Noms des 464 intervalles historiques (463 sans l'unisson)
selon Manuel Op de Coul (programmeur du calculateur d'échelles « Scala »)

464 intervalles pour tourner autour des 12 intervalles de la gamme de 12 1/2 tons divisant l'8ve + commas et shismas ? Ce qui est étonnant à constater est que tous les théoriciens qui donnent leur nom à un intervalle, reste dans l'ensemble des nombres quotients (de l'ensemble Q), même pour Euler qui pourtant connaissait l'existence des nombres réels (de l'ensemble R) résultats des opérations puissance/racine et qui selon les archéologues était en usage par les Phéniciens du XIe siècle avant notre ère pour être enseigné aux enfants à l'école (ces opérations racines ont été découvertes tracées sur un ensemble de tablettes d'ardoise).

Dans cette liste de 464 intervalles (tournant autour des 12 intervalles de la gamme de 12 1/2 tons divisant l'8ve + commas et shismas), certains noms sont fantaisistes et certaines désignations sont fausses. Ce qui pousse à se demander : à quoi musicalement servirent ces intervalles ? À rien musicalement. Mais comme la musique était une science comme l'astronomie devait être enseignée et apprise par les étudiants, cette coutume donna un grand nombre d'ouvrages écrit par des non-musiciens jusqu'au XIXe siècle où la musique retourna dans son domaine de l'art et son phénomène vibratoire dans les laboratoires de physique acoustique. C'est dans la 2de moitié du XXe siècle que la réunification de la composition musicale et de la physique acoustique s'est produite, notamment grâce au physicien Émile Leipp à Paris dont son ouvrage « Acoustique et Musique » avec les cahiers du GAM [Groupe Acoustique et Musique] sont une mine de savoir pour le compositeur.

Aujourd'hui, la démarche est différente : au contraire d'attacher l'infinité des intervalles aux 12 intervalles de la gamme octaviante (= échelle octaviante irrégulière, sans être un mode), nous considérons chacun de ces 463 intervalles pour ce qu'ils sonnent. Scalairisés, ça leur donne de se faire entendre dans tout le champ de l'audibilité humaine. La scalairisation est l'enchantement de pouvoir entendre tous les intervalles différents pour eux-mêmes, non plus pour le mode qu'ils servent à constituer.

A : classement Shadow-Sky du + grand au + petit intervalle (sans unisson)

| intervalle en cents | fraction | résultat | nom français |

1467 | 7/3 | 2,33.. | dixième
1404 | 9/4 | 2,25 | neuvième majeure
1365 | 11/5 | 2.2 | neuvième neutre
1200 | 2/1 | 2 | octave

B : classement Op de Coul par odre de grandeur du numérateur :

15/7 septimal minor ninth, BP ninth
15/8 classic major seventh
15/11 undecimal augmented fourth
15/13 tridecimal 5/4-tone
15/14 major diatonic semitone

16/7 septimal major ninth
16/9 Pythagorean minor seventh
16/11 undecimal semi-diminished fifth
16/13 tridecimal neutral third
16/15 minor diatonic semitone

17/8 septendecimal minor ninth
17/9 septendecimal major seventh
17/10 septendecimal diminished seventh
17/12 2nd septendecimal tritone
17/14 supraminor third
17/15 septendecimal whole tone
17/16 17th harmonic PAS DE NOM

18/11 undecimal neutral sixth
18/13 tridecimal augmented fourth
18/17 Arabic lute index finger, 18th harmonic

19/10 undevicesimal major seventh
19/12 undevicesimal minor sixth
19/15 undevicesimal ditone
19/16 19th harmonic NO !!! but 19/18
19/17 quasi-meantone
19/18 undevicesimal semitone

20/9 small ninth
20/11 large minor seventh
20/13 tridecimal semi-augmented fifth
20/17 septendecimal augmented second
20/19 small undevicesimal semitone

21/11 undecimal major seventh
21/16 narrow fourth
21/17 submajor third
21/20 minor semitone

22/13 tridecimal major sixth
22/15 undecimal diminished fifth
22/21 undecimal minor semitone

23/12 vicesimotertial major seventh
23/16 23rd harmonic NO !!! but 23/22
23/18 vicesimotertial major third

24/13 tridecimal neutral seventh
24/17 1st septendecimal tritone
24/19 smaller undevicesimal major third
24/23 vicesimotertial minor semitone

25/9 classic augmented eleventh, BP twelfth
25/12 classic augmented octave
25/14 middle minor seventh
25/16 classic augmented fifth
25/18 classic augmented fourth
25/21 BP second, quasi-tempered minor third
25/22 undecimal acute whole tone
25/24 classic chromatic semitone, minor chroma

26/15 tridecimal semi-augmented sixth
26/25 tridecimal 1/3-tone

27/14 septimal major seventh
27/16 Pythagorean major sixth
27/17 septendecimal minor sixth
27/20 acute fourth
27/22 neutral third, Zalzal wosta of al-Farabi
27/23 vicesimotertial minor third
27/25 large limma, BP small semitone
27/26 tridecimal comma

28/15 grave major seventh
28/17 submajor sixth
28/25 middle second
28/27 Archytas' 1/3-tone 28th harmonic

29/16 29th harmonic NO !!!! not 29/16 = 1,8125, and 29/28 = 1,0357

30/17 septendecimal minor seventh
30/19 smaller undevicesimal minor sixth

31/16 31st harmonic NO !!!
31/30 31st-partial chroma = 31st harmonic

32/15 minor ninth
32/17 17th subharmonic subharmonic?
32/19 19th subharmonic subharmonic?
32/21 wide fifth
32/23 23rd subharmonic subharmonic?
32/25 classic diminished fourth
32/27 Pythagorean minor third
32/29 29th subharmonic subharmonic?
32/31 Greek enharmonic 1/4-tone

33/25 2 pentatones
33/26 tridecimal major third
33/28 undecimal minor third
33/32 undecimal comma, al-Farabi's 1/4-tone

34/21 supraminor sixth
34/27 septendecimal major third

35/18 septimal semi-diminished octave
35/24 septimal semi-diminished fifth
35/27 9/4-tone, septimal semi-diminished fourth
35/32 septimal neutral second
35/34 septendecimal 1/4-tone

36/19 smaller undevicesimal major seventh
36/25 classic diminished fifth
36/35 septimal diesis, 1/4-tone

37/32 37th harmonic NO !!!

39/32 39th harmonic, Zalzal wosta of Ibn Sina NO !!!

40/21 acute major seventh
40/27 grave fifth
40/39 tridecimal minor diesis

41/29 quasi-tempered tritone

42/25 quasi-tempered major sixth

44/25 undecimal grave minor seventh
44/27 neutral sixth

45/32 diatonic tritone
45/44 1/5-tone

46/45 23rd-partial chroma

48/25 classic diminished octave
48/35 septimal semi-augmented fourth

49/25 BP eighth
49/30 larger approximation to neutral sixth
49/36 Arabic lute acute fourth
49/40 larger approximation to neutral third
49/45 BP minor semitone
49/48 slendro diesis, septimal 1/6-tone

50/27 grave major seventh
50/33 3 pentatones
50/49 Erlich's decatonic comma, tritonic diesis

51/50 17th-partial chroma 51th-harmonic

52/33 tridecimal minor sixth

54/35 septimal semi-augmented fifth
54/49 Zalzal's mujannab

55/36 undecimal semi-augmented fifth
55/48 undecimal semi-augmented whole tone
55/49 quasi-equal major second

56/55 undecimal diesis

57/56 Hendrix comma

60/49 smaller approximation to neutral third

63/25 quasi-equal major tenth, BP eleventh
63/32 octave - septimal comma
63/34 submajor seventh
63/40 narrow minor sixth
63/50 quasi-equal major third

64/33 33rd subharmonic
64/35 septimal neutral seventh
64/37 37th subharmonic
64/39 39th subharmonic
64/45 2nd tritone
64/49 2 septatones or septatonic major third
64/63 septimal comma, Archytas' comma

65/64 13th-partial chroma

68/35 23/4-tone
68/63 supraminor second

72/49 Arabic lute grave fifth
72/55 undecimal semi-diminished fourth
72/59 Ibn Sina's neutral third

74/73 approximation to Pythagorean comma

75/49 BP fifth
75/64 classic augmented second

77/76 approximation to 53-tone comma

80/49 smaller approximation to neutral sixth
80/63 wide major third

81/44 2nd undecimal neutral seventh
81/50 acute minor sixth
81/64 Pythagorean major third
81/68 Persian wosta
81/70 Al-Hwarizmi's lute middle finger
81/80 syntonic comma, Didymus comma 81th-harmonic

88/81 2nd undecimal neutral second

89/84 quasi-equal semitone

91/59 15/4-tone
91/90 medium tridecimal comma

96/95 19th-partial chroma

98/55 quasi-equal minor seventh

99/70 2nd quasi-equal tritone
99/98 small undecimal comma

100/63 quasi-equal minor sixth
100/81 grave major third
100/99 Ptolemy's comma

105/64 septimal neutral sixth
105/104 small tridecimal comma

121/120 undecimal seconds comma

125/64 classic augmented seventh, octave - minor diesis
125/72 classic augmented sixth
125/96 classic augmented third
125/108 semi-augmented whole tone
125/112 classic augmented semitone

126/125 small septimal comma, Starling comma

128/75 diminished seventh
128/81 Pythagorean minor sixth
128/105 septimal neutral third
128/121 undecimal semitone
128/125 minor diesis, diesis

131/90 13/4-tone

135/128 major chroma, major limma

140/99 quasi-equal tritone

144/125 classic diminished third
144/143 Grossma

145/144 29th-partial chroma

153/125 7/4-tone
153/152 Ganassi's comma

160/81 octave - syntonic comma

161/93 19/4-tone

162/149 Persian neutral second

168/89 quasi-equal major seventh

169/168 Schulter's comma

176/175 valinorsma

192/125 classic diminished sixth

196/195 mynucuma

216/125 semi-augmented sixth

225/128 augmented sixth
225/224 septimal kleisma

231/200 5/4-tone

241/221 Meshaqah's 3/4-tone

243/125 octave - maximal diesis
243/128 Pythagorean major seventh
243/160 acute fifth
243/200 acute minor third
243/224 Archytas' 2/3-tone
243/242 neutral third comma

245/243 minor BP diesis, Sensamagic comma

246/239 Meshaqah's 1/4-tone

248/243 tricesoprimal comma

250/153 17/4-tone
250/243 maximal diesis

256/135 octave - major chroma
256/225 diminished third
256/243 limma, Pythagorean minor second
256/245 septimal minor semitone
256/255 septendecimal kleisma

261/256 vicesimononal comma

270/161 Kirnberger's sixth

272/243 Persian whole tone

273/256 Ibn Sina's minor second

275/273 Garibert comma

320/243 grave fourth

325/324 marveltwin

351/350 ratwolf comma

352/351 minthma

361/360 Dudon comma

364/363 gentle comma

375/256 double augmented fourth
375/343 BP major semitone, minor BP chroma

385/384 undecimal kleisma

400/243 grave major sixth

405/256 wide augmented fifth
405/392 greenwoodma

441/440 Werckmeister's undecimal septenarian schisma

512/343 3 septatones or septatonic fifth
512/375 double diminished fifth
512/405 narrow diminished fourth

513/512 undevicesimal comma, Boethius' comma

525/512 Avicenna enharmonic diesis

540/539 Swets' comma

625/324 octave - major diesis
625/567 BP great semitone, major BP chroma

640/637 huntma

648/625 major diesis

675/512 wide augmented third

676/675 island comma

686/675 senga

687/500 11/4-tone

715/714 septendecimal bridge comma

729/400 acute minor seventh
729/512 Pythagorean tritone
729/640 acute major second
729/704 undecimal major diesis
729/728 squbema

736/729 vicesimotertial comma

749/500 ancient Chinese quasi-equal fifth

750/749 ancient Chinese tempering

800/729 grave whole tone

847/845 Cuthbert comma

875/864 keema

896/891 undecimal semicomma

1001/1000 fairytale comma

1024/675 narrow diminished sixth
1024/729 Pythagorean diminished fifth

1029/1000 keega
1029/1024 gamelan residue

1053/1024 tridecimal major diesis

1125/1024 double augmented prime

1188/1183 kestrel comma

1215/1024 wide augmented second

1216/1215 Eratosthenes' comma

1280/729 grave minor seventh

1288/1287 triaphonisma

1344/1331 hemimin

1575/1573 Nicola

1728/1715 Orwell comma

1732/1731 approximation to 1 cent

1875/1024 double augmented sixth

2025/1024 2 tritones

2048/1125 double diminished octave
2048/1215 narrow diminished seventh
2048/1875 double diminished third
2048/2025 diaschisma

2058/2057 xenisma

2080/2079 ibnsinma

2187/1280 acute major sixth
2187/2000 Gorgo limma
2187/2048 apotome
2187/2176 septendecimal comma

2200/2197 Parizek comma

2401/2400 Breedsma

2430/2401 nuwell comma

2560/2187 grave minor third

3025/3024 Lehmerisma

3125/3072 small diesis, magic comma
3125/3087 major BP diesis

3136/3125 middle second comma

3375/2048 double augmented fifth

3645/2048 wide augmented sixth

4000/3969 septimal semicomma
4000/3993 undecimal schisma

4096/2187 Pythagorean diminished octave
4096/2401 4 septatones or septatonic major sixth
4096/3375 double diminished fourth
4096/3645 narrow diminished third
4096/4095 tridecimal schisma, Sagittal schismina

4225/4224 leprechaun comma

4375/4374 ragisma

4608/4235 Arabic neutral second

5120/5103 Beta 5, Garibaldi comma

5625/4096 double augmented third

6144/3125 octave - small diesis
6144/6125 porwell comma

6561/4096 Pythagorean augmented fifth
6561/5120 acute major third
6561/6125 BP major link
6561/6400 Mathieu superdiesis

6655/6561 Triple BP comma

8192/5625 double diminished sixth
8192/6561 Pythagorean diminished fourth
8192/8019 undecimal minor diesis

9801/9800 kalisma, Gauss' comma

10125/8192 double augmented second

10240/6561 grave minor sixth

10648/10647 harmonisma

10935/8192 fourth + schisma, 5-limit approximation to ET fourth

10976/10935 hemimage

10985/10976 cantonisma

15625/15309 great BP diesis
15625/15552 kleisma, semicomma majeur

16384/10125 double diminished seventh
16384/10935 fifth - schisma, 5-limit approximation to ET fifth

16875/16384 double augmentation diesis, Negri comma
16875/16807 small BP diesis, mirkwai comma

17496/16807 septimal major diesis

18225/16807 minimal BP chroma

19657/19656 greater harmonisma

19683/10000 octave - minimal diesis
19683/10240 acute major seventh
19683/16384 Pythagorean augmented second
19683/19600 cataharry comma

20000/19683 minimal diesis

20480/19683 grave minor second

21875/19683 maximal BP chroma

23232/23231 lesser harmonisma

28672/28431 Secorian

32768/16875 octave - double augmentation diesis

32768/19683 Pythagorean diminished seventh
32768/16807 5 septatones or septatonic diminished octave

32805/32768 schisma

33075/32768 mirwomo comma

43923/43904 hemigail

50421/50000 trimyna

52973/50000 Mersenne's quasi-equal semitone

59049/32768 Pythagorean augmented sixth
59049/57344 Harrison's comma

64827/64000 Squalentine

65536/32805 octave - schisma
65536/59049 Pythagorean diminished third
65536/65219 orgonisma

65625/65536 horwell comma

78125/73728 Woolhouse semitone

78732/78125 medium semicomma, Sensi comma

83349/78125 BP minor link

118098/117649 stearnsma

123201/123200 chalmersia

147456/78125 Woolhouse major seventh

151263/151250 odiheim

177147/131072 Pythagorean augmented third

179200/177147 tolerma

194481/194480 supraminor scintillisma

234375/234256 sesdecal

250047/250000 Landscape comma

262144/177147 Pythagorean diminished sixth

321489/320000 varunisma

390625/196608 octave - Würschmidt's comma
390625/388962 dimcomp comma

393216/390625 Würschmidt's comma

413343/390625 BP small link

420175/419904 wizma

531441/262144 Pythagorean augmented seventh
531441/524288 Pythagorean comma, ditonic comma

823543/819200 quince
823543/820125 complementary BP diesis

1048576/531441 Pythagorean diminished ninth

1594323/1048576 Pythagorean double augmented fourth
1594323/1562500 Unicorn comma

1600000/1594323 Amity comma, kleisma - schisma

1638400/1594323 Immunity comma

2097152/1594323 Pythagorean double diminished fifth

2109375/2097152 semicomma, Fokker's comma

4782969/4194304 Pythagorean double augmented prime

8388608/4782969 Pythagorean double diminished octave

9765625/9565938 Fifives comma

14348907/8388608 Pythagorean double augmented fifth

16777216/14348907 Pythagorean double diminished fourth

29360128/29296875 Freivald comma

33554432/33480783 Beta 2, septimal schisma

34171875/33554432 Ampersand's comma

43046721/33554432 Pythagorean double augmented second

48828125/47775744 Sycamore comma

67108864/43046721 Pythagorean double diminished seventh
67108864/66430125 Misty comma, diaschisma - schisma

129140163/67108864 Pythagorean double augmented sixth
129140163/128000000 gravity comma

131072000/129140163 roda

134217728/119574225 whole tone - 2 schismas, 5-limit approximation to ET whole tone
134217728/129140163 Pythagorean double diminished third

201768035/201326592 wadisma

387420489/268435456 Pythagorean double augmented third

536870912/387420489 Pythagorean double diminished sixth

854296875/843308032 Blackjack comma

1162261467/536870912 Pythagorean double augmented seventh
1162261467/1073741824 Pythagorean-19 comma

1220703125/1162261467 Trithagorean comma
1220703125/1207959552 ditonma

1224440064/1220703125 parakleisma

! 6115295232/6103515625 Vishnu comma

! 274877906944/274658203125 semithirds comma

! 7629394531250/7625597484987 ennealimmal comma

! 19073486328125/19042491875328 '19-tone' comma

! 450359962737049600/450283905890997363 monzisma

! 36893488147419103232/36472996377170786403 '41-tone' comma

! 19383245667680019896796723/19342813113834066795298816 Mercator's comma