All Scales

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
scale #4095 scale #4095

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
scale #4094scale #4094 
000000
scale #2048scale #2048 
6 Decatonic Scales 6 Duotonic Scales

988884
scale #4062scale #4062 
100000
scale #2112scale #2112 

898884
scale #4090scale #4090 
010000
scale #2560scale #2560 

889884
scale #4086scale #4086 
001000
scale #2304scale #2304 

888984
scale #4078scale #4078 
000100
scale #2176scale #2176 

888894
scale #4092scale #4092 
000010
scale #3072scale #3072 

888885
scale #4030scale #4030 
000001
scale #2080scale #2080 
19 Nonatonic Scales 19 Tritonic Scales

876663
scale #3962scale #3962 
210000
scale #2576scale #2576 

777663
scale #4054scale #4054scale #4058scale #4058 
111000
scale #2624scale #2624scale #2564scale #2564 

767763
scale #3950scale #3950scale #3958scale #3958 
101100
scale #2320scale #2320scale #2312scale #2312 

766773
scale #4046scale #4046scale #4060scale #4060 
100110
scale #3136scale #3136scale #3080scale #3080 

766674
scale #3998scale #3998scale #4028scale #4028 
100011
scale #3104scale #3104scale #3088scale #3088 

686763
scale #4074scale #4074 
020100
scale #2688scale #2688 

677673
scale #3263scale #3263scale #4084scale #4084 
011010
scale #3328scale #3328scale #3074scale #3074 

676764
scale #4014scale #4014scale #4026scale #4026 
010101
scale #2592scale #2592scale #2568scale #2568 

676683
scale #4088scale #4088 
010020
scale #3584scale #3584 

668664
scale #4022scale #4022 
002001
scale #2336scale #2336 

667773
scale #3295scale #3295scale #4076scale #4076 
001110
scale #3200scale #3200scale #3076scale #3076 

666963
scale #3822scale #3822 
000300
scale #2184scale #2184 
43 Octatonic Scales 43 Tetratonic Scales

765442
scale #3930scale #3930 
321000
scale #2640scale #2640 

665542
scale #3926scale #3926scale #3946scale #3946 
221100
scale #2704scale #2704scale #2692scale #2692 

656542
scale #3798scale #3798 
212100
scale #2628scale #2628 

655552
scale #3918scale #3918scale #3954scale #3954 
211110
scale #3138scale #3138scale #3336scale #3336 

654553
scale #3898scale #3898scale #3932scale #3932 
210111
scale #3152scale #3152scale #3112scale #3112 

654463
scale #3960scale #3960 
210021
scale #3600scale #3600 

645652
scale #3804scale #3804 
201210
scale #3144scale #3144 

644563
scale #3996scale #3996 
200121
scale #3168scale #3168 

644464
scale #3900scale #3900 
200022
scale #3120scale #3120 

566452
scale #4050scale #4050 
122010
scale #3330scale #3330 

565552
scale #4042scale #4042scale #4052scale #4052 
121110
scale #3392scale #3392scale #3082scale #3082 

556543
scale #3766scale #3766scale #3802scale #3802 
112101
scale #2632scale #2632scale #2596scale #2596 

556453
scale #3990scale #3990scale #4020scale #4020 
112011
scale #3360scale #3360scale #3090scale #3090 

555562
scale #4038scale #4038scale #4056scale #4056 
111120
scale #3648scale #3648scale #3588scale #3588 

555553
scale #3886scale #3886scale #3956scale #3956scale #3994scale #3994scale #4012scale #4012
111111
scale #3344scale #3344scale #3106scale #3106scale #3232scale #3232scale #3092scale #3092

554563
scale #3982scale #3982scale #4024scale #4024 
110121
scale #3616scale #3616scale #3592scale #3592 

546652
scale #3942scale #3942 
102210
scale #3204scale #3204 

546553
scale #3894scale #3894scale #3948scale #3948 
102111
scale #3216scale #3216scale #3108scale #3108 

545752
scale #3814scale #3814scale #3820scale #3820 
101310
scale #3208scale #3208scale #3140scale #3140 

545662
scale #4044scale #4044 
101220
scale #3264scale #3264 

474643
scale #4010scale #4010 
030201
scale #2720scale #2720 

465562
scale #4066scale #4066scale #4072scale #4072 
021120
scale #3712scale #3712scale #3586scale #3586 

465472
scale #4080scale #4080 
021030
scale #3840scale #3840 

464743
scale #3818scale #3818 
020301
scale #2696scale #2696 

464644
scale #3770scale #3770 
020202
scale #2600scale #2600 

456562
scale #4068scale #4068 
012120
scale #3456scale #3456 

456553
scale #4006scale #4006scale #4018scale #4018 
012111
scale #3202scale #3202scale #3332scale #3332 

448444
scale #3510scale #3510 
004002
scale #2340scale #2340 
66 Heptatonic Scales 66 Pentatonic Scales

654321
scale #3434scale #3434 
432100
scale #2708scale #2708 

554331
scale #3914scale #3914scale #3922scale #3922 
332110
scale #3394scale #3394scale #3338scale #3338 

544431
scale #3670scale #3670scale #3796scale #3796 
322210
scale #3400scale #3400scale #3146scale #3146 

544332
scale #3674scale #3674scale #3764scale #3764 
322111
scale #3368scale #3368scale #3154scale #3154 

543342
scale #3866scale #3866scale #3928scale #3928 
321121
scale #3664scale #3664scale #3604scale #3604 

533442
scale #3698scale #3698scale #3740scale #3740 
311221
scale #3352scale #3352scale #3170scale #3170 

532353
scale #3868scale #3868scale #3896scale #3896 
310132
scale #3632scale #3632scale #3608scale #3608 

454422
scale #3498scale #3498 
232201
scale #2724scale #2724 

453432
scale #3882scale #3882scale #3924scale #3924 
231211
scale #3408scale #3408scale #3114scale #3114 

445332
scale #3734scale #3734scale #3794scale #3794 
223111
scale #3362scale #3362scale #3346scale #3346 

444441
scale #3910scale #3910scale #3938scale #3938 
222220
scale #3650scale #3650scale #3716scale #3716 

444342
scale #3988scale #3988scale #3862scale #3862scale #3944scale #3944 
222121
scale #3424scale #3424scale #3728scale #3728scale #3602scale #3602 

443532
scale #3690scale #3690scale #3756scale #3756 
221311
scale #3240scale #3240scale #3156scale #3156 

443352
scale #3854scale #3854scale #3952scale #3952 
221131
scale #3856scale #3856scale #3848scale #3848 

442443
scale #3768scale #3768 
220222
scale #3624scale #3624 

435432
scale #3482scale #3482scale #3500scale #3500 
213211
scale #3236scale #3236scale #3220scale #3220 

434541
scale #3782scale #3782scale #3812scale #3812 
212320
scale #3652scale #3652scale #3464scale #3464 

434442
scale #3638scale #3638scale #3800scale #3800scale #3890scale #3890scale #3916scale #3916
212221
scale #3656scale #3656scale #3620scale #3620scale #3340scale #3340scale #3266scale #3266

434343
scale #3884scale #3884scale #3892scale #3892 
212122
scale #3376scale #3376scale #3248scale #3248 

433452
scale #3980scale #3980scale #3992scale #3992 
211231
scale #3680scale #3680scale #3596scale #3596 

424641
scale #3686scale #3686scale #3788scale #3788 
202420
scale #3272scale #3272scale #3268scale #3268 

424542
scale #3692scale #3692 
202321
scale #3224scale #3224 

354351
scale #4034scale #4034scale #4048scale #4048 
132130
scale #3904scale #3904scale #3842scale #3842 

353442
scale #3978scale #3978scale #4008scale #4008 
131221
scale #3744scale #3744scale #3594scale #3594 

345342
scale #3986scale #3986scale #4004scale #4004 
123121
scale #3488scale #3488scale #3458scale #3458 

344532
scale #3750scale #3750scale #3786scale #3786 
122311
scale #3210scale #3210scale #3396scale #3396 

344451
scale #4036scale #4036scale #4040scale #4040 
122230
scale #3776scale #3776scale #3590scale #3590 

344433
scale #3738scale #3738scale #3762scale #3762 
122212
scale #3234scale #3234scale #3348scale #3348 

344352
scale #3974scale #3974scale #4016scale #4016 
122131
scale #3872scale #3872scale #3844scale #3844 

343542
scale #3810scale #3810scale #3816scale #3816 
121321
scale #3720scale #3720scale #3618scale #3618 

336333
scale #3506scale #3506scale #3508scale #3508 
114112
scale #3364scale #3364scale #3218scale #3218 

335442
scale #3878scale #3878scale #3940scale #3940 
113221
scale #3472scale #3472scale #3460scale #3460 

262623
scale #3754scale #3754 
040402
scale #2728scale #2728 

254442
scale #4002scale #4002 
032221
scale #3714scale #3714 

254361
scale #4064scale #4064 
032140
scale #3968scale #3968 
80 Hexatonic Scales

543210
scale #3402scale #3402 

443211
scale #3370scale #3370scale #3410scale #3410 

433221
scale #3426scale #3426scale #3432scale #3432scale #3666scale #3666scale #3732scale #3732

432321
scale #3416scale #3416scale #3668scale #3668 

422232
scale #3636scale #3636scale #3672scale #3672 

421242
scale #3696scale #3696scale #3864scale #3864 

420243
scale #3640scale #3640 

343230
scale #3906scale #3906 

342231
scale #3850scale #3850scale #3920scale #3920 

333321
scale #3466scale #3466scale #3496scale #3496scale #3658scale #3658scale #3748scale #3748

333231
scale #3606scale #3606scale #3792scale #3792scale #3858scale #3858scale #3912scale #3912

332232
scale #3610scale #3610scale #3760scale #3760scale #3860scale #3860scale #3880scale #3880

324222
scale #3372scale #3372scale #3378scale #3378 

323430
scale #3654scale #3654scale #3780scale #3780 

323421
scale #3274scale #3274scale #3404scale #3404 

322431
scale #3682scale #3682scale #3724scale #3724 

322332
scale #3628scale #3628scale #3634scale #3634scale #3688scale #3688scale #3736scale #3736

322242
scale #3852scale #3852scale #3888scale #3888 

313431
scale #3468scale #3468scale #3480scale #3480scale #3660scale #3660scale #3684scale #3684

303630
scale #3276scale #3276 

242412
scale #3242scale #3242scale #3412scale #3412 

241422
scale #3626scale #3626scale #3752scale #3752 

234222
scale #3490scale #3490scale #3730scale #3730 

233331
scale #3718scale #3718scale #3778scale #3778scale #3874scale #3874scale #3908scale #3908

233241
scale #3846scale #3846scale #3936scale #3936scale #3972scale #3972scale #3984scale #3984

232341
scale #3808scale #3808scale #3976scale #3976 

225222
scale #3474scale #3474scale #3492scale #3492 

224322
scale #3428scale #3428scale #3476scale #3476 

224232
scale #3504scale #3504scale #3876scale #3876 

224223
scale #3250scale #3250scale #3380scale #3380 

223431
scale #3622scale #3622scale #3784scale #3784 

143250
scale #4032scale #4032 

143241
scale #3970scale #3970scale #4000scale #4000 

142422
scale #3722scale #3722scale #3746scale #3746 

060603
scale #2730scale #2730 

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.

example dual tritonic from nonatonic

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).

example nonatone scale on tonic note C showing interval transitions

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.

the six components of the consonancy in six bars

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).

No comments:

Post a Comment