In binary, it's easy to see that the first bit is on or off. In decimal, you can easily tell this by whether the number is odd or even. All even numbers have the first bit off; therefore all scales are represented by an odd number.

We could have cut some corners in our discussion of scales by omitting the root tone (always assumed to be on) to work with 11 bits instead of 12, but there are compelling reasons for keeping the 12-bit number for our scales, such as simplifying the analysis of symmetry, simplifying the calculation of modes, and performing analysis of sonorities that do not include the root, where an even number is appropriate.

function hasRootTone(\$scale) { // returns true if the first bit is not a zero return (1 & \$scale) != 0; } \$allscales = array_filter(\$allscales, 'hasRootTone');

scales remaining: 2048

• #### A scale does not have any leaps greater than n semitones.

For the purposes of this exercise we are saying n = 4, a.k.a. a major third. Any collection of tones that has an interval greater than a major third is not considered a "scale". This configuration is consistent with Zeitler's constant used to generate his comprehensive list of scales.

function doesNotHaveFourConsecutiveOffBits(\$scale) { \$c = 0; for (\$i=0; \$i<12; \$i++) { if (!(\$scale & (1 << (\$i)))) { \$c++; if (\$c >= 4) { return false; } } else { \$c = 0; } } return true; } \$allscales = array_filter(\$allscales, 'doesNotHaveFourConsecutiveOffBits');

scales remaining: 1490

Now that we've whittled our set of tones to only the ones we'd call a "scale", let's count how many there are with each number of tones.

function countOnBits(\$scale) { \$c = 0; for (\$i=0; \$i<12; \$i++) { if (\$scale & (1 << (\$i))) { \$c++; } } return \$c; } \$howmany = array_fill(1, 12, 0); foreach (\$allscales as \$scale) { \$howmany[countOnBits(\$scale)]++; }
number of tones how many scales
10
20
31
431
5155
6336
7413
8322
9165
1055
1111
121

## Symmetry

There are two kinds of symmetry of interest to scale analysis. They are rotational symmetry and reflective symmetry.

### Rotational Symmetry

Rotational symmetry is the symmetry that occurs by transposing a scale up or down by an interval, and observing whether the transposition and the original have an identical pattern of steps.

The set of 12 tones has 5 axes of symmetry. The twelve can be divided by 1, 2, 3, 4, and 6.

Any scale containing this kind of symmetry can reproduce its own tones by transposition, and is also called a "mode of limited transposition" (Messaien)

function symmetries(\$scale) { \$rotateme = \$scale; \$symmetries = array(); for (\$i = 0; \$i < 12; \$i++) { \$rotateme = rotate_bitmask(\$rotateme); if (\$rotateme == \$scale) { \$symmetries[] = \$i; } } return \$symmetries; }

Below are all the scales that have rotational symmetry.

axes of symmetryinterval of repetitionscales
1,2,3,4,5,6,7,8,9,10,11 semitone
2,4,6,8,10 whole tone
3,6,9 minor thirds
4,8 major thirds
6 tritones

number of notes in scale Placement of rotational symmetries
1234567891011
100000000000
200000000000
300010001000
400100300100
500000000000
6010301003010
700000000000
8002001000200
900030003000
1000000500000
1100000000000
1211111111111

### A curious numeric pattern

You'll notice that in the list of scales exhibiting rotational symmetry, many have a decimal representation that ends with the digit 5. A closer look shows that the pattern exists for all scales that have symmetry on axes of 6, 4, and 2 - where all the notes in the scale are in pairs a tritone apart. In the binary schema, each pair of notes that are a tritone apart add up to a multiple of 10, except for the root pair that adds up to 65. Therefore, all scales with rotational symmetry on the tritone will end with the digit 5.

1 + 64
= 65

2 + 128
= 130

4 + 256
= 260

8 + 512
= 520

16 + 1024
= 1040

32 + 2048
= 2080

### Messiaen's Modes - and their truncations

The French composer Messiaen studied scales with rotational symmetry, because they are useful in creating sonorities with an ambiguous tonic. He famously named seven such modes, and asserted that they are the only ones that may exist; all other symmetrical scales are truncations of those. We can prove that Messiaen was correct, by putting every symmetrical scale into a chart of their truncation relationships.

In all of Messiaen's modes, the modes are regarded as identical, which makes sense because the whole point was to create ambiguous tonality. Therefore when we speak of one of the Messiaenic modes, we're talking about a whole modal family of scales, like this one which he labeled Mode 7:

Truncation is the process of removing notes from a scale, without affecting its symmetry. The concept is best explained by example. Let's take the mode which Messiaen called the Second Mode:

That scale is symmetrical along the axis of a 3-semitone interval, i.e. by transposing it up a minor third. When we remove tones to create a truncation, we must preserve that symmetry, by removing four notes from the scale, like this:

In addition to the 3-semitone symmetry, that scale is also symmetrical along the axis of a 6-semitone interval. We can create two different truncations of that preserve that symmetry:

Technically, all of Messiaen's modes are truncated forms of , the 12-tone scale, which is symmetrical at all intervals.

### Hierarchy of truncations

Modal FamilyScalesis truncation of
*

Messiaen's Modes of Limited Transposition
M1
*, M3, M6
M2 *, M7
M3 *
M4 M7
M5 M6, M4
M6 M7
M7 *
Truncations
T1
M2, T5, T4
T2 M3
T3
M1, T2
T4 M2, M6, M4
T5 M2, M6, M4
T6 T4, T5, M5, M1

### Going Further

• In this study we are ignoring sonorities that aren't like a "scale" by disallowing large interval leaps (see the first section). If we ignore those rules, what other truncated sets exist? (spoiler hint: not very many!)
• Does the naming of certain scales by Messiaen seem haphazard? Why did he name M5, which is merely a truncation of both M6 and M4, which are in turn truncations of M7?

### Reflective Symmetry

A scale can be said to have reflective symmetry if it has the same pattern of steps whether ascending or descending. A reflectively symmetrical scale, symmetrical on the axis of the root tone, is called a palindromic scale.

function isPalindromic(\$scale) { for (\$i=1; \$i<=5; \$i++) { if ( (bool)(\$scale & (1 << \$i)) !== (bool)(\$scale & (1 << (12 - \$i))) ) { return false; } } return true; }

Here are all the scales that are palindromic:

### Chirality

An object is chiral if it is distinguishable from its mirror image, and can't be transformed into its mirror image by rotation. Chirality is an important concept in knot theory, and also has applications in molecular chemistry.

The concept of chirality is apparent when you consider "handedness" of a shape, such as a glove. A pair of gloves has a left hand and a right hand, and you can't transform a left glove into a right one by flipping it over or rotating it. The glove, therefore, is a chiral object. Conversely, a sock can be worn by the right or left foot; the mirror image of a sock would still be the same sock. Therefore a sock is not a chiral object.

Palindromic scales are achiral. But not all non-palindromic scales are chiral. For example, consider the scales and (below). One is the mirror image (reflection) of the other. They are not palindromic scales. They are also achiral, because one can transform into the other by rotation.

The major scale is achiral. If you reflect the major scale, you get the phrygian scale; these two scales are modes (rotations) of each other. Therefore neither of them are chiral. Similarly, none of the other ecclesiastical major modes are chiral.

What makes chirality interesting? For one thing chirality indicates that a scale does not have any symmetry along any axis. If you examine all achiral scales, they will all have some axis of reflective symmetry - it's just not necessarily the root tone (which would make it palindromic).

Some achiral scales, and their axes of symmetry
function isChiral(\$scale) { \$reflected = reflect_bitmask(\$scale); for (\$i = 0; \$i < 12; \$i++) { \$reflected = rotate_bitmask(\$reflected, 1, 1); if (\$reflected == \$scale) { return false; } } return true; }

A chiral object and its mirror image are called enantiomorphs. (source) function enantiomorph(\$scale) { \$scale = reflect_bitmask(\$scale); \$scale = rotate_bitmask(\$scale, -1, 1); return \$scale; }
Named ScaleChirality / Enantiomorphs
585 (diminished seventh)achiral
1123 (iwato)
1186 (insen)
1365 (whole tone, second mode of limited transposition)achiral
1371 (altered)achiral
1387 (locrian)achiral
1389 (half diminished)achiral
1451 (phrygian)achiral
1453 (aeolian, natural minor)achiral
1459 (phrygian dominant, spanish romani)
1485 (aolian #4, romani scale)
1709 (dorian)achiral
1717 (mixolydian)achiral
1741 (ukranian dorian, romanian scale, altered dorian)
1749 (acoustic, lydian dominant)achiral
2257 (hirajoshi)
2733 (heptatonia seconda, ascending melodic minor, jazz minor)achiral
2741 (major, ionian)achiral
2773 (lydian)achiral
2483 (double harmonic)achiral
2457 (augmented)achiral
2477 (harmonic minor)
2509 (hungarian minor)achiral
2901 (lydian augmented)achiral
2731 (major neapolitan)achiral
2475 (minor neapolitan)
3669 (prometheus)
1235 (tritone scale)
1755 (octatonic, second mode of limited transposition)achiral
3549 (third mode of limited transposition)achiral
2535 (fourth mode of limited transposition)achiral
2275 (fifth mode of limited transposition)achiral
3445 (sixth mode of limited transposition)achiral
3055 (seventh mode of limited transposition)achiral
3765 (bebop dominant)achiral
4095 (chromatic 12-tone)achiral

### Going Further

• Reflective symmetry can occur with an axis on other notes of the scale. Are those interesting?
• The reflection axis can be on a tone, or between two tones. Is that interesting?
• Is there a more optimal or elegant way to find the reflective symmetry axes of a scale?
• Are there chiral enantiomorph pairs that are both named scales?

### Combined Symmetry

Using similar methods to those above, we can identify all the scales that are both palindromic and have rotational symmetry. Here they are:

## Imperfection

Imperfection is a concept invented (so far as I can tell) by William Zeitler, to describe the presence or absense of perfect fifths in the scale tones. Any tone in the scale that does not have the perfect fifth above it represented in the scale is an "imperfect" tone. The number of imperfections is a metric that plausibly correlates with the perception of dissonance in a sonority.

The only scale that has no imperfections is the 12-tone chromatic scale.

This table differs from the one at allthescales.org, because this script does not de-duplicate modes. If an imperfection exists in a scale, it will also exist in all the sibling modes of that scale. Hence the single imperfect tone in the 11-tone scale is found 11 times (in 11 scales that are all modally related), whereas Zeitler only counts it as one.

number of notes in scale # of Imperfections
0123456
10000000
20000000
30001000
400816700
50530754050
60669146102121
70784196112140
808841686200
9097284000
10010450000
1101100000
121000000

### Going Further

• Looking at the table, there are 5 scales of 5 tones with one imperfection. There are 6 scales of 6 tones with only one imperfection. The same patten continues right up to 11 tones. What causes this pattern?
• The 7-note scale with only one imperfection is the major diatonic scale (plus all its modes). Is it a coincidence that we find this scale sonically pleasing?

## Interval Spectrum / Richness

Howard Hanson, in the book "Harmonic Materials", posits that the character of a sonority is defined by the intervals that are within it. Furthermore, in his system an interval is equivalent to its inverse, i.e. the sonority of a perfect 5th is the same as the perfect 4th. He categorizes all intervals as being one of six classes, and gives each a letter: p m n s d t, ordered from most consonant (p) to most dissonant (t). When an interval appears more than once in a sonority, it is superscripted with a number, like p2.

P - the Perfects (5 or 7)

This is the interval of a perfect 5th, or perfect 4th.

M - The Major Third (4 or 8)

This is the interval of a major 3rd, or minor 6th

N - The Minor Third (3 or 9)

This is the interval of a minor 3rd, or a major 6th

S - the second (2 or 10)

This is the interval of a major 2nd, or minor 7th

D - the Diminished (1 or 11)

Intervals of a minor 2nd, or a major 7th

T - the Tritone (6 semitones)

For example, in Hanson's analysis, a C major triad has the sonority pmn, because it contains one perfect 5th between the C and G (p), a major third between C and E (m), and a minor third between the E and G (n). Interesting to note that a minor triad of A-C-E is also a pmn sonority, just with the arrangement of the minor and major intervals swapped. The diminished seventh chord has the sonority n4t2 because it contains four different minor thirds, and two tritones.

We can count the appearances of an interval using a method called "cyclic autocorrelation". To find out how many of interval X appear in a scale, we rotate a copy of the scale by X degrees, and then count the number of positions where an "on" bit appears in both the original and copy.

function find_spectrum(\$scale) { \$spectrum = array(); \$rotateme = \$scale; for (\$i=0; \$i<6; \$i++) { \$rotateme = rotate_bitmask(\$rotateme, \$direction = 1, \$amount = 1); \$spectrum[\$i] = countOnBits(\$scale & \$rotateme); } // special rule: if there is a tritone in the sonority, it will show up twice, so we divide by 2 \$spectrum[5] = \$spectrum[5] / 2; return \$spectrum; }

All members of the same modal family will have the same interval spectrum, because the spectrum doesn't care about rotation, and modes are merely rotations of each other. And there will be many examples of different families that have the same interval spectrum.

Below is a table of all the named scales, and their interval spectrum. You'll notice that modal families will all have the same spectrum; hence every named mode of the diatonic scale (lydian, dorian, phrygian etc.) has the same spectrum of p6m3n4s5d2t

 273 (augmented triad) m3 585 (diminished seventh) n4t2 1123 (iwato) p3m2nsd2t 1186 (insen) pmn2st 1365 (whole tone, second mode of limited transposition) m6s6t3 1371 (altered) p4m4n4s5d2t2 1387 (locrian) p6m3n4s5d2t 1389 (half diminished) p4m4n4s5d2t2 1451 (phrygian) p6m3n4s5d2t 1453 (aeolian, natural minor) p6m3n4s5d2t 1459 (phrygian dominant, spanish romani) p4m4n5s3d3t2 1485 (aolian #4, romani scale) p4m5n3s4d3t2 1709 (dorian) p6m3n4s5d2t 1717 (mixolydian) p6m3n4s5d2t 1741 (ukranian dorian, romanian scale, altered dorian) p4m4n5s3d3t2 1749 (acoustic, lydian dominant) p4m4n4s5d2t2 2257 (hirajoshi) p3m2nsd2t 2733 (heptatonia seconda, ascending melodic minor, jazz minor) p4m4n4s5d2t2 2741 (major, ionian) p6m3n4s5d2t 2773 (lydian) p6m3n4s5d2t 2483 (double harmonic) p4m5n4s2d4t2 2457 (augmented) p3m6n3d3 2477 (harmonic minor) p4m4n5s3d3t2 2509 (hungarian minor) p4m5n4s2d4t2 2901 (lydian augmented) p4m4n4s5d2t2 2731 (major neapolitan) p2m6n2s6d2t3 2475 (minor neapolitan) p4m5n3s4d3t2 3669 (prometheus) p4m4n3s5d3t2 1235 (tritone scale) p2m2n4s2d2t3 1755 (octatonic, second mode of limited transposition) p4m4n8s4d4t4 3549 (third mode of limited transposition) p6m9n6s6d6t3 2535 (fourth mode of limited transposition) p6m4n4s4d6t4 2275 (fifth mode of limited transposition) p4m2s2d4t3 3445 (sixth mode of limited transposition) p4m6n4s6d4t4 3055 (seventh mode of limited transposition) p8m8n8s8d8t5 3765 (bebop dominant) p7m4n5s6d4t2 4095 (chromatic 12-tone) p12m12n12s12d12t6

### Going Further

• Is there an optimal or elegant way to find all scales with a given spectrum?
• What patterns appear in interval distribution?
• Which are the most common, and least common spectra?

## Proximity

We would consider two scales to be "near" each other, if they bear many similarities; sharing many of the same notes, such that it would take few transformations to turn one into the other.

This distance measured by transformation is almost the same thing as a "Levenshtein Distance". The normal definition of a Levenshtein Distance on a string of characters allows three kinds of transformation: insertion, deletion, and substitution. Our scale transformations are different from a string transformation, in that scales can't do substitution, instead we move tones up or down with rules for handling collision. This difference means we can't employ all the elegant formulas bequeathed to us by Vladimir Levenshtein.

We can transform a scale in three ways:

• Move a tone up or down by a semitone
• Remove a tone

It is simple to generate all the scales at a distance of 1, just by performing all possible transformations to every interval above the root.

function findNearby(\$scale) { \$near = array(); for (\$i=1; \$i<12; \$i++) { if (\$scale & (1 << (\$i))) { \$copy = \$scale; \$off = \$copy ^ 1 << (\$i); \$near[] = \$off; \$copy = \$off | 1 << (\$i - 1); \$near[] = \$copy; \$copy = \$off | 1 << (\$i + 1); \$near[] = \$copy; } else { \$copy = \$scale; \$copy = \$copy | 1 << (\$i); \$near[] = \$copy; } } return \$near; }

### Example

Here are all the scales that are a distance of 1 from the major scale, aka , shown here as a simple C major scale:

 add a tone at C# lower the D to C# raise the D to D# delete the D add a tone at D# lower the E to D# 2733 (heptatonia seconda, ascending melodic minor, jazz minor) raise the E to F same as deleting E delete the E lower the F to E same as deleting F raise the F to F# 2773 (lydian) delete the F add a tone at F# lower the G to F# raise the G to G# delete the G add a tone at G# lower the A to G# raise the A to A# delete the A add a tone at A# 3765 (bebop dominant) lower the B to A# 1717 (mixolydian) raise the B to C same as deleting B delete the B

Evidently since every scale will have a collection of neighbours similar to this one, a complete graph of scale proximity is large, but not unfathomably so.

We can also easily calculate the Levenshtein distance between two scales. Apply an XOR operation to expose changed bits between two scales. Adjacent on bits in the XOR where the bits were different in the original scale identify a "move" operation. Any bits remaining are either an addition or deletion.

function levenshtein_scale(\$scale1, \$scale2) { \$distance = 0; \$d = \$scale1 ^ \$scale2; for (\$i=0; \$i<12; \$i++) { if ( (\$d & (1 << (\$i))) && (\$d &(1 << (\$i+1))) && (\$scale1 & (1 << (\$i))) != (\$scale1 & (1 << (\$i+1))) ) { \$distance++; \$d = \$d & ( ~ (1 << (\$i))); \$d = \$d & ( ~ (1 << (\$i+1))); } } for (\$i=0; \$i<12; \$i++) { if ((\$d & (1 << (\$i)))) { \$distance++; } } return \$distance; }

## Modes

There is a lot of confusion about what is a "mode", chiefly because the word is used slightly differently in various contexts.

When we say "C major", the word "major" refers to a specific pattern of whole- and half-steps. The "C" tells us to begin that pattern on the root tone of "C".

Modes are created when you use the same patterns of whole- and half-steps, but you begin on a different step. For instance, the "D Dorian" mode uses all the same notes as C major (the white keys on a piano), but it begins with D. Compared with the Major (also known as "Ionian" mode), the Dorian sounds different, because relative to the root note D, it has a minor third and a minor seventh.

The best way to understand modes is to think of a toy piano where the black keys are just painted on - all you have are the white keys: C D E F G A B. Can you play a song that sounds like it's in a minor key? You can't play a song in C minor, because that would require three flats. So instead you play the song with A as the root (A B C D E F G). That scale is a mode of the major scale, called the Aeolian Mode.

When you play that minor scale, you're not playing "C minor", you're playing the relative minor of C, which is "A minor". Modes are relatives of each other if they have the same pattern of steps, starting on different steps.

To compute a mode of the current scale, we "rotate" all the notes down one semitone. then if the rotated notes have an on bit in the root, then it is a mode of the original scale.

101010110101 = 2741 - major scale, "ionian" mode
110101011010 = 3418 - rotated down 1 semitone - not a scale
011010101101 = 1709 - rotated down 2 semitones - "dorian"
101101010110 = 2902 - rotated down 3 semitones - not a scale
010110101011 = 1451 - rotated down 4 semitones - "phrygian"
101011010101 = 2773 - rotated down 5 semitones - "lydian"
110101101010 = 3434 - rotated down 6 semitones - not a scale
011010110101 = 1717 - rotated down 7 semitones - "mixolydian"
101101011010 = 2906 - rotated down 8 semitones - not a scale
010110101101 = 1453 - rotated down 9 semitones - "aeolian"
101011010110 = 2774 - rotated down 10 semitones - not a scale
010101101011 = 1387 - rotated down 11 semitones - "locrian"

function modes(\$scale) { \$rotateme = \$scale; \$modes = array(); for (\$i = 0; \$i < 12; \$i++) { \$rotateme = rotate_bitmask(\$rotateme); if ((\$rotateme & 1) == 0) { continue; } \$modes[] = \$rotateme; } return \$modes; } function rotate_bitmask(\$bits, \$direction = 1, \$amount = 1) { for (\$i = 0; \$i < \$amount; \$i++) { if (\$direction == 1) { \$firstbit = \$bits & 1; \$bits = \$bits >> 1; \$bits = \$bits | (\$firstbit << 11); } else { \$firstbit = \$bits & (1 << 11); \$bits = \$bits << 1; \$bits = \$bits & ~(1 << 12); \$bits = \$bits | (\$firstbit >> 11); } } return \$bits; }

When we do this to every scale, we see modal relationships between scales, and we also discover symmetries when a scale is a mode of itself on another degree.

### Modal Siblings

We can present the list of all modes grouped in sets of modal siblings

Number of tones in the scale Number of modal families
10
20
31
49
531
659
759
842
919
106
111
121

### A complete list of all modal families

#### Modal families with 4 tones (tetratonic)

 pm3nd p2m2nd pm3nd m3s2t pmn2st p2mn2s m2s2t2 pmn2st n4t2 585 (diminished seventh)

#### Modal families with 5 tones (pentatonic)

 pm3ns2d2t p2m2n2sd2t p2m3n2sd2 p2m2s2d2t2 p2m2n2sd2t pm3ns2d2t p2m3n2sd2 p3m2n2s2d pm3n2s2dt p3mn2s2dt pmn4sdt2 p2mn3s2dt p3m2nsd2t 1123 (iwato)2257 (hirajoshi) pm2n2s2dt2 p2mn3s2dt p2m4n2d2 p2m4n2d2 p2m3ns2dt p2m2n3sdt pm2n2s2dt2 p2m3n2d2t p2m2n3sdt pmn4sdt2 pm3n2s2dt p3m2nsd2t p2m3ns2dt p3mn2s2dt p3m2n2s2d m4s4t2 p2m2n2s3t p4mn2s3

#### Modal families with 6 tones (hexatonic)

 p2m3n2s3d4t p3m2n3s3d3t p2m3n3s3d3t p3m2n2s3d3t2 p2m2n4s2d3t2 p2m3n3s3d3t p4m2ns2d4t2 p3m2n2s3d3t2 p3m2n3s3d3t p2m4n3s2d3t p3m4n3s2d3 p2m4ns4d2t2 p3m3n3s3d2t pm4n2s4d2t2 p3m3n2s2d3t2 p4m2n3s3d2t p2m2n4s3d2t2 p3m4n2s2d3t pm4n2s4d2t2 p3m3n2s2d3t2 p3m4n3sd3t p3m4n2s2d3t p4m2n2s2d3t2 p4m3n2s3d2t p4m2n3s3d2t p3m4n3sd3t p3m3n3s3d2t p3m4n3s2d3 p4m2s2d4t3 2275 (fifth mode of limited transposition) p4m2n2s2d3t2 p3m3n2s2d3t2 p3m3n2s2d3t2 p2m4ns4d2t2 p2m4n3s2d3t p3m3n3s3d2t p2m2n5s2d2t2 p2m2n4s3d2t2 p3m4n3sd3t p2m3n4s2d2t2 p2m2n5s2d2t2 p3m4n3sd3t p3m3n3s3d2t p4m2n3s3d2t p2m3n4s2d2t2 p4m2n3s3d2t p4m3n2s3d2t p2m4n2s4dt2 p4m2n3s4dt p3m4n3s2d2t p5m2n3s4d p2m2n4s2d2t3 1235 (tritone scale) p3m2n4s2d2t2 p3m2n4s2d2t2 p4m2n3s4dt p3m6n3d3 2457 (augmented) p3m4n3s2d2t p2m2n4s2d2t3 p2m4n2s4dt2 m6s6t3 1365 (whole tone, second mode of limited transposition)

#### Modal families with 7 tones (heptatonic)

 p3m4n4s4d5t p3m4n4s4d5t p3m4n3s5d4t2 p3m3n5s4d4t2 p2m4n4s5d4t2 p4m4n3s3d5t2 p4m3n4s4d4t2 p3m3n5s4d4t2 p4m4n3s3d5t2 p3m4n3s5d4t2 p4m3n4s4d4t2 p3m4n5s3d4t2 p4m4n4s4d4t p5m3n3s4d4t2 p4m4n3s5d3t2 p5m3n4s5d3t p4m4n4s3d4t2 p5m3n4s5d3t p4m4n4s4d4t p5m3n2s3d5t3 p4m3n4s3d4t3 p4m4n4s3d4t2 p4m3n4s3d4t3 p4m4n3s5d3t2 3669 (prometheus) p3m4n5s3d4t2 p5m3n2s3d5t3 p5m3n3s4d4t2 p4m3n4s4d4t2 p3m5n3s4d4t2 p4m5n4s3d4t p3m5n3s4d4t2 p4m4n4s3d4t2 p5m4n4s4d3t p4m3n5s4d3t2 p4m6n4s2d4t p3m5n4s4d3t2 p4m5n4s3d4t p4m4n2s4d4t3 p5m3n4s4d3t2 p3m4n4s4d3t3 p4m5n3s4d3t2 p2m6n2s6d2t3 2731 (major neapolitan) p3m5n4s4d3t2 p5m4n3s3d4t2 p3m4n4s4d3t3 p4m3n5s4d3t2 p5m4n3s3d4t2 p4m5n4s2d4t2 2483 (double harmonic)2509 (hungarian minor) p4m5n3s4d3t2 1485 (aolian #4, romani scale)2475 (minor neapolitan) p4m6n4s2d4t p5m3n4s4d3t2 p5m4n4s4d3t p4m4n4s3d4t2 p3m3n6s3d3t3 p3m3n6s3d3t3 p4m4n5s3d3t2 p4m4n4s5d2t2 1371 (altered)1389 (half diminished)1749 (acoustic, lydian dominant)2733 (heptatonia seconda, ascending melodic minor, jazz minor)2901 (lydian augmented) p4m4n5s3d3t2 1459 (phrygian dominant, spanish romani)1741 (ukranian dorian, romanian scale, altered dorian)2477 (harmonic minor) p6m3n4s5d2t 1387 (locrian)1451 (phrygian)1453 (aeolian, natural minor)1709 (dorian)1717 (mixolydian)2741 (major, ionian)2773 (lydian)

#### Modal families with 8 tones (octatonic)

 p4m5n5s6d6t2 p4m5n6s5d6t2 p4m5n5s6d6t2 p5m5n5s5d6t2 p5m5n5s6d5t2 p5m4n6s6d5t2 p5m6n5s4d6t2 p5m5n5s6d5t2 p5m5n5s5d6t2 p5m5n4s5d6t3 p5m4n6s5d5t3 p4m5n6s5d5t3 p5m5n5s5d5t3 p4m6n4s7d4t3 p4m5n6s5d5t3 p6m5n4s4d6t3 p5m5n5s5d5t3 p5m4n6s5d5t3 p5m5n4s5d6t3 p6m4n4s5d6t3 p5m5n5s5d5t3 p6m5n5s5d5t2 p5m5n6s4d5t3 p6m5n5s6d4t2 p5m6n6s4d5t2 p6m5n4s5d5t3 p7m4n5s6d4t2 3765 (bebop dominant) p6m5n5s6d4t2 p6m5n5s5d5t2 p6m4n4s4d6t4 2535 (fourth mode of limited transposition) p6m5n4s5d5t3 p5m5n6s4d5t3 p5m5n5s5d5t3 p5m7n5s4d5t2 p4m7n4s6d4t3 p5m7n5s4d5t2 p6m6n5s4d5t2 p5m5n6s5d4t3 p6m5n6s5d4t2 p4m6n4s6d4t4 3445 (sixth mode of limited transposition) p5m5n6s5d4t3 p4m4n8s4d4t4 1755 (octatonic, second mode of limited transposition)

#### Modal families with 9 tones

 p6m6n6s7d8t3 p6m6n7s7d7t3 p6m6n7s7d7t3 p6m7n7s6d7t3 p6m7n6s8d6t3 p6m7n7s6d7t3 p7m7n6s6d7t3 p7m6n7s7d6t3 p7m6n7s7d6t3 p7m7n6s6d7t3 p7m6n6s6d7t4 p6m7n6s7d6t4 p6m6n8s6d6t4 p6m7n6s7d6t4 p7m6n6s6d7t4 p8m6n6s7d6t3 p7m7n7s6d6t3 p7m7n7s6d6t3 p6m9n6s6d6t3 3549 (third mode of limited transposition)

#### Modal families with 10 tones

 p8m8n8s8d9t4 p8m8n8s9d8t4 p8m8n9s8d8t4 p8m9n8s8d8t4 p9m8n8s8d8t4 p8m8n8s8d8t5 3055 (seventh mode of limited transposition)

#### Modal families with 11 tones

 p10m10n10s10d10t5

#### Modal families with 12 tones

 p12m12n12s12d12t6 4095 (chromatic 12-tone)