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.

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 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. Then in 2017, in the preparation of this study, Ian Ring invented an even easier way to determine the Rahn prime using a much simpler algorithm. In Ian's algorithm, The Rahn Prime Form 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

395 

{0,1,3,7,8}

355 

{0,1,5,6,8}

811 

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

691 

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

843 

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

717 

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

919 

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

743 

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

815 

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

755 

{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
3-12 273 Augmented Triad 

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 Thaptic 
4-24 277 Mixolyric 
337 Koptic 
1093 Lydic 
1297 Aeolic 
4-27 293 Raga Haripriya 
593 Saric 
649 Byptic 
1097 Aeraphic 
329 Mynic 
553 Rothic 
581 Eporic 
1169 Raga Mahathi 
4-26 297 Mynic 
549 Raga Bhavani 
657 Epathic 
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 
369 Laditonic 
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 
453 Raditonic 
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 
2197 Raga Hamsadhvani 
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 (Kafi) 
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 
2341 Raga Priyadharshini 
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 Aeoloritonic 
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 
2609 Raga Bhinna Shadja 
5-28 333 Bogitonic 
837 Epaditonic 
1107 Mogitonic 
1233 Ionoditonic 
2601 Raga Chandrakauns 
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 Aeolanitonic 
5-32 595 Sogitonic 
665 Raga Mohanangi 
805 Rothitonic 
1225 Raga Samudhra Priya 
2345 Raga Chandrakauns 
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 
1109 Kataditonic 
1301 Koditonic 
1349 Tholitonic 
1361 Bolitonic 
5-34 597 Kung 
681 Kyemyonjo 
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 
497 Kadimic 
2191 Thydimic 
3143 Polimic 
3619 Thanimic 
3857 Ponimic 
6-Z40 303 Golimic 
753 Aeronimic 
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 
3155 Ladimic 
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 
567 Aeoladimic 
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 
2603 Gadimic 
3349 Aeolocrimic 
373 Epagimic 
1117 Raptimic 
1303 Epolimic 
1873 Dathimic 
2699 Sythimic 
3397 Sydimic 
6-Z17 407 Zylimic 
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 
933 Dadimic 
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 
2467 Raga Padi 
2617 Pylimic 
3281 Raga Vijayavasanta 
6-Z44 615 Phrothimic 
825 Thyptimic 
915 Raga Kalagada 
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 
2275 Messiaen Mode 5 
3185 Messiaen Mode 5 Inverse 
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 
1737 Raga Madhukauns 
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 Sothimic 
435 Raga Purna Pancama 
795 Aeologimic 
1635 Sygimic 
2265 Raga Rasamanjari 
2445 Zadimic 
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 
2757 Raga Nishadi 
429 Koptimic 
1131 Honchoshi Plagal Form 
1443 Raga Phenadyuti 
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 
1381 Padimic 
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 
1325 Phradimic 
1355 Raga Bhavani 
1705 Raga Manohari 
2725 Raga Nagagandhari 
6-31 691 Raga Kalavati 
811 Radimic 
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 Dominant Seventh 
2709 Raga Kumud 
6-30 715 Messiaen Truncated Mode 2 
1625 Lythimic 
2405 Katalimic 
845 Raga Neelangi 
1235 Messiaen Truncated Mode 2 
2665 Aeradimic 
6-Z50 723 Ionadimic 
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 
3985 Thadian 
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 
7-10 607 Kadian 
761 Ponian 
1993 Katoptian 
2351 Gynian 
3223 Thyphian 
3659 Polian 
3877 Thanian 
637 Debussy's Heptatonic 
1001 Badian 
1183 Sadian 
2639 Dothian 
3367 Moptian 
3731 Aeryrian 
3913 Bonian 
7-8 381 Kogian 
1119 Rarian 
2001 Gydian 
2607 Aerolian 
3351 Karian 
3723 Myptian 
3909 Rydian 
7-6 415 Aeoladian 
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 Ronian 
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 
2359 Gadian 
3017 Gacrian 
3227 Aeolocrian 
3661 Mixodorian 
635 Epolian 
985 Mela Sucaritra 
1615 Sydian 
2365 Sythian 
2855 Epocrain 
3475 Kylian 
3785 Epagian 
7-11 379 Aeragian 
1583 Salian 
1969 Stylian 
2237 Epothian 
2839 Lyptian 
3467 Katonian 
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 
3761 Raga Madhuri 
7-24 687 Aeolythian 
1401 Pagian 
1509 Ragian 
1941 Aeranian 
2391 Molian 
3243 Mela Rupavati 
3669 Mothian 
981 Mela Kantamani 
1269 Katythian 
1341 Madian 
1359 Aerygian 
2727 Mela Manavati 
3411 Enigmatic 
3753 Phraptian 
7-23 701 Mixonyphian 
1199 Magian 
1513 Stathian 
1957 Pyrian 
2647 Dadian 
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 
911 Radian 
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 
2267 Padian 
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 Zadian 
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 Socrian 
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 
1821 Aeradian 
2283 Aeolyptian 
2787 Zyrian 
3189 Aeolonian 
3441 Thacrian 
7-29 727 Phradian 
1483 Mela Bhavapriya 
1721 Mela Vagadhisvari 
1837 Dalian 
2411 Aeolorian 
2789 Zolian 
3253 Mela Naganandini 
941 Mela Jhankaradhvani 
1259 Stadian 
1447 Mela Ratnangi 
1693 Dogian 
2677 Thodian 
2771 Marva That 
3433 Thonian 
7-28 747 Lynian 
1431 Phragian 
1629 Synian 
1881 Katorian 
2421 Malian 
2763 Mela Suvarnangi 
3429 Marian 
861 Rylian 
1239 Epaptian 
1491 Mela Namanarayani 
1869 Katyrian 
2667 Byrian 
2793 Eporian 
3381 Katanian 
7-30 855 Porian 
1395 Asian (a) 
1485 Minor Romani 
1845 Lagian 
2475 Neapolitan Minor 
2745 Mela Sulini 
3285 Mela Citrambari 
939 Mela Senavati 
1383 Pynian 
1437 Sabach ascending 
1653 Minor Romani Inverse 
2517 Harmonic Lydian 
2739 Mela Suryakanta 
3417 Golian 
7-33 1367 Leading Whole-Tone Inverse 
1373 Storian 
1397 Major Locrian 
1493 Lydian Minor 
1877 Aeroptian 
2731 Neapolitan Major 
3413 Leading Whole-tone 
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 Locrian Double-flat 3 Double-flat 7 
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 Ionorian 
1627 Zyptian 
1739 Mela Sadvidhamargini 
1753 Hungarian Major 
2413 Locrian Natural 2 
2861 Katothian 
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 Makam Huzzam 
1645 Dorian Flat 5 
1715 Harmonic Minor Inverse 
2485 Harmonic Major 
2765 Lydian Diminished 
2905 Aeolian Flat 1 
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 
4003 Sadyllic 
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 
1277 Zadyllic 
1343 Zalyllic 
2719 Zocryllic 
3407 Katocryllic 
3751 Aerathyllic 
3923 Stoptyllic 
4009 Phranyllic 
8-10 765 
1215 
2025 
2655 
3375 
3735 
3915 
4005 
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 
3569 Aeoladyllic 
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 Aeolacryllic 
2415 Lothyllic 
3045 Raptyllic 
3255 Daryllic 
3675 Monyllic 
3885 Styryllic 
1005 Radyllic 
1275 Stagyllic 
1695 Phrodyllic 
2685 Ionoryllic 
2895 Aeoryllic 
3495 Banyllic 
3795 Epothyllic 
3945 Lydyllic 
8-12 763 Doryllic 
1631 Rynyllic 
2009 Stacryllic 
2429 Kadyllic 
2863 Aerogyllic 
3479 Rothyllic 
3787 Kagyllic 
3941 Stathyllic 
893 Dadyllic 
1247 Aeodyllic 
2003 Podyllic 
2671 Aerolyllic 
3049 Phrydyllic 
3383 Zoptyllic 
3739 Epanyllic 
3917 Katoptyllic 
8-Z15 863 Pyryllic 
1523 Zothyllic 
1997 Raga Cintamani 
2479 Harmonic and Neapolitan Minor Mixed 
2809 Gythyllic 
3287 Phrathyllic 
3691 Badyllic 
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 Gaptyllic 
999 Ionodyllic 
2511 Aeroptyllic 
2547 Raga Ramkali 
3303 Mylyllic 
3321 Epagyllic 
3699 Galyllic 
3897 Kalyllic 
8-6 495 Bocryllic 
1935 Mycryllic 
2295 Kogyllic 
3015 Laptyllic 
3195 Raryllic 
3555 Pylyllic 
3645 Zycryllic 
3825 Pynyllic 
8-Z29 751 
1913 
1943 
2423 
3019 
3259 
3557 
3677 
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 
2967 Madyllic 
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 Aeranyllic 
1779 Zynyllic 
1947 Byptyllic 
2487 Dothyllic 
2937 Phragyllic 
3021 Stodyllic 
3291 Lygyllic 
3693 Stadyllic 
987 Aeraptyllic 
1659 Maqam Shadd'araban 
1743 Epigyllic 
2541 Algerian 
2877 Phrylyllic 
2919 Molyllic 
3507 Maqam Hijaz 
3801 Maptyllic 
8-22 1391 Aeradyllic 
1469 Epiryllic 
1781 Gocryllic 
1963 Epocryllic 
2743 Staptyllic 
3029 Ionocryllic 
3419 Magen Abot 1 
3757 Raga Mian Ki Malhar 
1403 Espla's Scale 
1517 Sagyllic 
1711 Adonai Malakh 
1973 Zyryllic 
2749 Katagyllic 
2903 Gothyllic 
3499 Hamel 
3797 Rocryllic 
8-17 891 Ionilyllic 
1647 Polyllic 
1971 Aerynyllic 
2493 Manyllic 
2871 Stanyllic 
3033 Doptyllic 
3483 Mixotharyllic 
3789 Eporyllic 
8-16 943 Aerygyllic 
1511 Styptyllic 
1949 Mathyllic 
2519 Dathyllic 
2803 Raga Bhatiyar 
3307 Boptyllic 
3449 Bacryllic 
3701 Bagyllic 
983 Thocryllic 
1487 Mothyllic 
1853 Maryllic 
2539 Half-Diminished Bebop 
2791 Mixothyllic 
3317 Katynyllic 
3443 Verdi's Scala Enigmatica 
3769 Eponyllic 
8-23 1455 Phrygiolian 
1515 Phrygian/Locrian Mixed 
1725 Minor Bebop 
1965 Raga Mukhari 
2775 Godyllic 
2805 Ishikotsucho 
3435 Prokofiev 
3765 Dominant Bebop 
8-9 975 Messiaen Mode 4 
2535 Messiaen Mode 4 
3315 Tcherepnin Octatonic Mode 1 
3705 Messiaen Mode 4 Inverse 
8-19 887 Sathyllic 
1847 Thacryllic 
1907 Lynyllic 
2491 Layllic 
2971 Aeolynyllic 
3001 Lonyllic 
3293 Saryllic 
3533 Thadyllic 
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 
1757 
1771 
1883 
2779 Shostakovich 
2933 
2989 Bebop Minor 
3437 
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 Maqam Nahawand 
8-25 1495 Messiaen Mode 6 
1885 Saptyllic 
2795 Van der Horst Octatonic 
3445 Messiaen Mode 6 Inverse 
8-28 1755 Octatonic 
2925 Diminished 

Modal families with 9 tones

Forte Class NamePrime and its rotationsInverse and its rotations
9-1 511 Polygic 
2303 Stanygic 
3199 Thaptygic 
3647 Eporygic 
3871 Aerynygic 
3983 Thyptygic 
4039 Ionogygic 
4067 Aeolarygic 
4081 Manygic 
9-2 767 Raptygic 
2041 Aeolacrygic 
2431 Gythygic 
3263 Pyrygic 
3679 Rycrygic 
3887 Phrathygic 
3991 Badygic 
4043 Phrocrygic 
4069 Starygic 
1021 Ladygic 
1279 Sarygic 
2687 Thacrygic 
3391 Aeolynygic 
3743 Thadygic 
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 
2879 Stadygic 
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 
2527 Phradygic 
3059 Madygic 
3311 Mixodygic 
3577 Loptygic 
3703 Katalygic 
3899 Katorygic 
3997 Dogygic 
1015 Ionodygic 
1855 Gaptygic 
2555 Bythygic 
2975 Aeroptygic 
3325 Mixolygic 
3535 Mylygic 
3815 Galygic 
3955 Pothygic 
4025 Kalygic 
9-7 1471 Radygic 
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 
4021 Raga Pahadi 
9-5 991 Aeolygic 
1999 Zacrygic 
2543 Dydygic 
3047 Panygic 
3319 Tholygic 
3571 Dyrygic 
3707 Rynygic 
3833 Dycrygic 
3901 Bycrygic 
1007 Epitygic 
1951 Marygic 
2551 Thocrygic 
3023 Mothygic 
3323 Lacrygic 
3559 Thophygic 
3709 Katynygic 
3827 Bodygic 
3961 Zathygic 
9-8 1503 Padygic 
1917 Sacrygic 
2007 Stonygic 
2799 Epilygic 
3051 Stalygic 
3447 Kynygic 
3573 Kaptygic 
3771 Stophygic 
3933 Ionidygic 
1527 Aeolyrigic 
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 Scale 
1903 Rocrygic 
1979 Aeradygic 
2939 Goptygic 
2999 Chromatic and Permuted Diatonic Dorian Mixed 
3037 Nine Tone Scale 
3517 Epocrygic 
3547 Sadygic 
3821 Epyrygic 
9-12 1911 Messiaen Mode 3 
3003 Genus Chromaticum 
3549 Messiaen Mode 3 Inverse 

Modal families with 10 tones

Forte Class NamePrime and its rotationsInverse and its rotations
10-1 1023 Dodyllian 
2559 Zogyllian 
3327 Madyllian 
3711 Dycryllian 
3903 Aeogyllian 
3999 Dydyllian 
4047 Thogyllian 
4071 Rygyllian 
4083 Bathyllian 
4089 Katoryllian 
10-2 1535 Mixodyllian 
2045 Katogyllian 
2815 Aeradyllian 
3455 Ryptyllian 
3775 Loptyllian 
3935 Kataphyllian 
4015 Phradyllian 
4055 Dagyllian 
4075 Katyllian 
4085 Sydyllian 
10-3 1791 Aerygyllian 
2043 Maqam Tarzanuyn 
2943 Dathyllian 
3069 Maqam Shawq Afza 
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 
2031 Gadyllian 
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 
3055 Messiaen Mode 7 
3575 Symmetrical Decatonic 
3835 Katodyllian 
3965 Messiaen Mode 7 Inverse 

Modal families with 11 tones

Forte Class NamePrime and its rotationsInverse and its rotations
11-1 2047 Monatic 
3071 Solatic 
3583 Zylatic 
3839 Mixolatic 
3967 Soratic 
4031 Godatic 
4063 Eptatic 
4079 Ionatic 
4087 Aeolatic 
4091 Thydatic 
4093 Aerycratic 

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

585 
4,8 major thirds

273 

819 
6 tritones

325 

455 

715 

845 

975 

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:


585 

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 1755  that preserve that symmetry:

Technically, all of Messiaen's modes are truncated forms of 4095 , 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

585 
M2, T5, T4
T2 M3
T3

273 
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:


273 

337 

433 

497 

585 

681 

745 

793 

857 

953 

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 1105  and 325  (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
273 Augmented Triad achiral
585 Diminished Seventh achiral
661 Major Pentatonic achiral
859 Ultralocrian 2905 Aeolian Flat 1 
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 
1621 Scriabin's Prometheus 1357 Takemitsu Linea Mode 2 
1643 Locrian Natural 6 2765 Lydian Diminished 
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 Aeoloritonic 
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 Makam Huzzam 
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 Inverse achiral
3549 Messiaen Mode 3 Inverse 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:


273 

585 

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:


273 

325 

403 

455 

585 

611 

715 

793 

819 

845 

871 

923 

975 

715 

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 585  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, 281  and 275  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)
273 Augmented Triad m3000300
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 Inverse p4m6n4s6d4t4464644
3549 Messiaen Mode 3 Inverse 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:


693 

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:


273 

585 

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.

3 tones


273 

4 tones


585 

325 

293 

297 

277 

291 

275 

5 tones


661 

597 

595 

341 

587 

403 

339 

333 

355 

331 

307 

327 

309 

301 

299 

283 

313 

295 

279 

6 tones


819 

715 

723 

717 

693 

691 

685 

683 

427 

679 

667 

619 

455 

663 

615 

423 

363 

603 

411 

347 

359 

365 

407 

605 

599 

371 

591 

349 

399 

343 

335 

315 

311 

303 

317 

287 

7 tones


859 

871 

855 

731 

823 

747 

743 

727 

699 

733 

719 

695 

471 

443 

439 

631 

701 

755 

463 

687 

375 

431 

671 

623 

607 

415 

379 

367 

381 

351 

319 

8 tones


975 

951 

943 

887 

879 

891 

927 

759 

751 

863 

495 

735 

831 

763 

765 

703 

479 

639 

447 

383 

9 tones


991 

959 

895 

767 

511 

10 tones

11 tones

12 tones

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.


341 

511 

661 

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:


273 

291 

293 

297 

325 

585 

661 

819 

293 

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 span the edge of two different generic intervals, but there mustn't be any overlap (as we'll see below). Stated another way, in a proper scale, there should never be a generic 4th that is smaller than a generic 3rd. 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:


275 

277 

339 

341 

403 

587 

595 

597 

683 

685 

691 

693 

715 

717 

723 

859 

275 

595 

339 

587 

715 

683 

859 

Lastly, there are all the other scales that aren't proper at all. 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
3 tones1.299 273 Augmented Triad 
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 Phrygiolian  1463  1467 Spanish Phrygian  1495 Messiaen Mode 6  1755 Octatonic 
9 tones2.799 1407 Tharygic  1471 Radygic  1503 Padygic  1519 Locrian/Aeolian Mixed  1759 Pylygic  1775 Lyrygic  1911 Messiaen Mode 3 
10 tones2.866 1535 Mixodyllian  1791 Aerygyllian  1919 Rocryllian  1983 Soryllian  2015 Messiaen Mode 7 
11 tones2.933 2047 Monatic 
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.

Number of hemitones found in all scales
Number of hemitones
tones in scale 0123456789101112
31000000000000
4191200000000000
51580600000000000
61301501401500000000
70021140210420000000
800007016884000000
9000000847290000
10000000004510000
1100000000001100
120000000000001

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

Number of tritones found in all scales
Number of tritones
tones in scale 0123456789101112
31000000000000
471680000000000
55407530500000000
61121021466960000000
70014112196847000000
800006216884800000
9000000847290000
10000000004510000
1100000000001100
120000000000001

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
  • Add 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 2741 , 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 D2737 Raga Hari Nata 
add a tone at D#2749 Katagyllic 
lower the E to E♭2733 Melodic Minor Ascending 
raise the E to Fsame as deleting E
delete the E2725 Raga Nagagandhari 
lower the F to Esame as deleting F
raise the F to F#2773 Lydian 
delete the F2709 Raga Kumud 
add a tone at F#2805 Ishikotsucho 
lower the G to G♭2677 Thodian 
raise the G to G#2869 Major Augmented 
delete the G2613 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 A2229 Raga Nalinakanti 
add a tone at A#3765 Dominant Bebop 
lower the B to B♭1717 Mixolydian 
raise the B to Csame 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. The software used to generate this analysis is an open source project at GitHub. 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