This page shows, as polygons, all 351 basic scales of varying orders 1 to 12. Explanations and guides to the diagrams and captions begin after the graphics. Click on a polygon to see a more musically familiar representation, and click again to return to the polygonal view.
First up is the single, unique, dodecaphonic with all its symmetries.
1 dodecatonic Scale | |
CCCCC6 |
For scales of orders 11, 10, 9, 8 and 7 (which is to say for those scales comprising 11, 10, 9, 8 and 7 distinct tones) each subset is shown in (vertical) parallel with its corresponding 'dual' or 'complement' scales of order 1, 2, 3, 4 and 5 respectively.
1 Undecatonic Scales | 1 Unatonic Scales | ||||||||
AAAAA5 |
000000 |
6 Decatonic Scales | 6 Duotonic Scales | ||||||||
988884 |
100000 |
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898884 |
010000 |
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889884 |
001000 |
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888984 |
000100 |
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888894 |
000010 |
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888885 |
000001 |
19 Nonatonic Scales | 19 Tritonic Scales | ||||||||
876663 |
210000 |
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777663 |
111000 |
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767763 |
101100 |
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766773 |
100110 |
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766674 |
100011 |
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686763 |
020100 |
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677673 |
011010 |
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676764 |
010101 |
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676683 |
010020 |
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668664 |
002001 |
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667773 |
001110 |
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666963 |
000300 |
43 Octatonic Scales | 43 Tetratonic Scales | ||||||||
765442 |
321000 |
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665542 |
221100 |
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656542 |
212100 |
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655552 |
211110 |
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654553 |
210111 |
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654463 |
210021 |
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645652 |
201210 |
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644563 |
200121 |
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644464 |
200022 |
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566452 |
122010 |
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565552 |
121110 |
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556543 |
112101 |
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556453 |
112011 |
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555562 |
111120 |
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555553 |
111111 |
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554563 |
110121 |
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546652 |
102210 |
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546553 |
102111 |
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545752 |
101310 |
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545662 |
101220 |
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474643 |
030201 |
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465562 |
021120 |
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465472 |
021030 |
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464743 |
020301 |
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464644 |
020202 |
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456562 |
012120 |
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456553 |
012111 |
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448444 |
004002 |
66 Heptatonic Scales | 66 Pentatonic Scales | ||||||||
654321 |
432100 |
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554331 |
332110 |
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544431 |
322210 |
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544332 |
322111 |
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543342 |
321121 |
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533442 |
311221 |
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532353 |
310132 |
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454422 |
232201 |
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453432 |
231211 |
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445332 |
223111 |
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444441 |
222220 |
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444342 |
222121 |
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443532 |
221311 |
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443352 |
221131 |
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442443 |
220222 |
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435432 |
213211 |
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434541 |
212320 |
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434442 |
212221 |
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434343 |
212122 |
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433452 |
211231 |
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424641 |
202420 |
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424542 |
202321 |
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354351 |
132130 |
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353442 |
131221 |
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345342 |
123121 |
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344532 |
122311 |
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344451 |
122230 |
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344433 |
122212 |
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344352 |
122131 |
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343542 |
121321 |
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336333 |
114112 |
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335442 |
113221 |
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262623 |
040402 |
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254442 |
032221 |
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254361 |
032140 |
80 Hexatonic Scales | ||||
543210 |
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443211 |
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433221 |
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432321 |
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422232 |
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421242 |
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420243 |
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343230 |
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342231 |
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333321 |
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333231 |
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332232 |
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324222 |
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323430 |
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323421 |
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322431 |
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322332 |
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322242 |
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313431 |
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303630 |
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242412 |
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241422 |
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234222 |
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233331 |
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233241 |
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232341 |
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225222 |
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224322 |
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224232 |
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224223 |
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223431 |
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143250 |
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143241 |
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142422 |
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060603 |
Duality/Complementarity
An N-note scale (or chord), where N < 12, obviously omits 12 - N of the 12 possible pitch classes. But the omitted notes are just another pitch class set and we say that one is the complement (which seems to be the preferred term in musicological circles) or the dual (the preferred term in geometric circles) of the other. This means for example that scales with 9 notes are complemented, one for one, with corresponding scales of 3 notes.
However, in representing these correspondences we have a slight problem. Our polygonal representation of a scale demands that the polygon always has its zero node (the centre-top vertex at 12 o'clock) set to represent the 'always on' tonic note of the scale (or root note of the chord). Clearly an 'omitted note' used to form a complementary scale could never be in this position. Consequently we will always need to rotate a complementary polygon until the 12 o'clock position is occupied. Although easily done, such a rotation does somewhat obscure the visual connection between the pairs.
Thus for example for the 19 possible scales of order 9, each row shows either a (symmetric/invertible) scale polygon or a reflected-pair of (unsymmetric/uninvertible) nonatonic scales on the left. Corresponding tritonic scales on the right are constructed by forming a scale from the notes absent from the one in the left hand side, rotated so that the new scale polygon has its tonic note at 12 o'clock. This applies also to 11/1, 10/2, 8/4 and 7/5 note scales.
For the 80 distinct hexatonic scales which contain 6 of the 12 possible tones, dual (or complementary) scales formed from the other 6 notes absent from the original scale are of course also hexatonic scales and there is thus no need for parallel columns.
Modality and Consonance
The 351 polygons are grouped in 200 separate rows in these tables; you may wish to refer to the enumerations for explanations. They break down as 1 dodecatonic row, then dual pairs of 1 undecatonic/unatonic row (each), 6 decatonic/duotonic rows, 12 nonatonic/tritonic rows, 28 octatonic/tetratonic rows, and 35 heptatonic/pentatonic rows. This accounts for the first 165 rows - 1 dodecatonic plus two parallel (left and right) sets of 82 rows (1 + 6 + 12 + 28 + 35). The final 35 are in the separate rows of the self-dual hexatonic scales.
Labelling
The first nonatonic scale shown comprises the tonic note, then three consecutive semitones, a whole tone, three more semitones, then two whole tones taking you to the tonic (an octave above). For example if the tonic is C, the nine note scale would be C, C#, D, D#, F, F#, G, G#, A# and the last tone jump (the final 2 in the label) taking you to C again. These labels will always add up to 12 (where, when required, A counts as 10, B as 11 and C as 12).
Consonancies
Each of the 200 rows is accompanied by, on the extreme left of the row, its unique six column 'rearranged interval vector' (shown both as graph and label) which conveys the numbers of occurrences of - from left to right - interval classes 5 (perfect 4ths and perfect 5ths), 2 (major 2nds and minor 7ths), 3 (minor 3rds and major 6ths), 4 (major 3rds and minor 6ths), 1 (minor 2nds and major 7ths) and 6 (tritones).
This ordering is - we claim - 'natural' because our ears and brains (contrariness or cussedness notwithstanding) are most easily 'pleased' by exposure to interval class 5, known as the dominant (or - in the other direction - subdominant). Class 3, the major 2nd - also known as the supertonic - is the 5th of the 5th, or the dominant of the dominant. Likewise the major 6th (aka the submediant) is the dominant of the dominant of the dominant. And the major 3rd, aka the mediant, is the dominant of the dominant of the dominant of the dominant - you get the picture, it's the cycle of fifths. We refer to this labelling as the Consonancy, a simple permutation (by swapping the first and fifth numeric components) of Forte's interval vector (note that this is, itself, a reduced autocorrelation).
The next illustration shows all of the intervals present in the nonatonic scale above, with consonancy 876666. There are six bars - one for each of the digits in the vector.
For example, the first bar (annotated '8 of IV/V') shows the eight occasions where the interval of a perfect fourth (i.e. interval class 5) is found between pairs of scale notes, and the third bar (annotated '6 of iii/VI') shows the six occasions where the interval of a minor third (i.e. interval class 3) turns up. The bottom nine eighth notes, or quavers, in each bar are the nine notes of the scale in sequence. There is a rest when the relevant interval up from that scale note is missing from the scale. For example, there is a rest in the third place of bar two because the third note of the scale (D) would require an E for its relative major second, but as there's no E in this scale it counts as a deficit in the major second component.
Reflections and Rotations
All scales are shown in their principal minimal interval form, which is to say that particular interval sequence which sorts to the first place, alphabetically. This is principally to keep our polyginal images fully consistent with the musicological Normal Form used to represent pitch class sets.
Consider the western diatonic major scale comprising seven notes. There are seven ways its heptagon can be drawn onto a clock face, such that the 12 o'clock position is always occupied. One of them may be elected as a 'standard' and then each of the remaining six possible orientations is a simple rotation (clockwise or anticlockwise, it makes no difference) from this standard. There are 7 possible ways of writing the interval steps in a standard western diatonic scale (its seven modes), are the Ionian (2212221), Dorian (2122212), Phrygian (1222122), Lydian (2221221), Mixolydian (2212212), Aeolian (2122122) and Locrian (1221222). And of these it is the Locrian (ironically the least favoured mode) which - alphabetically with its 1221222 - sorts first and thus gets the honour of representing all tonal/diatonic/heptatonic/mainstream music. The Ionian (the major) sorts into sixth place (almost at the back of the queue) and the Aeolian (minor) into third.
Blue polygons are symmetric and scale-invertable, whereas pink polygons - always in reflecting (but rotated into the aforementioned principal minimal interval form) pairs - are the unsymmetric scales (where inversions of musical phrases invert out of the original scale and into the reflected one). However, reflections of these unsymmetric polygons will also need to be rotated to place them in their (musicologically prime) conventional orientations. This means, unfortunately, that adjacent pink pairs of unsymmetric polygons will not be obvious reflections of each other (in, say, the 12 o'clock - 6 o'clock axis, a 'horizontal left-right flip'). But they are.
For those rows where four scales show up (only in scales of orders 4 to 8) one of the pairs of (mutually reflecting) polygons is one of Allen Forte's alternate Z-forms (which he uses to designate scales or chords with exactly the same interval vector as the other pair).
The global order of all rows in the tables is from scales with the highest to the lowest consonancy (that reduced rearrangement of autocorrelation sequenced as the 'cycle of fifths' described above). This guarantees - in particular - that the standard western diatonic scale will be in the first row of the heptatonic scales alongside its complementary pentatonic (the standard 'oriental' pentatonic) in the same row to the right. Highly consonant scales are those scales which carry the most perfect fourths and fifths (interval class 5) and least minor seconds and tritones (interval classes 1 and 6). Consonancies aren't vectors, they're just labels constructed from things which resemble vectors (but which, also, aren't actual vectors). They can certainly be ordered because they are six character, alphabetically orderable, strings.
Note that the octatonic/tetratonic and heptatonic/pentatonic lists are rendered at half the size of the others since we may need up to 8 'clocks' per line (plus a pair of consonance graphs). All images are, however, sourced at the same size (272x256 pixels).
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