The Exciting Universe Of Music Theory
presents

# A Study of Scales

Where we discuss every possible combination of notes

## 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.

Here is 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!

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.

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.

### Interval Pattern

Another popular way of representing a scale is by its interval pattern. When I was learning the major scale, I was taught to say aloud: "tone, tone, semitone, tone, tone, tone, semitone". Many music theorists like to represent a scale this way because it's accurate and easy to understand: "TTSTTTS". Having a scale's interval pattern has merit as an intermediary step can make some kinds of analysis simpler. Expressed numerically - which is more convenient for computation - the major scale is [2,2,1,2,2,2,1].

### Pitch Class Sets

Yet another way to represent a scale is as a "pitch class set", where the tones are assigned numbers 0 to 11 (sometimes using "T" and "E" for 10 and 11), and the set enumerates the ones present in the scale. A pitch class set for the major scale is notated like this: {0,2,4,5,7,9,11}. The "scales" we'll study here are a subset of Pitch Classes (ie those that have a root, and obey Zeitler's Rules1) and we can use many of the same mathematical tricks to manipulate them.

## 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.

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.

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.

number of tones how many scales
10
20
31
431
5155
6336
7413
8322
9165
1055
1111
121

## 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. It's as if you take the bracelet diagram that we've been using throughout this study, and twist it like a dial so that a different note is at the top, in the root position.

```			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"
```

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.

### Prime Form

Often when discussing the properties of a scale, those properties (like interval distribution or evenness) are the same for all related scales, ie a scale, all its the modes, its inverse, and the modes of its inverse. In order to simplify things, it is useful to declare that one of those is the "prime form", so when doing analysis we discard all of them except one.

It's important to emphasize - because this point is sometimes missed - that the interval distribution in a scale is the same for all the scale's modes produced by rotation, but also for the scale's inverse produced by reflection. The prime form of a scale is chosen to represent the entire group of scales with equivalent interval patterns. In this study, prime scales are marked with a star .

In discussing Prime Form of a scale, we are undeniably treading into the topic of Pitch Class Sets, a more generalized study involving every possible combination of tones, regardless of the rules that make it a scale.

### Forte vs Rahn

There are two dominant strategies for declaring the Prime Form of a set of tones; one was defined by Allen Forte2, and another similar one (with only subtle differences) described later by John Rahn3. While I have deep admiration for Forte's theoretical work, I prefer the Rahn prime formula, for the simple reason that the Rahn primes are the easier to calculate.

The calculation of the Prime Form according to Forte requires some inelegant cyclomatic complexity. Rahn's algorithm is slightly easier to do manually, and is simply the one with the lowest value when expressed in bits, as we have done in this study. This connection between Rahn's prime forms and the bit representation of a scale was proven in 2017 by brute force calculation of every possible scale and prime form according to all three algorithms.

In a necessarily succinct overview of this topic, I'll demonstrate the differences between Forte's algorithm and Rahn's. The prime form for pitch class sets is identical except for 6 sets. Both algorithms look at the distance between the first and last tones of the set, preferring the one with the smaller interval. In the case of a tie, this is where Forte and Rahn differ: Forte begins at the start of the set working toward the end, whereas Rahn starts at the end working toward the beginning.

Here are the six sets where Forte and Rahn disagree on which one is prime.

ForteRahn

{0,1,3,7,8}

{0,1,5,6,8}

{0,1,3,5,8,9}

{0,1,4,5,7,9}

{0,1,3,6,8,9}

{0,2,3,6,7,9}

{0,1,2,4,7,8,9}

{0,1,2,5,6,7,9}

{0,1,2,3,5,8,9}

{0,1,4,5,6,7,9}

{0,1,2,4,5,7,9,10}

{0,1,3,4,5,7,8,10}

### A complete list of all modal families

#### Modal families with 3 tones

Forte Class NamePrime and its rotationsInverse and its rotations

#### Modal families with 4 tones

Forte Class NamePrime and its rotationsInverse and its rotations
4-19 275 Dalic
305 Gonic
785 Aeoloric
2185 Dygic
281 Lanic
401 Epogic
547 Pyrric
2321 Zyphic
4-20 291 Raga Lavangi
393 Lothic
561 Phratic
2193 Major Seventh
4-24 277 Mixolyric
337 Kmhmu 4 Tone Type 2
1093 Lydic
1297 Aeolic
4-27 293 Raga Haripriya
593 Saric
649 Byptic
1097 Aeraphic
329 Lonic
581 Bolic
1169 Raga Mahathi
4-26 297 Karen 4 Tone Type 3
549 Lahuzu 4 Tone Type 4
657 Lahuzu 4 Tone Type 3
1161 Bi Yu
4-25 325 Messiaen Truncated Mode 6
1105 Messiaen Truncated Mode 6 Inverse
4-28 585 Diminished Seventh

#### Modal families with 5 tones

Forte Class NamePrime and its rotationsInverse and its rotations
5-13 279 Poditonic
1809 Ranitonic
2187 Ionothitonic
3141 Kanitonic
285 Zaritonic
465 Zoditonic
1095 Phrythitonic
2595 Rolitonic
3345 Zylitonic
5-Z38 295 Gyritonic
625 Ionyptitonic
905 Bylitonic
2195 Zalitonic
3145 Stolitonic
457 Staptitonic
569 Mothitonic
583 Aeritonic
2339 Raga Kshanika
3217 Molitonic
5-Z37 313 Goritonic
551 Aeoloditonic
913 Aeolyritonic
2323 Doptitonic
3209 Aeraphitonic
5-15 327 Syptitonic
1137 Stonitonic
2211 Raga Gauri
3153 Zathitonic
5-Z17 283 Aerylitonic
433 Raga Zilaf
1571 Lagitonic
2189 Zagitonic
2833 Dolitonic
5-27 299 Raga Chitthakarshini
689 Raga Nagasvaravali
1417 Raga Shailaja
1573 Raga Guhamanohari
425 Raga Kokil Pancham
565 Aeolyphritonic
1165 Gycritonic
1315 Pyritonic
2705 Raga Mamata
5-26 309 Palitonic
849 Aerynitonic
1101 Stothitonic
1299 Aerophitonic
2697 Katagitonic
345 Gylitonic
555 Aeolycritonic
1425 Ryphitonic
1605 Zanitonic
2325 Pynitonic
5-29 331 Raga Chhaya Todi
709 Raga Shri Kalyan
1201 Mixolydian Pentatonic
1577 Raga Chandrakauns
2213 Raga Desh
421 Han-kumoi
653 Dorian Pentatonic
1129 Raga Jayakauns
1187 Kokin-joshi
2641 Raga Hindol
5-31 587 Pathitonic
601 Bycritonic
713 Thoptitonic
1609 Thyritonic
589 Ionalitonic
617 Katycritonic
841 Phrothitonic
1171 Raga Manaranjani I
2633 Bartók Beta Chord
5-25 301 Raga Audav Tukhari
721 Raga Dhavalashri
1099 Dyritonic
1673 Thocritonic
2597 Raga Rasranjani
361 Bocritonic
557 Raga Abhogi
1163 Raga Rukmangi
1681 Raga Valaji
2629 Raga Shubravarni
5-20 355 African Pentatonic 4
395 Phrygian Pentatonic
1585 Raga Khamaji Durga
2225 Ionian Pentatonic
2245 Raga Vaijayanti
397 Aeolian Pentatonic
419 Hon-kumoi-joshi
1123 Iwato
2257 Lydian Pentatonic
5-28 333 Bogitonic
1107 Mogitonic
1233 Ionoditonic
2601 Marga Hindolam
357 Banitonic
651 Golitonic
1113 Locrian Pentatonic 2
1617 Phronitonic
2373 Dyptitonic
5-21 307 Raga Megharanjani
787 Aeolapritonic
817 Zothitonic
2201 Ionagitonic
2441 Kyritonic
409 Laritonic
563 Thacritonic
803 Loritonic
2329 Styditonic
2449 Zacritonic
5-30 339 Zaptitonic
789 Zogitonic
1221 Epyritonic
1329 Epygitonic
2217 Kagitonic
405 Raga Bhupeshwari
675 Altered Pentatonic
1125 Ionaritonic
1305 Dynitonic
2385 Karen 5tone Type 2
5-32 595 Sogitonic
665 Raga Mohanangi
805 Rothitonic
1225 Raga Samudhra Priya
2345 Gothitonic
613 Phralitonic
659 Raga Rasika Ranjani
809 Dogitonic
1177 Garitonic
2377 Bartók Gamma Chord
5-22 403 Raga Reva
611 Anchihoye
793 Mocritonic
2249 Raga Multani
2353 Raga Girija
5-33 341 Bothitonic
1301 Koditonic
1349 Tholitonic
1361 Ngapauk auk Pyan
5-34 597 Kung
1173 Dominant Pentatonic
1317 Chaio
1353 Raga Harikauns
5-35 661 Major Pentatonic
677 Scottish Pentatonic
1189 Suspended Pentatonic
1193 Minor Pentatonic
1321 Blues Minor

#### Modal families with 6 tones

Forte Class NamePrime and its rotationsInverse and its rotations
6-Z37 287 Gynimic
2191 Thydimic
3143 Polimic
3619 Thanimic
3857 Ponimic
6-Z40 303 Golimic
753 Kytrimic
1929 Aeolycrimic
2199 Dyptimic
3147 Ryrimic
3621 Gylimic
489 Phrathimic
573 Saptimic
1167 Aerodimic
2631 Macrimic
3363 Rogimic
3729 Starimic
6-Z39 317 Korimic
977 Kocrimic
1103 Lynimic
2599 Malimic
3347 Synimic
3721 Phragimic
377 Kathimic
559 Lylimic
1937 Galimic
2327 Epalimic
3211 Epacrimic
3653 Sathimic
6-Z41 335 Zanimic
965 Ionothimic
1265 Pynimic
2215 Ranimic
3625 Podimic
485 Stoptimic
655 Kataptimic
1145 Zygimic
2375 Aeolaptimic
3235 Pothimic
3665 Stalimic
6-Z42 591 Gaptimic
633 Kydimic
969 Ionogimic
2343 Tharimic
3219 Ionaphimic
3657 Epynimic
6-Z38 399 Zynimic
483 Kygimic
2247 Raga Vijayasri
2289 Mocrimic
3171 Zythimic
3633 Daptimic
6-15 311 Stagimic
881 Aerothimic
1811 Kyptimic
2203 Dorimic
2953 Ionylimic
3149 Phrycrimic
473 Aeralimic
571 Kynimic
1607 Epytimic
2333 Stynimic
2851 Katoptimic
3473 Lathimic
6-14 315 Stodimic
945 Raga Saravati
1575 Zycrimic
2205 Ionocrimic
2835 Ionygimic
3465 Katathimic
441 Thycrimic
1827 Katygimic
2331 Dylimic
2961 Bygimic
3213 Eponimic
6-22 343 Ionorimic
1393 Mycrimic
1477 Raga Jaganmohanam
1813 Katothimic
2219 Phrydimic
3157 Zyptimic
469 Katyrimic
1141 Rynimic
1309 Pogimic
1351 Aeraptimic
2723 Raga Jivantika
3409 Katanimic
6-Z46 599 Thyrimic
697 Lagimic
1481 Zagimic
1829 Pathimic
2347 Raga Viyogavarali
3221 Bycrimic
629 Aeronimic
937 Stothimic
1181 Katagimic
1319 Phronimic
2707 Banimic
3401 Palimic
6-21 349 Borimic
1111 Sycrimic
1489 Raga Jyoti
1861 Phrygimic
3349 Aeolocrimic
373 Epagimic
1117 Raptimic
1303 Epolimic
1873 Dathimic
2699 Sythimic
3397 Sydimic
6-Z17 407 All-Trichord Hexachord
739 Rorimic
1817 Phrythimic
2251 Zodimic
2417 Kanimic
3173 Zarimic
467 Raga Dhavalangam
797 Katocrimic
1223 Phryptimic
2281 Rathimic
2659 Katynimic
3377 Phralimic
6-Z47 663 Phrynimic
741 Gathimic
1209 Raga Bhanumanjari
1833 Ionacrimic
2379 Raga Gurjari Todi
3237 Raga Brindabani Sarang
669 Gycrimic
1191 Pyrimic
1257 Blues Scale
2643 Raga Hamsanandi
3369 Mixolimic
6-Z45 605 Dycrimic
745 Kolimic
1175 Epycrimic
1865 Thagimic
2635 Gocrimic
3365 Katolimic
6-16 371 Rythimic
791 Aeoloptimic
1841 Thogimic
2233 Donimic
2443 Panimic
3269 Raga Malarani
413 Ganimic
931 Raga Kalakanthi
1127 Eparimic
2513 Aerycrimic
2611 Raga Vasanta
3353 Phraptimic
6-Z43 359 Bothimic
907 Tholimic
1649 Bolimic
2227 Raga Gaula
2501 Ralimic
3161 Kodimic
461 Raga Syamalam
839 Ionathimic
1139 Aerygimic
2617 Pylimic
3281 Raga Vijayavasanta
6-Z44 615 Schoenberg Hexachord
825 Thyptimic
2355 Raga Lalita
2505 Mydimic
3225 Ionalimic
627 Mogimic
807 Raga Suddha Mukhari
921 Bogimic
2361 Docrimic
2451 Raga Bauli
3273 Raga Jivantini
6-18 423 Sogimic
909 Katarimic
1251 Sylimic
2259 Raga Mandari
2673 Mythimic
3177 Rothimic
459 Zaptimic
711 Raga Chandrajyoti
1593 Zogimic
2277 Kagimic
2403 Lycrimic
3249 Raga Tilang
6-Z48 679 Lanimic
917 Dygimic
1253 Zolimic
1337 Epogimic
2387 Paptimic
3241 Dalimic
6-7 455 Messiaen Mode 5 Rotation 2
2275 Messiaen Mode 5
3185 Messiaen Mode 5 Rotation 1
6-Z24 347 Barimic
1457 Raga Kamalamanohari
1579 Sagimic
1733 Raga Sarasvati
2221 Raga Sindhura Kafi
2837 Aelothimic
437 Ronimic
1133 Stycrimic
1307 Katorimic
1699 Raga Rasavali
2701 Hawaiian
2897 Rycrimic
6-27 603 Aeolygimic
729 Stygimic
1611 Dacrimic
2349 Raga Ghantana
2853 Baptimic
621 Pyramid Hexatonic
873 Bagimic
1179 Sonimic
1683 Raga Malayamarutam
2637 Raga Ranjani
2889 Thoptimic
6-Z23 365 Marimic
1115 Superlocrian Hexamirror
1675 Raga Salagavarali
1745 Raga Vutari
2605 Rylimic
2885 Byrimic
6-Z19 411 Lygimic
867 Phrocrimic
1587 Raga Rudra Pancama
2253 Raga Amarasenapriya
2481 Raga Paraju
2841 African Pentatonic 3
435 Raga Purna Pancama
795 Aeologimic
1635 Sygimic
2265 Raga Rasamanjari
2865 Solimic
6-Z49 667 Rodimic
869 Kothimic
1241 Pygimic
1619 Prometheus Neapolitan
2381 Takemitsu Linea Mode 1
2857 Stythimic
6-Z25 363 Soptimic
1419 Raga Kashyapi
1581 Raga Bagesri
1713 Raga Khamas
2229 Raga Nalinakanti
429 Koptimic
1131 Honchoshi Plagal Form
1677 Raga Manavi
2613 Raga Hamsa Vinodini
2769 Dyrimic
6-Z28 619 Double-Phrygian Hexatonic
857 Aeolydimic
1427 Lolimic
1613 Thylimic
2357 Raga Sarasanana
2761 Dagimic
6-Z26 427 Raga Suddha Simantini
1379 Kycrimic
1421 Raga Trimurti
1589 Raga Rageshri
2261 Raga Caturangini
2737 Raga Hari Nata
6-34 683 Stogimic
1369 Boptimic
1429 Bythimic
1621 Scriabin's Prometheus
2389 Eskimo Hexatonic 2
853 Epothimic
1237 Salimic
1333 Lyptimic
1357 Takemitsu Linea Mode 2
1363 Gygimic
2729 Aeragimic
6-33 685 Raga Suddha Bangala
1195 Raga Gandharavam
1385 Phracrimic
1445 Raga Navamanohari
1685 Zeracrimic
2645 Raga Mruganandana
725 Raga Yamuna Kalyani
1205 Raga Siva Kambhoji
1355 Aeolorimic
1705 Raga Manohari
2725 Raga Nagagandhari
6-31 691 Raga Kalavati
1433 Dynimic
1637 Syptimic
2393 Zathimic
2453 Raga Latika
821 Aeranimic
851 Raga Hejjajji
1229 Raga Simharava
1331 Raga Vasantabhairavi
2473 Raga Takka
2713 Porimic
6-32 693 Arezzo Major Diatonic Hexachord
1197 Minor Hexatonic
1323 Ritsu
1449 Raga Gopikavasantam
1701 Mixolydian Hexatonic
2709 Raga Kumud
6-30 715 T4 Prime Mode
1625 Hungarian Major No5
2405 T4 First Rotation
845 Raga Neelangi
1235 Tritone Scale
2665 Messiaen Mode 2 Truncation 1
813 Larimic
1227 Thacrimic
1689 Lorimic
2409 Zacrimic
2661 Stydimic
6-Z29 717 Raga Vijayanagari
843 Molimic
1203 Pagimic
1641 Bocrimic
2469 Raga Bhinna Pancama
2649 Aeolythimic
6-20 819 Augmented Inverse
2457 Augmented
6-35 1365 Whole Tone

#### Modal families with 7 tones

Forte Class NamePrime and its rotationsInverse and its rotations
7-3 319 Epodian
1009 Katyptian
2207 Mygian
3151 Pacrian
3623 Aerocrian
3859 Aeolarian
3977 Kythian
505 Sanian
575 Ionydian
2335 Epydian
3215 Katydian
3655 Mathian
3875 Aeryptian
7-9 351 Epanian
1521 Stanian
1989 Dydian
2223 Konian
3159 Stocrian
3627 Kalian
3861 Phroptian
501 Katylian
1149 Bydian
1311 Bynian
2703 Galian
3399 Zonian
3747 Myrian
3921 Pythian
761 Ponian
1993 Katoptian
2351 Gynian
3223 Thyphian
3659 Polian
3877 Thanian
637 Debussy's Heptatonic
2639 Dothian
3367 Moptian
3731 Aeryrian
3913 Bonian
7-8 381 Kogian
1119 Rarian
2001 Gydian
2607 Aerolian
3351 Crater Scale
3723 Myptian
3909 Rydian
995 Phrathian
2255 Dylian
2545 Thycrian
3175 Eponian
3635 Katygian
3865 Starian
499 Ionaptian
799 Lolian
2297 Thylian
2447 Thagian
3271 Mela Raghupriya
3683 Dycrian
3889 Parian
7-Z12 671 Stycrian
997 Rycrian
1273 Heptatonic Blues
2383 Katorian
3239 Mela Tanarupi
3667 Kaptian
3881 Morian
7-Z36 367 Aerodian
1777 Saptian
1931 Stogian
2231 Macrian
3013 Thynian
3163 Rogian
3629 Boptian
493 Rygian
1147 Epynian
1679 Kydian
2621 Ionogian
2887 Gaptian
3491 Tharian
3793 Aeopian
7-16 623 Sycrian
889 Borian
1939 Dathian
3017 Gacrian
3227 Aeolocrian
3661 Mixodorian
635 Epolian
985 Mela Sucaritra
1615 Sydian
2365 Sythian
2855 Epocrian
3475 Kylian
3785 Epagian
7-11 379 Aeragian
1583 Salian
1969 Zorian
2237 Epothian
2839 Lyptian
3467 Sudhvidhamagini
3781 Gyphian
445 Gocrian
1135 Katolian
1955 Sonian
2615 Thoptian
3025 Epycrian
3355 Bagian
3725 Kyrian
7-14 431 Epyrian
1507 Zynian
1933 Mocrian
2263 Lycrian
2801 Zogian
3179 Daptian
3637 Raga Rageshri
491 Aeolyrian
1423 Doptian
1597 Aeolodian
2293 Gorian
2759 Mela Pavani
3427 Zacrian
7-24 687 Aeolythian
1401 Pagian
1509 Ragian
1941 Aeranian
2391 Molian
3243 Mela Rupavati
3669 Mothian
981 Mela Kantamani
1269 Katythian
1359 Aerygian
2727 Mela Manavati
3411 Enigmatic
3753 Phraptian
7-23 701 Mixonyphian
1199 Magian
1513 Stathian
1957 Pyrian
3371 Aeolylian
3733 Gycrian
757 Ionyptian
1213 Gyrian
1327 Zalian
1961 Soptian
2711 Stolian
3403 Bylian
3749 Raga Sorati
7-Z18 755 Phrythian
815 Bolian
1945 Zarian
2425 Rorian
2455 Bothian
3275 Mela Divyamani
3685 Kodian
829 Lygian
979 Mela Dhavalambari
1231 Logian
2537 Laptian
2663 Lalian
3379 Verdi's Scala Enigmatica Descending
3737 Phrocrian
7-7 463 Zythian
967 Mela Salaga
2279 Dyrian
2531 Danian
3187 Koptian
3313 Aeolacrian
3641 Thocrian
487 Dynian
2291 Zydian
2503 Mela Jhalavarali
3193 Zathian
3299 Syptian
3697 Ionarian
7-19 719 Kanian
971 Mela Gavambodhi
1657 Ionothian
2407 Zylian
2533 Podian
3251 Mela Hatakambari
3673 Ranian
847 Ganian
973 Mela Syamalangi
1267 Katynian
2471 Mela Ganamurti
2681 Aerycrian
3283 Mela Visvambhari
3689 Katocrian
7-13 375 Sodian
1815 Godian
1905 Katacrian
2235 Bathian
2955 Thorian
3165 Mylian
3525 Zocrian
477 Stacrian
1143 Styrian
1863 Pycrian
2619 Ionyrian
2979 Gyptian
3357 Phrodian
3537 Katogian
7-Z17 631 Zygian
953 Mela Yagapriya
1831 Pothian
2363 Kataptian
2963 Bygian
3229 Aeolaptian
3529 Stalian
7-Z38 439 Bythian
1763 Katalian
1819 Pydian
2929 Aeolathian
2957 Thygian
3181 Rolian
475 Aeolygian
1595 Dacrian
1735 Mela Navanitam
2285 Aerogian
2845 Baptian
2915 Aeolydian
3505 Stygian
7-27 695 Sarian
1465 Mela Ragavardhani
1765 Lonian
1835 Byptian
2395 Zoptian
2965 Darian
3245 Mela Varunapriya
949 Mela Mararanjani
1261 Modified Blues
1339 Kycrian
1703 Mela Vanaspati
2717 Epygian
2899 Kagian
3497 Phrolian
7-25 733 Donian
1207 Aeoloptian
1769 Blues Heptatonic II
1867 Solian
2651 Panian
2981 Ionolian
3373 Lodian
749 Aeologian
1211 Ceiling Scale
1687 Phralian
1897 Ionopian
2653 Sygian
2891 Phrogian
3493 Rathian
7-21 823 Stodian
883 Ralian
1843 Ionygian
2459 Ionocrian
2489 Mela Gangeyabhusani
2969 Tholian
3277 Mela Nitimati
827 Mixolocrian
947 Mela Gayakapriya
1639 Aeolothian
2461 Sagian
2521 Mela Dhatuvardhani
2867 Major Romani
3481 Katathian
7-26 699 Aerothian
1497 Mela Jyotisvarupini
1623 Lothian
1893 Ionylian
2397 Stagian
2859 Phrycrian
3477 Kyptian
885 Sathian
1245 Lathian
1335 Elephant Scale
1875 Persichetti Scale
2715 Kynian
2985 Epacrian
3405 Stynian
7-Z37 443 Kothian
1591 Rodian
1891 Thalian
2269 Pygian
2843 Sorian
2993 Stythian
3469 Monian
7-15 471 Dodian
1479 Mela Jalarnava
2283 Aeolyptian
2787 Zyrian
3189 Aeolonian
3441 Thacrian
1483 Mela Bhavapriya
1837 Dalian
2411 Aeolorian
2789 Zolian
3253 Mela Naganandini
1447 Mela Ratnangi
1693 Dogian
2677 Thodian
2771 Marva That
3433 Thonian
7-28 747 Lynian
1431 Phragian
1629 Synian
1881 Korian
2421 Malian
2763 Mela Suvarnangi
3429 Marian
861 Rylian
1239 Epaptian
1491 Namanarayani
1869 Katyrian
2667 Byrian
2793 Eporian
3381 Katanian
7-30 855 Porian
1395 Locrian Dominant
1485 Minor Romani
1845 Mixolydian Augmented
2475 Neapolitan Minor
2745 Mela Sulini
3285 Lydian #6
939 Mela Senavati
1383 Pynian
1437 Sabach ascending
1653 Minor Romani Inverse
2517 Harmonic Lydian
2739 Mela Suryakanta
3417 Golian
1373 Storian
1397 Major Locrian
1493 Lydian Minor
1877 Aeroptian
2731 Neapolitan Major
7-20 743 Chromatic Hypophrygian Inverse
919 Chromatic Phrygian Inverse
1849 Chromatic Hypodorian Inverse
2419 Raga Lalita
2507 Todi That
3257 Mela Calanata
3301 Chromatic Mixolydian Inverse
925 Chromatic Hypodorian
935 Chromatic Dorian
1255 Chromatic Mixolydian
2515 Chromatic Hypolydian
2675 Chromatic Lydian
3305 Chromatic Hypophrygian
3385 Chromatic Phrygian
7-22 871 Hungarian Romani Minor 4th Mode
923 Ultraphrygian
1651 Asian
2483 Double Harmonic
2509 Double Harmonic Minor
2873 Ionian Augmented Sharp 2
3289 Lydian Sharp 2 Sharp 6
7-31 731 Alternating Heptamode
1627 Hungarian Major 4th Mode
1753 Hungarian Major
2413 Harmonic Minor Flat 5
2861 Hungarian Major 5th Mode
2917 Nohkan Flute Scale
877 Moravian Pistalkova
1243 Epylian
1691 Kathian
1747 Mela Ramapriya
2669 Jeths' Mode
2893 Lylian
2921 Pogian
7-32 859 Ultralocrian
1459 Phrygian Dominant
1643 Locrian Natural 6
1741 Lydian Diminished
2477 Harmonic Minor
2777 Aeolian Harmonic
2869 Major Augmented
875 Locrian Double-flat 7
1435 Phrygian Flat 4
1645 Dorian Flat 5
1715 Harmonic Minor Inverse
2485 Harmonic Major
2765 Lydian Flat 3
2905 Lydian Augmented Sharp 2
7-34 1371 Superlocrian
1389 Minor Locrian
1461 Major-Minor
1707 Dorian Flat 2
1749 Acoustic
2733 Melodic Minor Ascending
2901 Lydian Augmented
7-35 1387 Locrian
1451 Phrygian
1453 Aeolian
1709 Dorian
1717 Mixolydian
2741 Major
2773 Lydian

#### Modal families with 8 tones

Forte Class NamePrime and its rotationsInverse and its rotations
8-2 383 Logyllic
2033 Stolyllic
2239 Dacryllic
3167 Thynyllic
3631 Gydyllic
3863 Eparyllic
3979 Dynyllic
4037 Ionyllic
509 Ionothyllic
1151 Mythyllic
2623 Aerylyllic
3359 Bonyllic
3727 Tholyllic
3911 Katyryllic
4049 Stycryllic
8-3 639 Ionaryllic
1017 Dythyllic
2367 Laryllic
3231 Kataptyllic
3663 Sonyllic
3879 Pathyllic
3987 Loryllic
4041 Zaryllic
8-4 447 Thyphyllic
2019 Palyllic
2271 Poptyllic
3057 Phroryllic
3183 Mixonyllic
3639 Paptyllic
3867 Storyllic
3981 Phrycryllic
507 Moryllic
1599 Pocryllic
2301 Bydyllic
2847 Phracryllic
3471 Gyryllic
3783 Phrygyllic
3939 Dogyllic
4017 Dolyllic
8-11 703 Aerocryllic
1529 Kataryllic
2021 Katycryllic
2399 Zanyllic
3247 Aeolonyllic
3671 Aeonyllic
3883 Kyryllic
3989 Sythyllic
1013 Stydyllic
1343 Zalyllic
2719 Zocryllic
3407 Katocryllic
3751 Aerathyllic
3923 Stoptyllic
4009 Phranyllic
8-10 765 Mixonyphyllic
1215 Aeolanyllic
2025 Mixolydyllic
2655 Thocryllic
3375 Kygyllic
3735 Ionagyllic
3915 Gogyllic
8-7 831 Rodyllic
1011 Kycryllic
2463 Ionathyllic
2553 Aeolaptyllic
3279 Pythyllic
3687 Zonyllic
3891 Ryryllic
3993 Ioniptyllic
8-5 479 Kocryllic
1991 Phryptyllic
2287 Lodyllic
3043 Ionayllic
3191 Bynyllic
3643 Kydyllic
3869 Bygyllic
503 Thoptyllic
1823 Phralyllic
2299 Phraptyllic
2959 Dygyllic
3197 Gylyllic
3527 Ronyllic
3811 Epogyllic
3953 Thagyllic
8-13 735 Sylyllic
1785 Tharyllic
1995 Sideways Scale
2415 Lothyllic
3045 Raptyllic
3255 Daryllic
3675 Monyllic
3885 Styryllic
1275 Stagyllic
1695 Phrodyllic
2685 Ionoryllic
2895 Aeragyllic
3495 Banyllic
3795 Epothyllic
3945 Lydyllic
8-12 763 Doryllic
1631 Rynyllic
2009 Stacryllic
2863 Aerogyllic
3479 Rothyllic
3787 Kagyllic
3941 Stathyllic
893 Pycryllic
1247 Mygyllic
2003 Lolyllic
2671 Lylyllic
3049 Aeronyllic
3383 Daptyllic
3739 Ioninyllic
3917 Epaphyllic
8-Z15 863 Pyryllic
1523 Zothyllic
1997 Raga Cintamani
2479 Harmonic and Neapolitan Minor Mixed
2809 Gythyllic
3287 Phrathyllic
3893 Phrocryllic
1003 Ionyryllic
1439 Rolyllic
1661 Gonyllic
2549 Rydyllic
2767 Katydyllic
3431 Zyptyllic
3763 Modyllic
3929 Aeolothyllic
8-21 1375 Bothyllic
1405 Goryllic
1525 Sodyllic
2005 Gygyllic
2735 Gynyllic
3415 Ionaptyllic
3755 Phryryllic
3925 Thyryllic
8-8 927 Koptyllic
999 Bylyllic
2511 Epyryllic
2547 Raga Ramkali
3303 Soptyllic
3321 Ionycryllic
3699 Aeolylyllic
3897 Locryllic
8-6 495 Bocryllic
1935 Mycryllic
2295 Kogyllic
3015 Laptyllic
3195 Raryllic
3555 Pylyllic
3645 Zycryllic
3825 Pynyllic
8-Z29 751 Epacryllic
1913 Zagyllic
1943 Malyllic
2423 Thorcryllic
3019 Mydyllic
3259 Loptyllic
3557 Thycryllic
3677 Katylyllic
989 Phrolyllic
1271 Kolyllic
1871 Aeolyllic
2683 Thodyllic
2983 Zythyllic
3389 Socryllic
3539 Aeoryllic
3817 Zoryllic
8-14 759 Katalyllic
1839 Zogyllic
1977 Dagyllic
2427 Katoryllic
3261 Dodyllic
3531 Neveseri
3813 Aeologyllic
957 Phronyllic
1263 Stynyllic
1959 Katolyllic
2679 Rathyllic
3027 Rythyllic
3387 Aeryptyllic
3561 Pothyllic
3741 Zydyllic
8-18 879 Aeolocryllic
1779 Aerythyllic
1947 Ionoyllic
2487 Phroptyllic
2937 Aeolathyllic
3021 Gyptyllic
3291 Kodyllic
3693 Epaptyllic
987 Aeraptyllic
1659 Magyllic
1743 Epigyllic
2541 Algerian
2877 Phrylyllic
2919 Molyllic
3507 Ponyllic
3801 Maptyllic
1469 Epiryllic
1781 Lydian/Mixolydian Mixed
1963 Epocryllic
2743 Staptyllic
3029 Ionocryllic
3419 Magen Abot 1
3757 Raga Mian Ki Malhar
1403 Espla's Scale
1517 Spanish Octamode 4th Rotation
1973 Spanish Octamode 6th Rotation
2749 Spanish Octamode 1st Rotation
2903 Spanish Octamode 10th Rotation
3499 Hamel
3797 Spanish Octamode 8th Rotation
8-17 891 Ionilyllic
1647 Polyllic
1971 Aerynyllic
2493 Manyllic
2871 Stanyllic
3033 Doptyllic
3483 Mugham Shüshtär
3789 Eporyllic
8-16 943 Aerygyllic
1511 Styptyllic
1949 Mathyllic
2519 Dathyllic
2803 Raga Bhatiyar
3307 Boptyllic
3449 Bacryllic
3701 Bagyllic
983 Epygyllic
1487 Lycryllic
1853 Phrynyllic
2539 Half-Diminished Bebop
2791 Ionyptyllic
3317 Lanyllic
3443 Verdi's Scala Enigmatica
3769 Aeracryllic
8-23 1455 Quartal Octamode
1515 Phrygian/Locrian Mixed
1725 Minor Bebop
1965 Raga Mukhari
2775 Quartal Octamode 10th Rotation
2805 Ichikotsuchô
3435 Prokofiev
3765 Dominant Bebop
8-9 975 Messiaen Mode 4 Rotation 3
2535 Messiaen Mode 4
3315 Tcherepnin Octatonic Mode 1
3705 Messiaen Mode 4 Rotation 2
8-19 887 Sathyllic
1847 Thacryllic
1907 Lynyllic
2491 Layllic
2971 Aeolynyllic
3001 Lonyllic
3293 Saryllic
955 Ionogyllic
1655 Katygyllic
1895 Salyllic
2525 Aeolaryllic
2875 Ganyllic
2995 Raga Saurashtra
3485 Sabach
3545 Thyptyllic
8-24 1399 Syryllic
1501 Stygyllic
1879 Mixoryllic
1909 Epicryllic
2747 Stythyllic
2987 Neapolitan Major and Minor Mixed
3421 Aerothyllic
3541 Racryllic
8-20 951 Thogyllic
1767 Dyryllic
1851 Zacryllic
2523 Mirage Scale
2931 Zathyllic
2973 Panyllic
3309 Bycryllic
3513 Dydyllic
8-27 1463 Zaptyllic
1757 Ionyphyllic
1771 Stylyllic
1883 Mixopyryllic
2779 Shostakovich
2933 Dalyllic
2989 Bebop Minor
3437 Gathyllic
1499 Bebop Locrian
1723 JG Octatonic
1751 Aeolyryllic
1901 Ionidyllic
2797 Stalyllic
2909 Mocryllic
2923 Baryllic
3509 Stogyllic
8-26 1467 Spanish Phrygian
1719 Lyryllic
1773 Blues Scale II
1899 Moptyllic
2781 Gycryllic
2907 Magen Abot 2
2997 Major Bebop
3501 Gregorian Nr.4
8-25 1495 Messiaen Mode 6 Rotation 2
1885 Messiaen Mode 6 Rotation 1
2795 Van der Horst Octatonic
3445 Messiaen Mode 6
8-28 1755 Octatonic
2925 Diminished

#### Modal families with 9 tones

Forte Class NamePrime and its rotationsInverse and its rotations
9-1 511 Chromatic Nonamode
2303 Nonatonic Chromatic 2
3199 Nonatonic Chromatic 3
3647 Nonatonic Chromatic 4
3871 Nonatonic Chromatic 5
3983 Nonatonic Chromatic 6
4039 Nonatonic Chromatic 7
4067 Nonatonic Chromatic 8
4081 Nonatonic Chromatic Descending
9-2 767 Raptygic
2041 Aeolacrygic
2431 Gythygic
3263 Pyrygic
3679 Rycrygic
3887 Phrathygic
4043 Phrocrygic
4069 Starygic
1279 Sarygic
2687 Thacrygic
3391 Aeolynygic
3919 Lynygic
4007 Doptygic
4051 Ionilygic
4073 Sathygic
9-3 895 Aeolathygic
2035 Aerythygic
2495 Aeolocrygic
3065 Zothygic
3295 Phroptygic
3695 Kodygic
3895 Eparygic
3995 Ionygic
4045 Gyptygic
1019 Aeranygic
1663 Lydygic
2557 Dothygic
3487 Byptygic
3791 Stodygic
3943 Zynygic
4019 Lonygic
4057 Phrygic
9-6 1407 Tharygic
1533 Katycrygic
2037 Sythygic
2751 Sylygic
3423 Lothygic
3759 Darygic
3927 Monygic
4011 Styrygic
4053 Kyrygic
9-4 959 Katylygic
2023 Zodygic
3311 Mixodygic
3577 Loptygic
3703 Katalygic
3899 Katorygic
3997 Dogygic
1015 Ionodygic
1855 Marygic
2555 Bythygic
2975 Gaptygic
3325 Epygic
3535 Aeroptygic
3815 Mylygic
3955 Galygic
4025 Kalygic
1789 Blues Enneatonic II
2027 Boptygic
2783 Gothygic
3061 Apinygic
3439 Lythygic
3767 Chromatic Bebop
3931 Aerygic
4013 Raga Pilu
1531 Styptygic
1727 Sydygic
2029 Kiourdi
2813 Zolygic
2911 Katygic
3503 Zyphygic
3799 Aeralygic
3947 Ryptygic
9-5 991 Aeolygic
1999 Zacrygic
2543 Dydygic
3047 Panygic
3319 Tholygic
3571 Dyrygic
3707 Rynygic
3833 Dycrygic
3901 Bycrygic
1007 Ionycrygic
1951 Gonygic
2551 Zoptygic
3023 Aeracrygic
3323 Phrygygic
3559 Aerathygic
3709 Locrygic
3827 Dorygic
3961 Mixolydygic
1917 Sacrygic
2007 Stonygic
2799 Epilygic
3051 Stalygic
3447 Kynygic
3573 Kaptygic
3771 Stophygic
3933 Ionidygic
1527 Aeolyrygic
1887 Aerocrygic
2013 Mocrygic
2811 Barygic
2991 Zanygic
3453 Katarygic
3543 Aeolonygic
3819 Aeolanygic
3957 Porygic
9-10 1759 Pylygic
1787 Mycrygic
2011 Raphygic
2927 Rodygic
2941 Laptygic
3053 Zycrygic
3511 Epolygic
3803 Epidygic
3949 Koptygic
9-9 1519 Locrian/Aeolian Mixed
1967 Diatonic Dorian Mixed
1981 Houseini
2807 Zylygic
3031 Epithygic
3451 Garygic
3563 Ionoptygic
3773 Raga Malgunji
3829 Taishikicho
9-11 1775 Lyrygic
1915 Thydygic
1975 Ionocrygic
2935 Modygic
3005 Gycrygic
3035 Gocrygic
3515 Moorish Phrygian
3565 Aeolorygic
3805 Moptygic
1783 Youlan
1903 Diminishing Nonamode Basic
1979 Diminishing Nonamode 6th Rotation
2939 Diminishing Nonamode 2nd Rotation
2999 Diminishing Nonamode
3037 Nine Tone Scale
3517 Diminishing Nonamode 1st Rotation
3547 Diminishing Nonamode 9th Rotation
3821 Diminishing Nonamode 8th Rotation
9-12 1911 Messiaen Mode 3 Rotation 1
3003 Messiaen Mode 3 Rotation 2
3549 Messiaen Mode 3

#### Modal families with 10 tones

Forte Class NamePrime and its rotationsInverse and its rotations
10-1 1023 Chromatic Decamode
2559 Decatonic Chromatic 2
3327 Decatonic Chromatic 3
3711 Decatonic Chromatic 4
3903 Decatonic Chromatic 5
3999 Decatonic Chromatic 6
4047 Decatonic Chromatic 7
4071 Decatonic Chromatic 8
4083 Decatonic Chromatic 9
4089 Decatonic Chromatic Descending
10-2 1535 Mixodyllian
2045 Katogyllian
3455 Ryptyllian
3775 Loptyllian
3935 Kataphyllian
4055 Dagyllian
4075 Katyllian
4085 Sydyllian
10-3 1791 Aerygyllian
2043 Lythyllian
2943 Dathyllian
3069 Bacryllian
3519 Raga Sindhi-Bhairavi
3807 Bagyllian
3951 Mathyllian
4023 Styptyllian
4059 Zolyllian
4077 Gothyllian
10-4 1919 Rocryllian
2039 Danyllian
3007 Zyryllian
3067 Goptyllian
3551 Sagyllian
3581 Epocryllian
3823 Epinyllian
3959 Katagyllian
4027 Ragyllian
4061 Staptyllian
10-5 1983 Soryllian
3039 Godyllian
3063 Solyllian
3567 Epityllian
3579 Zyphyllian
3831 Ionyllian
3837 Minor Pentatonic With Leading Tones
3963 Aeoryllian
4029 Major/Minor Mixed
10-6 2015 Messiaen Mode 7 Rotation 4
3055 Messiaen Mode 7
3575 Messiaen Mode 7 Rotation 1
3835 Messiaen Mode 7 Rotation 2
3965 Messiaen Mode 7 Rotation 3

#### Modal families with 11 tones

Forte Class NamePrime and its rotationsInverse and its rotations
11-1 2047 Chromatic Undecamode
3071 Chromatic Undecamode 2
3583 Chromatic Undecamode 3
3839 Chromatic Undecamode 4
3967 Chromatic Undecamode 5
4031 Chromatic Undecamode 6
4063 Chromatic Undecamode 7
4079 Chromatic Undecamode 8
4087 Chromatic Undecamode 9
4091 Chromatic Undecamode 10
4093 Chromatic Undecamode 11

#### Modal families with 12 tones

Forte Class NamePrime and its rotationsInverse and its rotations
12-1 4095 Chromatic

## 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)

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
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 Olivier 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

This diagram illustrates that all symmetrical scales are truncations of the Messaienic modes (labeled "M" plus their number as assigned by Messaien). M1, M2, M3, and M7 are all truncations of the chromatic scale 4095; M2, M4, and M6 are truncations of M7; M1 and M5 are truncations of M6. M1 is also a truncation of M3, and M5 is also a truncation of M4. All the other rotationally symmetrical scales are labeled "T#" and are shown where they belong in the 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
• 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 an axis of reflection. If that axis falls on the root, then the scale will have the same interval pattern ascending and descending. A reflectively symmetrical scale, symmetrical on the axis of the root tone, is called a palindromic scale.

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

A chiral object and its mirror image are called enantiomorphs. (source)

ScaleChirality / Enantiomorph
585 Diminished Seventh achiral
661 Major Pentatonic achiral
859 Ultralocrian 2905 Lydian Augmented Sharp 2
1193 Minor Pentatonic achiral
1257 Blues Scale 741 Gathimic
1365 Whole Tone achiral
1371 Superlocrian achiral
1387 Locrian achiral
1389 Minor Locrian achiral
1397 Major Locrian achiral
1451 Phrygian achiral
1453 Aeolian achiral
1459 Phrygian Dominant 2485 Harmonic Major
1485 Minor Romani 1653 Minor Romani Inverse
1493 Lydian Minor achiral
1499 Bebop Locrian 2933 Dalyllic
1621 Scriabin's Prometheus 1357 Takemitsu Linea Mode 2
1643 Locrian Natural 6 2765 Lydian Flat 3
1709 Dorian achiral
1717 Mixolydian achiral
1725 Minor Bebop achiral
1741 Lydian Diminished 1645 Dorian Flat 5
1749 Acoustic achiral
1753 Hungarian Major 877 Moravian Pistalkova
1755 Octatonic achiral
2257 Lydian Pentatonic  355 African Pentatonic 4
2275 Messiaen Mode 5 achiral
2457 Augmented achiral
2475 Neapolitan Minor 2739 Mela Suryakanta
2477 Harmonic Minor 1715 Harmonic Minor Inverse
2483 Double Harmonic achiral
2509 Double Harmonic Minor achiral
2535 Messiaen Mode 4 achiral
2731 Neapolitan Major achiral
2733 Melodic Minor Ascending achiral
2741 Major achiral
2773 Lydian achiral
2777 Aeolian Harmonic 875 Locrian Double-flat 7
2869 Major Augmented 1435 Phrygian Flat 4
2901 Lydian Augmented achiral
2925 Diminished achiral
2989 Bebop Minor 1723 JG Octatonic
2997 Major Bebop achiral
3055 Messiaen Mode 7 achiral
3411 Enigmatic 2391 Molian
3445 Messiaen Mode 6 achiral
3549 Messiaen Mode 3 achiral
3669 Mothian 1359 Aerygian
3765 Dominant Bebop achiral
4095 Chromatic achiral

• 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:

## Balance

We assert a scale is "balanced", if the distribution of tones arranged around a 12-spoke wheel would balance on its centre. This is related to the well-known problem in mathematics known as the "balanced centrifuge problem".

There are 47 balanced scales. Here they are:

## Interval Spectrum / Richness / Interval Vector

Howard Hanson, in the book "Harmonic Materials"4, posits that the character of a sonority is defined by the intervals that are within it. This definition is also called an "Interval Vector"5 or "Interval Class Vector" in Pitch Class Set theory. 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. Hanson 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 autocorrelation6". 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.

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 - for example, and both have the spectrum "pm3nd", but they are not modes of each other.

Below is a table of some of 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

ScaleSpectrum (Hanson)Vector (modern)
585 Diminished Seventh n4t2004002
661 Major Pentatonic p4mn2s3032140
859 Ultralocrian p4m4n5s3d3t2335442
1193 Minor Pentatonic p4mn2s3032140
1257 Blues Scale p4m2n3s3d2t233241
1365 Whole Tone m6s6t3060603
1371 Superlocrian p4m4n4s5d2t2254442
1387 Locrian p6m3n4s5d2t254361
1389 Minor Locrian p4m4n4s5d2t2254442
1397 Major Locrian p2m6n2s6d2t3262623
1451 Phrygian p6m3n4s5d2t254361
1453 Aeolian p6m3n4s5d2t254361
1459 Phrygian Dominant p4m4n5s3d3t2335442
1485 Minor Romani p4m5n3s4d3t2343542
1493 Lydian Minor p2m6n2s6d2t3262623
1499 Bebop Locrian p5m5n6s5d4t3456553
1621 Scriabin's Prometheus p2m4n2s4dt2142422
1643 Locrian Natural 6 p4m4n5s3d3t2335442
1709 Dorian p6m3n4s5d2t254361
1717 Mixolydian p6m3n4s5d2t254361
1725 Minor Bebop p7m4n5s6d4t2465472
1741 Lydian Diminished p4m4n5s3d3t2335442
1749 Acoustic p4m4n4s5d2t2254442
1753 Hungarian Major p3m3n6s3d3t3336333
1755 Octatonic p4m4n8s4d4t4448444
2257 Lydian Pentatonic p3m2nsd2t211231
2275 Messiaen Mode 5 p4m2s2d4t3420243
2457 Augmented p3m6n3d3303630
2475 Neapolitan Minor p4m5n3s4d3t2343542
2477 Harmonic Minor p4m4n5s3d3t2335442
2483 Double Harmonic p4m5n4s2d4t2424542
2509 Double Harmonic Minor p4m5n4s2d4t2424542
2535 Messiaen Mode 4 p6m4n4s4d6t4644464
2731 Neapolitan Major p2m6n2s6d2t3262623
2733 Melodic Minor Ascending p4m4n4s5d2t2254442
2741 Major p6m3n4s5d2t254361
2773 Lydian p6m3n4s5d2t254361
2777 Aeolian Harmonic p4m4n5s3d3t2335442
2869 Major Augmented p4m4n5s3d3t2335442
2901 Lydian Augmented p4m4n4s5d2t2254442
2925 Diminished p4m4n8s4d4t4448444
2989 Bebop Minor p5m5n6s5d4t3456553
2997 Major Bebop p6m5n6s5d4t2456562
3055 Messiaen Mode 7 p8m8n8s8d8t5888885
3411 Enigmatic p4m4n3s5d3t2353442
3445 Messiaen Mode 6 p4m6n4s6d4t4464644
3549 Messiaen Mode 3 p6m9n6s6d6t3666963
3669 Mothian p4m4n3s5d3t2353442
3765 Dominant Bebop p7m4n5s6d4t2465472
4095 Chromatic p12m12n12s12d12t612121212126
• 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?

## Deep Scales

A "deep" scale is one for which the interval vector consists of unique values. There are only two Prime Deep Scales, and all their rotations and reflections will also be Deep. One of them is the major diatonic collection, and the other is the major scale with the leading tone omitted. Here they are:

## Evenness

Another interesting property of a scale is whether the notes are evenly spaced, or clumped together. The theory of musical scale evenness owes to "Diatonic Set Theory", the work of Richard Krantz and Jack Douhett7. In their paper, they explain how you can determine the "evenness" of a scale, first by establishing the intervals between each note and every other.

Generic interval is 2, Specific interval is 5
A crucial concept to understand in Diatonic Set Theory is the distinction between a generic interval and a specific interval. A specific interval is the number of semitones between two tones; for example between a C and a E, that's a specific interval of 4 semitones. The generic interval is the number of scale steps between two tones of a scale; for example in C major scale, the distance between C and E is 2.

To measure the evenness of the scale, the first step is to build the distribution spectra. The spectra shows the distinct specific intervals between notes, for each generic interval of the scale. Each spectrum is notated like this:

`<generic interval> = { specific interval, specific interval, ...}`

The number in angle brackets is the generic interval, ie we are asking "for notes that are this many steps away in the scale". The numbers in curly brackets are the specific intervals we find present for those steps, ie "between those steps we find notes that are this many semitones apart".

It's best explained with an example. Below is the scale bracelet diagram and distribution spectra for Scale 1449:

ScaleNotesDistribution Spectra
<1> = {1,2,3}
<2> = {3,4,5}
<3> = {5,7}
<4> = {7,8,9}
<5> = {9,10,11}

In line 1, the first spectrum, <1> indicates that we are looking at notes that are one scale step away from each other. We have notes that are one semitone apart (eg G and G#), two semitones (D# and F), and three semitones (C and D#). Duplicates of these are ignored; we merely want to know what intervals are present, not how many of them exist.

In line 2, the second spectrum, <2> indicates that we are looking at notes that are two scale steps away from each other. We see pairs that are three semitones apart (eg F and G#), four semitones (D# and G), and five semitones (C and F).

When there is more than one specific interval, the spectrum width is the difference between the largest and smallest value. For example for the <3> spectrum above, the specific intervals are {5,7} and so its width is 2, which is 7 minus 5.

The spectrum variation is the sum of all those widths, divided by the number of tones.

Once the distribution specta are built, we analyze them to discover interesting properties of the scale. For instance,

• If all the spectra have just one specific interval, then the scale has exactly equal distribution
• If the spectra have two intervals with a difference no greater than one, then the scale is maximally even - it's distributed as evenly as it can be with no room for improvement.
• If the spectra has any widths greater than 1, then it's not maximally even.
• If there are exactly two specific intervals in all the spectra, then the scale is said to have Myhill's property.

Ultimately, the measure of a scale's evenness is its Spectra Variation. We add up all the spectrum widths, and divide by the number of tones in the scale, to achieve an average width with respect to the scale size. If a scale has perfectly spaced notes with completely uniform evenness, then it has a spectra variation of zero. A higher variation means the scale distribution is less even.

The following four scales have a perfect score - a spectra variation of zero:

Obviously, it is possible to evenly distribute 6 tones around a 12-tone scale. But it is impossible to do that with a 5 tone (pentatonic) or 7 tone (heptatonic) scale. For such tone counts all we can hope to achieve is an optimally even distribution.

Below are all the prime scales (ie with rotations and reflections omitted), sorted from most even to least even. If you click to each scale detail page, you can read its spectra variation there.

## Myhill's Property

Myhill's Property is the quality of a pitch class set where the spectrum has exactly two specific intervals for every generic interval. There are 6 prime scales with Myhill's property.

There is a chapter all about the Myhill Property - and why it mattters - in the book.

## Propriety

In the section about evenness, we discussed the concepts of Generic Intervals and Specific Intervals. There is a property of a scale named "Propriety", which indicates whether the relation between generic and specific intervals is ambiguous or not. This property was discovered by David Rothenberg in 1978, so it is sometimes called "Rothenberg Propriety".

Rothenberg stated that there are three levels of propriety. At the most exclusive level, there are those whose specific intervals have an unambiguous relationship to the generic scale steps; these are called Strictly Proper. An easy example of a strictly proper scale is the 12-tone chromatic scale. If you hear an interval of 3 semitones, you know without any doubt that it is the generic distance of 3 scale steps. Any specific interval of 7 semitones is without any ambiguity going to be a generic interval of 7 scale steps. And so on.

Strictly proper scales are not common. Since all transformations of a scale have the same propriety, here we will only look at prime scales. Here are all the strictly proper ones:

Rothenberg defined that below these strictly proper scales, there is a strata of scales that are merely proper, bur not strictly so. To be proper, a specific interval can describe two different generic intervals, but there mustn't be any overlap. Stated another way, in a proper scale, there should never be a generic 4th that is smaller than a generic 3rd; but they might be the same size. The collection of proper scales is larger than the strict collection, but it's still an exclusive club. Here are all the prime scales that are proper but not strictly so:

Lastly, there are all the other scales that aren't proper at all; these are Improper Scales. Those scales will all have interval overlapping, where there the size of a generic interval does not assure that it is specifically larger or smaller than another generic interval.

You can judge the propriety of a scale by inspecting its distribution spectra. Look at this scale:

ScaleNotesDistribution Spectra
<1> = {1,2,3}
<2> = {3,4,5}
<3> = {5,7}
<4> = {7,8,9}
<5> = {9,10,11}

Observe the spectrum of each generic interval, and how the specific intervals fit into niches. The specific interval of 3 semitones could be <1> or <2>. The specific interval of 5 semitones could be <2> or <3>, and so on. The generic ranges meet and share common edges, but they do not overlap. That means this scale is proper, but it is not strictly proper.

Next, we'll look at an example of an improper scale, the Neapolitan Minor.

ScaleNotesDistribution Spectra
<1> = {1,2,3}
<2> = {2,3,4}
<3> = {4,5,6}
<4> = {6,7,8}
<5> = {8,9,10}
<6> = {9,10,11}

The impropriety of this scale is evident in two of its specific intervals. In Neapolitan Minor, it is possible to have a generic interval of two scale steps with a specific interval of 2 (between B and D flat), which is smaller than a generic interval of one step with a specific interval of 3 (between A flat and B). The fact that a 2nd can be larger than a 3rd means this scale is not proper. The same situation exists where a generic interval of 5 scale steps can have a specific interval of 10, while a generic interval of 6 can have an interval of 9. This "overlap" of specific intervals in the distribution spectra indicates that this scale is improper.

You might think, what's the big deal here? The deal is that when we list the strictly proper and proper scales, they include all the diatonic modes, common scales like whole tone and the more typical pentatonics, consonant scales that are typically used in music. The Propriety of a scale is a good indicator of "sounds good", and yet it's a measurement that has no basis in the harmonic series, which most other theories rely on for the notion of consonance. Propriety also has no reliance on the tuning system being comprised of 12 equal semitones. Because of this interesting observation of interval distribution patterns, propriety can be applied to tuning systems of more than (or less than) 12 tones, to pick out scales that are likely to have meaningful potential for music-making.

## Maximal Area

Maximal Area is a property invented by David Rappaport8. He observed that along with the maximally even sets, there are popular scales that share a similar composition of intervals, but not in their most evenly spaced configuration. Rappaport observed that when tones of a scale are arranged around a circle, the interior area of a polygon with vertices at each tone describes a "score" that favours popular scales. Every scale with maximal evenness will also have maximal area, but not all scales with maximal area are maximally even.

Note that the interior area for a scale is identical for all transpositions and inversions of a scale, so it suffices to measure the area for prime scales only. While for each cardinality there will be only one prime set that has maximal evenness, there may be multiple prime sets that share the same maximal area. Here they are.

CardinalityInterior AreaSets
4 tones2 585 Diminished Seventh
5 tones2.299 597 Kung  661 Major Pentatonic
6 tones2.598 1365 Whole Tone
7 tones2.665 1367 Leading Whole-Tone Inverse  1371 Superlocrian  1387 Locrian
8 tones2.732 1375 Bothyllic  1391 Aeradyllic  1399 Syryllic  1455 Quartal Octamode  1463 Zaptyllic  1467 Spanish Phrygian  1495 Messiaen Mode 6 Rotation 2  1755 Octatonic
9 tones2.799 1407 Tharygic  1471 Radygic  1503 Padygic  1519 Locrian/Aeolian Mixed  1759 Pylygic  1775 Lyrygic  1911 Messiaen Mode 3 Rotation 1
10 tones2.866 1535 Mixodyllian  1791 Aerygyllian  1919 Rocryllian  1983 Soryllian  2015 Messiaen Mode 7 Rotation 4
11 tones2.933 2047 Chromatic Undecamode
12 tones3 4095 Chromatic

## Hemitonia and Tritonia

One interesting approach to understanding scales is to look at the distribution of consonant and dissonant intervals in the scale. We did this already in the "spectrum analysis" above, but two intervals in particular are commonly regarded in more depth: tritones and semitones - which are deemed the most dissonant of intervals, and those that give a scale its spice and flavour.

A scale that contains semitones is called "hemitonic", and we will also refer to a hemitonic as the scale member that has the upper semitone neighbour (for example, mi-fa and ti-do in a major scale, the hemitones are mi and fa).

A scale that contains tritones is called "tritonic", and we will also refer to a tritonic as the scale member that has a tritone above it. This concept seems very close to the notion of "imperfection" (explained below)

Since hemitonia and tritonia are based on the interval spectrum, all the modes of a scale will have the same hemitonia. For example, the major scale is dihemitonic and ancohemitonic; thus so are dorian, phrygian, lydian, etc.

Number of tones# of Hemitonic Scales# of Tritonic Scales
300
41224
5140150
6335335
7413413
8322322
9165165
105555
111111
1211

It's well to know that so many scales are hemitonic and tritonic (having more than zero semitones or tritones), but obviously some scales are more hemitonic than others. A scale could have no hemitonics ("ahemitonic"), one ("unihemitonic"), two ("dihemitonic"), three ("trihemitonic") and so on. Let's look at the count of hemitones in our scales.

Fun fact: there are no scales with 11 hemitones. Do you understand why?

### Cohemitonia

Cohemitonia is the presense of two semitones consecutively in scale order. For instance, if a scale has three cromatically consecutive notes, then it is cohemitonic. We will also refer to the cehemitonic as the tone that has two semitone neighbours above it.

## 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 mutations to turn one into the other.

This distance measured by mutation 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 mutation: insertion, deletion, and substitution. Our scale mutations are different from a string mutation, 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 mutate 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 mutations to every interval above the root.

### 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# 2743 Staptyllic lower the D to D♭ 2739 Mela Suryakanta raise the D to D# 2745 Mela Sulini delete the D 2737 Raga Hari Nata add a tone at D# 2749 Spanish Octamode 1st Rotation lower the E to E♭ 2733 Melodic Minor Ascending raise the E to F same as deleting E delete the E 2725 Raga Nagagandhari lower the F to E same as deleting F raise the F to F# 2773 Lydian delete the F 2709 Raga Kumud add a tone at F# 2805 Ichikotsuchô lower the G to G♭ 2677 Thodian raise the G to G# 2869 Major Augmented delete the G 2613 Raga Hamsa Vinodini add a tone at G# 2997 Major Bebop lower the A to A♭ 2485 Harmonic Major raise the A to A# 3253 Mela Naganandini delete the A 2229 Raga Nalinakanti add a tone at A# 3765 Dominant Bebop lower the B to B♭ 1717 Mixolydian raise the B to C same as deleting B delete the B 693 Arezzo Major Diatonic Hexachord

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.

## 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 Zeitler's9, 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 only 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?

## Negative

One peculiar way we can manipulate a scale is to "flip its bits" -- so that every bit that is on becomes off, and all that were of are turned on. If you flip a scale with a root tone, you will get a non-scale without a root tone; so it's not so useful to speak of negating a scale, instead we negate an entire modal family to find the modal family that is its negative.

For example, one that's easy to conceptualize is the major scale, which (in C) occupies all the white keys on a piano. The negative of the major scale is all the notes that aren't in the major scale - just the black keys, which interestingly have the pattern of a major pentatonic (with F# as the root). In pitch class set theory, the negative of a set is called its "complement", and Dr Forte named complementary pairs with matching numbers.

## Glossary

TET
Twelve-tone Equal Temperament. The system in which our octave is split into twelve equal intervals.
achiral
Not having chirality, i.e. the mirror image can be achieved by rotation.
ancohemitonic
A scale that is not cohemitonic. This either means it contain no semitones (and thus is anhemitonic), or contain semitones (being hemitonic) where none of the semitones appear consecutively in scale order.
anhemitonic
A scale that does not include any semitones
atritonic
Containing no tritones
balance
Having tones distributed such that if they were equal weights distributed on a spokes of a 12-spoke wheel, the wheel would balance on its centre.
cardinality
Fancy way of saying "the number of things" in a group or set. Cardinal numbers are numbers used for counting, in contrast to ordinal numbers for denoting sequence, or nominal numbers that names or identifies something. If a scale has seven tones, then its cardinality is seven.
chiral
The quality of being different from ones own mirror-image, in a way that can not be achieved by rotation.
cohemitonic
Cohemitonic scales contain two or more semitones (making them hemitonic) such that two or more of the semitones appear consecutively in scale order. Example: the Hungarian minor scale
coherence
An unabiguous relationship between specific intervals and generic intervals. Also known as propriety.
complement relation
Having all the tones that are absent from another set
dicohemitonic
A scale that contains exactly two semitones consecutively in scale order
dihemitonic
A scale that contains exactly two semitones
distribution spectra
The collection of the spectrum of distribution of specific intervals for each generic interval of a scale
enantiomorph
The result of a transformation by reflection, i.e. with its interval pattern reversed, but specifically in the case of chiral scales.
generic interval
The number of scale steps between two tones
heliotonic
A scale which can be rendered with one notehead on each line and space, using nothing more than single or double alterations
hemitonic
A scale that has tones separated by one semitone
heptatonic
A scale with seven tones. For example, the major scale is heptatonic.
imperfection
A scale member where the perfect fifth above it is not in the scale
interval of equivalence
The interval at which the pitch class is considered equivalent. In TET, the interval of equivalence is 12, aka an octave.
interval pattern
The sequence of semitones, tones, and larger intervals, that describe a scale. For example, a major scale is "T T S T T T S". Expressed numerically, a major scale has the interval pattern [2,2,1,2,2,2]; the final interval is implied.
interval spectrum
A signature invented by Howard Hanson, describing all the intervals that can be found in a sonority
mutation
The alteration of a scale by addition or removal of a tone, or by shifting a tone up or down by a semitone.
normal form
the most compact way to arrange of pitches in a set, without altering the set by transposition
octatonic
A scale with eight tones.
palindromic
A scale that has the same interval pattern forward and backward.
pentatonic
A scale with five tones
pitch class set
An unordered set of pitches, usually described in integer form.
prime form
The most exemplary form of a pitch class set, being the transformation that is most condensed and left-packed.
propriety
An unabiguous relationship between specific intervals and generic intervals. Also known as coherence.
proximity
The number of transformations required to change one scale into another
ridge tone
A pitch that appears in every scale built upon the scale degrees of itself.
root
The lowest tone of the scale, signifying the tone upon which all others are measured as an interval above
scale
A set of tones starting on a root, contained within one octave, having no more than a major third leap
sonority
The whole of a sound, comprised of all component tones
specific interval
The number of semitones between two tones
spectra variation
The average of the spectra widths with respect to the number of tones in a scale.
spectrum width
The difference between the lowest and highest specific intervals for a given generic interval.
subset relation
Consisting of tones that are all present within another set
symmetry
Having the ability to transform into itself by reflection or rotation
tone
A single entity having a pitch, as in one member of a scale
trihemitonic
A scale that contains exactly three semitones
tritonic
Containing one or more tritones
truncation
A scale produced by removing tones from another scale
unhemitonic
A scale that contains only one semitone
z-relation
The relation between two pitch class sets that have the same interval vector, but are not transpositions or inversions of each other.

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 MIDI playback by MIDI.js. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org).

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

## References

• [1] http://allthescales.org/
• [2] Allan Forte, The Structure of Atonal Music (1973, ISBN 0-300-02120-8).
• [3] "The Integer Model Of Pitch", Basic Atonal Theory, John Rahn p.35 ISBN 0-02-873-160-3
• [4] Howard Hansen, "Harmonic Materials Of Modern Music", ISBN 978-0891972075
• [5] Paul Nelson, "Pitch Class Sets" http://composertools.com/Theory/PCSets.pdf
• [6] http://andrewduncan.net/cmt/
• [7] Circular Distributions and Spectra Variations in Music. http://archive.bridgesmathart.org/2005/bridges2005-255.pdf
• [8] Rappoport, David. "Maximal Area Sets and Harmony". Graphs and Combinatorics, Vol 23, 2007-06-01.
• [9] Zeitler, William, table of imperfections counted in scales. https://allthescales.org/intro.html#Perfection