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! decimal binary 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.

2741 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 1 0 2 0 3 1 4 31 5 155 6 336 7 413 8 322 9 165 10 55 11 11 12 1

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. 1 01010110101 = 2741 - major scale, "ionian" mode 1 1 0101011010 = 3418 - rotated down 1 semitone - not a scale 01 1 010101101 = 1709 - rotated down 2 semitones - "dorian" 101 1 01010110 = 2902 - rotated down 3 semitones - not a scale 0101 1 0101011 = 1451 - rotated down 4 semitones - "phrygian" 10101 1 010101 = 2773 - rotated down 5 semitones - "lydian" 110101 1 01010 = 3434 - rotated down 6 semitones - not a scale 0110101 1 0101 = 1717 - rotated down 7 semitones - "mixolydian" 10110101 1 010 = 2906 - rotated down 8 semitones - not a scale 010110101 1 01 = 1453 - rotated down 9 semitones - "aeolian" 1010110101 1 0 = 2774 - rotated down 10 semitones - not a scale 01010110101 1 = 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. Forte Rahn

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}

1719

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

1467

{0,1,3,4,5,7,8,10} A complete list of all modal families Modal families with 3 tones Forte Class Name Prime and its rotations Inverse and its rotations 3-12 273 Augmented Triad

Modal families with 4 tones Forte Class Name Prime and its rotations Inverse 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 Koptic

1093 Lydic

1297 Aeolic

4-27 293 Raga Haripriya

593 Saric

649 Byptic

1097 Aeraphic

329 Mynic 2

553 Rothic 2

581 Eporic 2

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 Name Prime and its rotations Inverse 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 Name Prime and its rotations Inverse 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 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

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 Schoenberg Hexachord

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 Name Prime and its rotations Inverse 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 Crater Scale

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 Namanarayani

1869 Katyrian

2667 Byrian

2793 Eporian

3381 Katanian

7-30 855 Porian

1395 Locrian Dominant

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 Alternating Heptamode

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 Name Prime and its rotations Inverse 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 Erkian

1215 Hibian

2025 Mivian

2655 Qojian

3375 Vecian

3735 Xupian

3915 Yuyian

4005 Zibian

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 Sideways Scale

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 Epoian

1913 Lofian

1943 Luxian

2423 Otuian

3019 Subian

3259 Ulian

3557 Wekian

3677 Xafian

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 Quartal Octamode

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 Ugrian

1757 Kunian

1771 Kuwian

1883 Lomian

2779 Shostakovich

2933 Sizian

2989 Bebop Minor

3437 Vopian

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 Name Prime and its rotations Inverse 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

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 Diminishing Nonamode

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 Name Prime and its rotations Inverse 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

2815 Aeradyllian

3455 Ryptyllian

3775 Loptyllian

3935 Kataphyllian

4015 Phradyllian

4055 Dagyllian

4075 Katyllian

4085 Rechberger's Decamode

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 Name Prime and its rotations Inverse 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 Name Prime and its rotations Inverse 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 symmetry interval of repetition scales 1,2,3,4,5,6,7,8,9,10,11 semitone

4095 2,4,6,8,10 whole tone

1365 3,6,9 minor thirds

585

1755

2925 4,8 major thirds

273

819

1911

2457

3003

3549 6 tritones

325

455

715

845

975

1105

1235

1495

1625

1885

2015

2275

2405

2535

2665

2795

3055

3185

3315

3445

3575

3705

3835

3965

number of notes in scale Placement of rotational symmetries 1 2 3 4 5 6 7 8 9 10 11 3 0 0 0 1 0 0 0 1 0 0 0 4 0 0 1 0 0 3 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 6 0 1 0 3 0 10 0 3 0 1 0 7 0 0 0 0 0 0 0 0 0 0 0 8 0 0 2 0 0 10 0 0 2 0 0 9 0 0 0 3 0 0 0 3 0 0 0 10 0 0 0 0 0 5 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 12 1 1 1 1 1 1 1 1 1 1 1 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:

3055

2015

3575

3835

3965 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:

1755

2925 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:

1625

715

2405

2665

845

1235 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 Family Scales is truncation of *

4095 Messiaen's Modes of Limited Transposition M1

1365 *, M3, M6 M2

1755

2925 *, M7 M3

3549

3003

1911 * M4

975

2535

3315

3705 M7 M5

455

2275

3185 M6, M4 M6

1885

1495

2795

3445 M7 M7

3055

2015

3575

3835

3965 * Truncations T1

585 M2, T5, T4 T2

2457

819 M3 T3

273 M1, T2 T4

1625

715

2405 M2, M6, M4 T5

2665

845

1235 M2, M6, M4 T6

325

1105 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

1017

1093

1189

1253

1301

1365

1461

1525

1613

1709

1773

1821

1885

1981

2045

2211

2275

2323

2387

2483

2547

2635

2731

2795

2843

2907

3003

3067

3143

3239

3303

3351

3415

3511

3575

3663

3759

3823

3871

3935

4031

4095 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) Scale Chirality / 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 Sizian 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

1365

1885

2275

2795

3003

3575

4095

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

1105

1235

1365

1495

1625

1651

1755

1885

1911

2015

2249

2275

2353

2405

2457

2483

2509

2535

715

2795

2873

2925

3003

3055

3185

3289

3315

3445

3549

3575

3705

3835

3965

4095

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 Scale Spectrum (Hanson) Vector (modern) 273 Augmented Triad m3 000300 585 Diminished Seventh n4t2 004002 661 Major Pentatonic p4mn2s3 032140 859 Ultralocrian p4m4n5s3d3t2 335442 1193 Minor Pentatonic p4mn2s3 032140 1257 Blues Scale p4m2n3s3d2t 233241 1365 Whole Tone m6s6t3 060603 1371 Superlocrian p4m4n4s5d2t2 254442 1387 Locrian p6m3n4s5d2t 254361 1389 Minor Locrian p4m4n4s5d2t2 254442 1397 Major Locrian p2m6n2s6d2t3 262623 1451 Phrygian p6m3n4s5d2t 254361 1453 Aeolian p6m3n4s5d2t 254361 1459 Phrygian Dominant p4m4n5s3d3t2 335442 1485 Minor Romani p4m5n3s4d3t2 343542 1493 Lydian Minor p2m6n2s6d2t3 262623 1499 Bebop Locrian p5m5n6s5d4t3 456553 1621 Scriabin's Prometheus p2m4n2s4dt2 142422 1643 Locrian Natural 6 p4m4n5s3d3t2 335442 1709 Dorian p6m3n4s5d2t 254361 1717 Mixolydian p6m3n4s5d2t 254361 1725 Minor Bebop p7m4n5s6d4t2 465472 1741 Lydian Diminished p4m4n5s3d3t2 335442 1749 Acoustic p4m4n4s5d2t2 254442 1753 Hungarian Major p3m3n6s3d3t3 336333 1755 Octatonic p4m4n8s4d4t4 448444 2257 Lydian Pentatonic p3m2nsd2t 211231 2275 Messiaen Mode 5 p4m2s2d4t3 420243 2457 Augmented p3m6n3d3 303630 2475 Neapolitan Minor p4m5n3s4d3t2 343542 2477 Harmonic Minor p4m4n5s3d3t2 335442 2483 Double Harmonic p4m5n4s2d4t2 424542 2509 Double Harmonic Minor p4m5n4s2d4t2 424542 2535 Messiaen Mode 4 p6m4n4s4d6t4 644464 2731 Neapolitan Major p2m6n2s6d2t3 262623 2733 Melodic Minor Ascending p4m4n4s5d2t2 254442 2741 Major p6m3n4s5d2t 254361 2773 Lydian p6m3n4s5d2t 254361 2777 Aeolian Harmonic p4m4n5s3d3t2 335442 2869 Major Augmented p4m4n5s3d3t2 335442 2901 Lydian Augmented p4m4n4s5d2t2 254442 2925 Diminished p4m4n8s4d4t4 448444 2989 Bebop Minor p5m5n6s5d4t3 456553 2997 Major Bebop p6m5n6s5d4t2 456562 3055 Messiaen Mode 7 p8m8n8s8d8t5 888885 3411 Enigmatic p4m4n3s5d3t2 353442 3445 Messiaen Mode 6 Inverse p4m6n4s6d4t4 464644 3549 Messiaen Mode 3 Inverse p6m9n6s6d6t3 666963 3669 Mothian p4m4n3s5d3t2 353442 3765 Dominant Bebop p7m4n5s6d4t2 465472 4095 Chromatic p12m12n12s12d12t6 12121212126 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

1387

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 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: Scale Notes Distribution Spectra

1449 <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

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

1365

4095 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

1365

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

1387

1371

1367

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

1755

1495

1467

1463

1455

1399

975

951

943

1391

887

1375

879

891

927

759

751

863

495

735

831

763

765

703

479

639

447

383 9 tones

1911

1775

1519

1759

1503

1471

991

959

1407

895

767

511 10 tones

2015

1983

1919

1791

1535

1023 11 tones

2047 12 tones

4095 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

1023

1387

2047 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

1365

1755

4095 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:

67

69

73

163

165

275

277

339

341

403

587

595

597

683

685

691

693

715

717

723

859

1367

1371

1387

1463

1467

1495

1775

1911

1983

2015

2047 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: Scale Notes Distribution Spectra

1449 <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. Scale Notes Distribution Spectra

2475 <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. Cardinality Interior Area Sets 3 tones 1.299 273 Augmented Triad 4 tones 2 585 Diminished Seventh 5 tones 2.299 597 Kung 661 Major Pentatonic 6 tones 2.598 1365 Whole Tone 7 tones 2.665 1367 Leading Whole-Tone Inverse 1371 Superlocrian 1387 Locrian 8 tones 2.732 1375 Bothyllic 1391 Aeradyllic 1399 Syryllic 1455 Quartal Octamode 1463 Ugrian 1467 Spanish Phrygian 1495 Messiaen Mode 6 1755 Octatonic 9 tones 2.799 1407 Tharygic 1471 Radygic 1503 Padygic 1519 Locrian/Aeolian Mixed 1759 Pylygic 1775 Lyrygic 1911 Messiaen Mode 3 10 tones 2.866 1535 Mixodyllian 1791 Aerygyllian 1919 Rocryllian 1983 Soryllian 2015 Messiaen Mode 7 11 tones 2.933 2047 Chromatic Undecamode 12 tones 3 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 3 0 0 4 12 24 5 140 150 6 335 335 7 413 413 8 322 322 9 165 165 10 55 55 11 11 11 12 1 1 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 0 1 2 3 4 5 6 7 8 9 10 11 12 3 1 0 0 0 0 0 0 0 0 0 0 0 0 4 19 12 0 0 0 0 0 0 0 0 0 0 0 5 15 80 60 0 0 0 0 0 0 0 0 0 0 6 1 30 150 140 15 0 0 0 0 0 0 0 0 7 0 0 21 140 210 42 0 0 0 0 0 0 0 8 0 0 0 0 70 168 84 0 0 0 0 0 0 9 0 0 0 0 0 0 84 72 9 0 0 0 0 10 0 0 0 0 0 0 0 0 45 10 0 0 0 11 0 0 0 0 0 0 0 0 0 0 11 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 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 0 1 2 3 4 5 6 7 8 9 10 11 12 3 1 0 0 0 0 0 0 0 0 0 0 0 0 4 7 16 8 0 0 0 0 0 0 0 0 0 0 5 5 40 75 30 5 0 0 0 0 0 0 0 0 6 1 12 102 146 69 6 0 0 0 0 0 0 0 7 0 0 14 112 196 84 7 0 0 0 0 0 0 8 0 0 0 0 62 168 84 8 0 0 0 0 0 9 0 0 0 0 0 0 84 72 9 0 0 0 0 10 0 0 0 0 0 0 0 0 45 10 0 0 0 11 0 0 0 0 0 0 0 0 0 0 11 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 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 D 2737 Raga Hari Nata add a tone at D# 2749 Katagyllic 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 Ishikotsucho 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 0 1 2 3 4 5 6 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 4 0 0 8 16 7 0 0 5 0 5 30 75 40 5 0 6 0 6 69 146 102 12 1 7 0 7 84 196 112 14 0 8 0 8 84 168 62 0 0 9 0 9 72 84 0 0 0 10 0 10 45 0 0 0 0 11 0 11 0 0 0 0 0 12 1 0 0 0 0 0 0 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