GKM 2009 Diastemata Intervals Psaltiki ALL 001

PSALTIKIMonophoniesFRACTION CENTS

Fractions and Equal TemperDIATONIC SCALES Epitropi FRACTIONS DiatonicNi Pa Bou Ga Di Ke Zo Ni 1 8/9 81/100 3/4 2/3 16/27 27/50 1/2 0.000 203.910 364.807 498.045 701.955 905.865 1,066.762 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 1 1/8 1 71/729 1 2/25 1 1/8 1 1/8 1 71/729 1 2/25 203.910 160.897 133.238 203.910 203.910 160.897 133.238

Epitropi 72 ET DiatonicNi Pa Bou Ga Di Ke Zo Ni 0 12 22 30 42 54 64 72 72 200.000 366.667 500.000 700.000 900.000 1,066.667 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 10 8 12 12 10 8 200.000 166.667 133.333 200.000 200.000 166.667 133.333

Didymos FRACTIONS DiatonicNi Pa Bou Ga Di Ke Zo Ni 1 8/9 4/5 3/4 2/3 16/27 8/15 1/2 0.000 203.910 386.314 498.045 701.955 905.865 1,088.269 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 1 1 1 1 1 1 1 1/8 1/9 1/15 1/8 1/8 1/9 1/15 203.910 182.404 111.731 203.910 203.910 182.404 111.731

Al-FarabiNi Pa Bou Ga 1 8/9 22/27 3/4 0.000 203.910 354.547 498.045 Ni-Pa 1 Pa-Bou 1 Bou-Ga 1 1/8 1/11 7/81 203.910 150.637 143.498

Di Ke Zo Ni

2/3 11/18 9/16 1/2

701.955 852.592 996.090 1,200.000

Ga-Di Di-Ke Ke-Zo Zo-Ni

1 1 1 1

1/8 1/11 7/81 1/8

203.910 150.637 143.498 203.910

Chrysanthos FRACTIONS diatonic (from Ni)Ni Pa Bou Ga Di Ke Zo Ni 1 8/9 22/27 3/4 2/3 16/27 44/81 1/2 0.000 203.910 354.547 498.045 701.955 905.865 1,056.502 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 1 1 1 1 1 1 1 1/8 1/11 7/81 1/8 1/8 1/11 7/81 203.910 150.637 143.498 203.910 203.910 150.637 143.498

Chrysanthos FRACTIONS diatonic (from Di)Di Ke Zo Ni Pa Bou Ga Di 1 8/9 22/27 3/4 2/3 11/18 9/16 1/2 0.000 203.910 354.547 498.045 701.955 852.592 996.090 1,200.000 Di-Ke Ke-Zo Zo-Ni Ni-Pa Pa-Bou Bou-Ga Ga-Di 1 1 1 1 1 1 1 1/8 1/11 7/81 1/8 1/11 7/81 1/8 203.910 150.637 143.498 203.910 150.637 143.498 203.910

Chrysanthos 68 ET DiatonicNi Pa Bou Ga Di Ke Zo Ni 0 12 21 28 40 52 61 68 68 211.765 370.588 494.118 705.882 917.647 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 9 7 12 12 9 7 211.765 158.824 123.529 211.765 211.765 158.824 123.529

Chrysanthos 66 ET DiatonicNi Pa Bou Ga Di Ke Zo Ni 0 12 20 27 39 51 59 66 66 218.182 363.636 490.909 709.091 927.273 1,072.727 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 8 7 12 12 8 7 218.182 145.455 127.273 218.182 218.182 145.455 127.273

Pangratios 24 ET DiatonicNi Pa Bou 0 4 7 24 200.000 350.000 Ni-Pa Pa-Bou 4 3 200.000 150.000

Ga Di Ke Zo Ni

10 14 18 21 24

500.000 700.000 900.000 1,050.000 1,200.000

Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni

3 4 4 3 3

150.000 200.000 200.000 150.000 150.000

CHROMATIC NEANES SCALES Euthemiades-Deberlis FRACTIONS chromatic neanesNi Pa Bou Ga Di Ke Zo Ni 1 15/16 4/5 3/4 2/3 5/8 8/15 1/2 0.000 111.731 386.314 (15/16)*(64/75) 498.045 [(15/16)^2]*(64/75) 701.955 813.686 1,088.269 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 1 1/15 1 11/64 1 1/15 1 1/8 1 1/15 1 11/64 1 1/15 111.731 274.582 111.731 203.910 111.731 274.582 111.731

Papadimitriou 2008 FRACTIONS chromatic neanesNi Pa Bou Ga Di Ke Zo Ni 1 25/27 4/5 3/4 2/3 50/81 8/15 1/2 0.000 133.238 386.314 (25/27)*(108/125) 498.045 (25/27)*(108/125)*(15/16) 701.955 835.193 1,088.269 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 1 2/25 1 17/108 1 1/15 1 1/8 1 2/25 1 17/108 1 1/15 133.238 253.076 111.731 203.910 133.238 253.076 111.731

Epitropi 72 ET Chromatic NeanesNi Pa Bou Ga Di Ke Zo Ni 0 8 22 30 42 50 64 72 72 133.333 366.667 500.000 700.000 833.333 1,066.667 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 8 14 8 12 8 14 8 133.333 233.333 133.333 200.000 133.333 233.333 133.333

Chrysanthos 64 ET Chromatic NeanesNi Pa Bou Ga Di Ke Zo 0 7 19 26 38 45 57 64 131.250 356.250 487.500 712.500 843.750 1,068.750 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo 7 12 7 12 7 12 131.250 225.000 131.250 225.000 131.250 225.000

Ni

64

1,200.000

Zo-Ni

7

131.250

Chrysanthos-Phokaefs 68 ET Chromatic NeanesNi Pa Bou Ga Di Ke Zo Ni 0 7 21 28 40 47 61 68 68 123.529 370.588 494.118 705.882 829.412 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 7 14 7 12 7 14 7 123.529 247.059 123.529 211.765 123.529 247.059 123.529

Chrysanthos UNCORRECTED ET Chromatic NeanesNi Pa Bou Ga Di Ke Zo Ni 0 7 19 26 38 45 59 66 68 123.529 335.294 458.824 670.588 794.118 1,041.176 1,164.706 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 7 12 7 12 7 14 7 123.529 211.765 123.529 211.765 123.529 247.059 123.529

CHROMATIC NECHEANES SCALES Euthemiades-Deberlis FRACTIONS chromatic neanesNi Pa Bou Ga Di Ke Zo Ni 1 15/16 4/5 3/4 2/3 5/8 8/15 1/2 0.000 111.731 386.314 (15/16)*(64/75) 498.045 [(15/16)^2]*(64/75) 701.955 813.686 1,088.269 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 1 1/15 1 11/64 1 1/15 1 1/8 1 1/15 1 11/64 1 1/15 111.731 274.582 111.731 203.910 111.731 274.582 111.731

Epitropi 1881 FRACTIONS chromatic necheanesPa Bou Ga Di Ke Zo Ni Pa 1 243/256 25/32 3/4 2/3 81/128 25/48 1/2 0.000 90.225 Pa-Bou 427.373 (243/256)*(200/243) Bou-Ga 498.045 (243/256)*(200/243)*(24/25) Ga-Di 701.955 Di-Ke 792.180 Ke-Zo 1,129.328 Zo-Ni 1,200.000 Ni-Pa 1 1 1 1 1 1 1 13/243 43/200 1/24 1/8 13/243 43/200 1/24 90.225 337.148 70.672 203.910 90.225 337.148 70.672

Epitropi 72 ET Chromatic Necheanes72 Pa Bou Ga Di Ke Zo Ni Pa 0 6 26 30 42 48 68 72 0.000 100.000 433.333 500.000 700.000 800.000 1,133.333 1,200.000 Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni Ni-Pa 6 20 4 12 6 20 4 100.000 333.333 66.667 200.000 100.000 333.333 66.667

Chrysanthos 68 ET Chromatic Necheanes68 Pa Bou Ga Di Ke Zo Ni Pa 0 7 25 28 40 47 65 68 0.000 123.529 441.176 494.118 705.882 829.412 1,147.059 1,200.000 Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni Ni-Pa 7 18 3 12 7 18 3 123.529 317.647 52.941 211.765 123.529 317.647 52.941

Pangratios 24 ET Chromatic24 Pa Bou Ga Di Ke Zo Ni Pa 0 2 8 10 14 16 22 24 0.000 100.000 400.000 500.000 700.000 800.000 1,100.000 1,200.000 Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni Ni-Pa 2 6 2 4 2 6 2 100.000 300.000 100.000 200.000 100.000 300.000 100.000

ENHARMONIC SCALES Epitropi 1881 FRACTIONS EnharmonicGa Di Ke Zo Ni 1 8/9 64/81 3/4 2/3 0.000 203.910 407.820 498.045 701.955 Ga-Di Di-Ke Ke-Zo Zo-Ni 1 1/8 1 1/8 1 13/243 1 1/8 203.910 203.910 90.225 203.910

Pa Bou Ga

16/27 128/243 1/2

905.865 1,109.775 1,200.000

Ni-Pa 1 1/8 Pa-Bou 1 1/8 Bou-Ga 1 13/243

203.910 203.910 90.225

Epitropi 1881 72 Enharmonic72 Ga Di Ke Zo Ni Pa Bou Ga 0 12 24 30 42 54 66 72 0.000 200.000 400.000 500.000 700.000 900.000 1,100.000 1,200.000 Ga-Di Di-Ke Ke-Zo Zo-Ni Ni-Pa 12 12 6 12 12 200.000 200.000 100.000 200.000 200.000

Chrysanthos 68 Enharmonic68 Ga Di Ke Zo Ni Pa Bou Ga 0 12 24 27 40 52 55 68 0.000 211.765 423.529 476.471 705.882 917.647 970.588 1,200.000 Ga-Di Di-Ke Ke-Zo Zo-Ni Ni-Pa 12 12 3 13 12 211.765 211.765 52.941 229.412 211.765

Chroa SPATHI SCALES Epitropi 72 ET SpatheionNi Pa Bou Ga Di Ke Zo Ni 0 12 26 30 34 54 64 72 72 200.000 433.333 500.000 566.667 900.000 1,066.667 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 14 4 4 20 10 8 200.000 233.333 66.667 66.667 333.333 166.667 133.333

Chrysanthos 68 ET Spatheion (GKM approx.)Ni Pa Bou Ga Di Ke Zo Ni 0 12 25 28 34 52 61 68 68 211.765 441.176 494.118 600.000 917.647 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 13 3 6 18 9 7 211.765 229.412 52.941 105.882 317.647 158.824 123.529

Chroa Zygos SCALES Epitropi 72 ET ZygosNi Pa Bou Ga Di Ke Zo Ni 0 18 22 38 42 54 64 72 72 300.000 366.667 633.333 700.000 900.000 1,066.667 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 18 4 16 4 12 10 8 300.000 66.667 266.667 66.667 200.000 166.667 133.333

Chrysanthos 68 ET Zygos (GKM visual approx., pg 115 et 170)Ni Pa Bou Ga Di Ke Zo Ni 0 14 21 35 40 52 61 68 68 247.059 370.588 617.647 705.882 917.647 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 14 7 14 5 12 9 7 247.059 123.529 247.059 88.235 211.765 158.824 123.529

Chroa Kliton SCALES Epitropi 72 ET KlitonNi Pa Bou Ga Di Ke Zo Ni 0 18 26 38 42 54 64 72 72 300.000 433.333 633.333 700.000 900.000 1,066.667 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 18 8 12 4 12 10 8 300.000 133.333 200.000 66.667 200.000 166.667 133.333

Chrysanthos 68 ET Kliton (GKM visual approx., pg 115 et 170)Ni Pa Bou Ga Di Ke Zo 0 12 25 36 40 52 61 68 211.765 441.176 635.294 705.882 917.647 1,076.471 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo 12 13 11 4 12 9 211.765 229.412 194.118 70.588 211.765 158.824

Ni

68

1,200.000

Zo-Ni

7

123.529

Gregorios 68 ET KlitonNi Pa Bou Ga Di Ke Zo Ni 0 12 26 37 40 52 61 68 68 211.765 458.824 652.941 705.882 917.647 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 14 11 3 12 9 7 211.765 247.059 194.118 52.941 211.765 158.824 123.529

Chrysanthos 68 ET Kliton (Giannopoulos)Ni Pa Bou Ga Di Ke Zo Ni 0 12 24 36 40 52 61 68 68 211.765 423.529 635.294 705.882 917.647 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 12 12 4 12 9 7 211.765 211.765 211.765 70.588 211.765 158.824 123.529

SALTIKI SCALESDiphoniesFRACTION CENTS

l Tempered units (including CENTS)TriphoniesFRACTION CENTS

TetraphoniesFRACTION

(Ni-Pa = 9/8, Pa-Bou = 800/729, Bou-Ga = 27/25, Ni-G(Ni-Bou)d (Pa-Ga)d (Bou-Di)d (Ga-Ke)d diphonic Bou-Di 4th diphonic Bou-Di 5th 1 19/81 1 5/27 1 43/200 1 17/64 1 175/477 1 10/21 364.807 294.135 337.148 407.820 541.058 674.295 (Ni-Ga)d (Pa-Di)d (Bou-Ke)d (Ga-Zo)d 1 1/3 1 1/3 1 175/477 1 7/18 498.045 498.045 541.058 568.717 (Ni-Di)d 1 (Pa-Ke)d 1 (Bou-Zo)d 1 1/2 1/2 1/2

(Ni-Pa = 12, Pa-Bou = 10, Bou-Ga = 8) / 72(Ni-Bou)d (Pa-Ga)d (Bou-Di)d (Ga-Ke)d diphonic Bou-Di 4th diphonic Bou-Di 5th 366.667 300.000 333.333 400.000 533.333 666.667 (Ni-Ga)d (Pa-Di)d (Bou-Ke)d (Ga-Zo)d 500.000 500.000 533.333 566.667 (Ni-Di)d (Pa-Ke)d (Bou-Zo)d

(Ni-Pa = 9/8, Pa-Bou = 10/9, Bou-Ga = 16/15, Ni-Ga=4/3)(Ni-Bou)d (Pa-Ga)d (Bou-Di)d (Ga-Ke)d diphonic Bou-Di 4th diphonic Bou-Di 5th 1 1/4 1 5/27 1 1/5 1 17/64 1 7/20 1 11/25 386.314 294.135 315.641 407.820 519.551 631.283 (Ni-Ga)d (Pa-Di)d (Bou-Ke)d (Ga-Zo)d 1 1/3 1 1/3 1 7/20 1 13/32 498.045 498.045 519.551 590.224 (Ni-Di)d 1 (Pa-Ke)d 1 (Bou-Zo)d 1 1/2 1/2 1/2

(Ni-Pa = 9/8, Pa-Bou = 12/11, Bou-Ga = 88/81, Ni-Ga=4/3)(Ni-Bou)d (Pa-Ga)d (Bou-Di)d 1 1 1 5/22 5/27 2/9 354.547 294.135 347.408 (Ni-Ga)d 1 (Pa-Di)d 1 (Bou-Ke)d 1 1/3 1/3 1/3 498.045 498.045 498.045 (Ni-Di)d 1 1/2 (Pa-Ke)d 1 5/11 (Bou-Zo)d 1 109/243

(Ga-Ke)d diphonic Bou-Di 4th diphonic Bou-Di 5th

1

5/22

354.547 551.318 694.816

(Ga-Zo)d 1

1/3

498.045

1 3/8 1 40/81

(Ni-Pa = 9/8, Pa-Bou = 12/11, Bou-Ga = 88/81, Ni-Ga=4/3)(Ni-Bou)d (Pa-Ga)d (Bou-Di)d (Ga-Ke)d diphonic Bou-Di 4th diphonic Bou-Di 5th 1 5/22 1 5/27 1 2/9 1 17/64 1 3/8 1 40/81 354.547 294.135 347.408 407.820 551.318 694.816 (Ni-Ga)d (Pa-Di)d (Bou-Ke)d (Ga-Zo)d 1 1/3 1 1/3 1 3/8 1 67/176 498.045 498.045 551.318 558.457 (Ni-Di)d 1 (Pa-Ke)d 1 (Bou-Zo)d 1 1/2 1/2 1/2

(Di-Ke = 9/8, Ke-Zo = 12/11, Zo-Ni = 88/81, Di-Ni=4/3)(Ni-Bou)d (Pa-Ga)d (Bou-Di)d 1 1 1 5/22 5/27 2/9 354.547 294.135 347.408 (Ni-Ga)d 1 (Pa-Di)d 1 1/3 1/3 498.045 498.045 (Ni-Di)d 1 1/2

diphonic Bou-Di 4th diphonic Bou-Di 5th

1 3/8 1 40/81

551.318 694.816



(Ni-Pa = 4, Pa-Bou = 3, Bou-Ga = 3) / 24(Ni-Bou)d (Pa-Ga)d 350.000 300.000 (Ni-Ga)d (Pa-Di)d 500.000 500.000 (Ni-Di)d (Pa-Ke)d

(Bou-Di)d (Ga-Ke)d diphonic Bou-Di 4th diphonic Bou-Di 5th

350.000 400.000 550.000 700.000

(Bou-Ke)d (Ga-Zo)d

550.000 550.000

(Bou-Zo)d

(Ni-Pa = 16/15, Pa-Bou = 75/64, Bou-Ga = 16/15, Ni-Ga=4/3(Ni-Bou)cM (Pa-Ga)cM (Bou-Di)cM (Ga-Ke)cM diphonic Bou-Di 4th diphonic Bou-Di 5th 1 1 1 1 1/4 1/4 1/5 1/5 386.314 (Ni-Ga)cM 1 1/3 386.314 (Pa-Di)cM 1 13/32 315.641 (Bou-Ke)cM 1 7/25 315.641 (Ga-Zo)cM 1 13/32 427.373 631.283 498.045 (Ni-Di)cM 1 590.224 (Pa-Ke)cM 1 427.373 (Bou-Zo)cM 1 590.224 1/2 1/2 1/2

1 7/25 1 11/25

(Ni-Pa = 27/25, Pa-Bou = 125/108, Bou-Ga = 16/15, Ga-Di =(Ni-Bou)cM (Pa-Ga)cM (Bou-Di)cM (Ga-Ke)cM diphonic Bou-Di 4th diphonic Bou-Di 5th 1 1/4 1 19/81 1 1/5 1 43/200 1 37/125 1 11/25 386.314 (Ni-Ga)cM 1 1/3 364.807 (Pa-Di)cM 1 7/18 315.641 (Bou-Ke)cM 1 37/125 337.148 (Ga-Zo)cM 1 13/32 448.879 631.283 498.045 (Ni-Di)cM 1 568.717 (Pa-Ke)cM 1 448.879 (Bou-Zo)cM 1 590.224 1/2 1/2 1/2

(Ni-Pa = 8, Pa-Bou = 14, Bou-Ga = 8, Ga-Di=12) / 72(Ni-Bou)cM (Pa-Ga)cM (Bou-Di)cM (Ga-Ke)cM diphonic Bou-Di 4th diphonic Bou-Di 5th 366.667 (Ni-Ga)cM 366.667 (Pa-Di)cM 333.333 (Bou-Ke)cM 333.333 (Ga-Zo)cM 466.667 666.667 500.000 (Ni-Di)cM 566.667 (Pa-Ke)cM 466.667 (Bou-Zo)cM 566.667

(Ni-Pa = 7, Pa-Bou = 12, Bou-Ga = 7, Ga-Di=12) / 64(Ni-Bou)cM (Pa-Ga)cM (Bou-Di)cM (Ga-Ke)cM diphonic Bou-Di 4th 356.250 (Ni-Ga)cM 356.250 (Pa-Di)cM 356.250 (Bou-Ke)cM 356.250 (Ga-Zo)cM 487.500 487.500 (Ni-Di)cM 581.250 (Pa-Ke)cM 487.500 (Bou-Zo)cM 581.250

diphonic Bou-Di 5th

712.500



(Ni-Pa = 16/15, Pa-Bou = 75/64, Bou-Ga = 16/15, Ni-Ga=4/3(Ni-Bou)cS (Pa-Ga)cS (Bou-Di)cS (Ga-Ke)cS diphonic diphonic diphonic diphonic Bou-Di 4th Bou-Di 5th Ni-Bou 4th Ni-Bou 5th 1 1 1 1 1/4 1/4 1/5 1/5 386.314 (Ni-Ga)cS 1 1/3 386.314 (Pa-Di)cS 1 13/32 315.641 (Bou-Ke)cS 1 7/25 315.641 (Ga-Zo)cS 1 13/32 519.551 631.283 590.224 772.627 498.045 (Ni-Di)cS 1 590.224 (Pa-Ke)cS 1 427.373 (Bou-Zo)cS 1 590.224 1/2 1/2 1/2

1 7/20 1 11/25 1 13/32 1 9/16

(Ni-Pa = 9/8, Pa-Bou = 256/243, Bou-Ga = 243/200, Ga-Di =(Pa-Ga)cS (Bou-Di)cS (Ga-Ke)cS (Di-Zo)cS Di-Zo 4th Di-Zo 5th Pa-Ga 4th Pa-Ga 5th 1 7/25 1 17/64 1 11/64 1 5/27 1 1/3 1 295/729 1 11/25 1 399/625 427.373 (Pa-Di)cS 1 1/3 407.820 (Bou-Ke)cS 1 217/512 274.582 (Ga-Zo)cS 1 19/81 294.135 (Di-Ni)cS 1 11/25 498.045 588.270 631.283 854.745 498.045 (Pa-Ke)cS 1 611.730 (Bou-Zo)cS 1 364.807 (Ga-Ni)cS 1 631.283 1/2 1/2 1/2

diphonic diphonic diphonic diphonic

(Ni-Pa = 12, Pa-Bou = 6, Bou-Ga = 20, Ga-Di=4) / 72(Pa-Ga)cS (Bou-Di)cS (Ga-Ke)cS (Di-Zo)cS Di-Zo 4th Di-Zo 5th Pa-Ga 4th Pa-Ga 5th 433.333 (Pa-Di)cS 400.000 (Bou-Ke)cS 266.667 (Ga-Zo)cS 300.000 (Di-Ni)cS 500.000 600.000 633.333 866.667 500.000 (Pa-Ke)cS 600.000 (Bou-Zo)cS 366.667 (Ga-Ni)cS 633.333


(Ni-Pa = 12, Pa-Bou = 7, Bou-Ga = 18, Ga-Di=3) / 68(Pa-Ga)cS (Bou-Di)cS (Ga-Ke)cS (Di-Zo)cS Di-Zo 4th Di-Zo 5th Pa-Ga 4th Pa-Ga 5th 441.176 (Pa-Di)cS 370.588 (Bou-Ke)cS 264.706 (Ga-Zo)cS 335.294 (Di-Ni)cS 547.059 670.588 652.941 882.353 494.118 (Pa-Ke)cS 582.353 (Bou-Zo)cS 388.235 (Ga-Ni)cS 652.941


(Ni-Pa = 4, Pa-Bou = 2, Bou-Ga = 8, Ga-Di = 2) / 24(Pa-Ga)c (Bou-Di)c (Ga-Ke)c (Di-Zo)c Di-Zo 4th Di-Zo 5th Pa-Ga 4th Pa-Ga 5th 400.000 400.000 300.000 300.000 500.000 600.000 600.000 800.000 (Pa-Di)c (Bou-Ke)c (Ga-Zo)c (Di-Ni)c 500.000 600.000 400.000 600.000 (Pa-Ke)c (Bou-Zo)c (Ga-Ni)c


(Ga-Di = 9/8, Di-Ke = 9/8, Ke-Zo = 256/243)(Ga-Ke)e (Di-Zo)e (Ke-Ni)e (Zo-Pa)e 1 17/64 1 5/27 1 5/27 1 17/64 407.820 294.135 294.135 407.820 (Ga-Zo)e (Di-Ni)e (Ke-Pa)e (Zo-Bou)e 1 1/3 1 1/3 1 1/3 1 217/512 498.045 498.045 498.045 611.730 (Ga-Ni)e (Di-Pa)e (Ke-Bou)e (Zo-Ga)e 1 1 1 1 1/2 1/2 1/2 1/2

(Ni-Bou)e diphonic Ni-Bou 4th diphonic Ni-Bou 5th

1 17/64 1 217/512 1 133/221

407.820 611.730 815.640

(Ga-Di = 12, Di-Ke = 12, Ke-Zo = 6) / 72(Ga-Ke)e (Di-Zo)e (Ke-Ni)e (Zo-Pa)e (Ni-Bou)e diphonic Ni-Bou 4th diphonic Ni-Bou 5th 400.000 300.000 300.000 400.000 400.000 600.000 800.000 (Ga-Zo)e (Di-Ni)e (Ke-Pa)e (Zo-Bou)e 500.000 500.000 500.000 600.000 (Ga-Ni)e (Di-Pa)e (Ke-Bou)e (Zo-Ga)e

(Ga-Di = 12, Di-Ke = 12, Ke-Zo = 3, Zo-Ni = 13) / 72(Ga-Ke)e (Di-Zo)e (Ke-Ni)e (Zo-Pa)e (Ni-Bou)e diphonic Ni-Bou 4th diphonic Ni-Bou 5th 423.529 264.706 282.353 441.176 264.706 476.471 529.412 (Ga-Zo)e (Di-Ni)e (Ke-Pa)e (Zo-Bou)e 476.471 494.118 494.118 494.118 (Ga-Ni)e (Di-Pa)e (Ke-Bou)e (Zo-Ga)e

(Ni-Pa = 12, Pa-Bou = 14, Bou-Ga = 4, Ga-Di = 4) / 72(Ni-Bou)s (Pa-Ga)s (Bou-Di)s (Ga-Ke)s diphonic Bou-Di 4th diphonic Bou-Di 5th 433.333 300.000 133.333 400.000 333.333 266.667 (Ni-Ga)s (Pa-Di)s (Bou-Ke)s (Ga-Zo)s 500.000 366.667 466.667 566.667 (Ni-Di)s (Pa-Ke)s (Bou-Zo)s

(Ni-Pa = 12, Pa-Bou = 13, Bou-Ga = 3, Ga-Di = 6, Di-Ke = 18(Ni-Bou)s (Pa-Ga)s (Bou-Di)s (Ga-Ke)s diphonic Bou-Di 4th diphonic Bou-Di 5th 441.176 282.353 158.824 423.529 370.588 317.647 (Ni-Ga)s (Pa-Di)s (Bou-Ke)s (Ga-Zo)s 494.118 388.235 476.471 582.353 (Ni-Di)s (Pa-Ke)s (Bou-Zo)s

(Ni-Pa = 12, Pa-Bou = 18, Bou-Ga = 16, Ga-Di = 4) / 72(Ni-Bou)z (Pa-Ga)z (Bou-Di)z (Ga-Ke)z diphonic Bou-Di 4th diphonic Bou-Di 5th 366.667 333.333 333.333 266.667 633.333 666.667 (Ni-Ga)z (Pa-Di)z (Bou-Ke)z (Ga-Zo)z 633.333 400.000 533.333 433.333 (Ni-Di)z (Pa-Ke)z (Bou-Zo)z

(Ni-Pa = 14, Pa-Bou = 7, Bou-Ga = 14, Ga-Di = 5) / 68(Ni-Bou)z (Pa-Ga)z (Bou-Di)z (Ga-Ke)z diphonic Bou-Di 4th diphonic Bou-Di 5th 370.588 370.588 335.294 300.000 582.353 670.588 (Ni-Ga)z (Pa-Di)z (Bou-Ke)z (Ga-Zo)z 617.647 458.824 547.059 458.824 (Ni-Di)z (Pa-Ke)z (Bou-Zo)z

(Ni-Pa = 12, Pa-Bou = 18, Bou-Ga = 16, Ga-Di = 4) / 72(Ni-Bou)k (Pa-Ga)k (Bou-Di)k (Ga-Ke)k diphonic Bou-Di 4th diphonic Bou-Di 5th 433.333 333.333 266.667 266.667 566.667 533.333 (Ni-Ga)k (Pa-Di)k (Bou-Ke)k (Ga-Zo)k 633.333 400.000 466.667 433.333 (Ni-Di)k (Pa-Ke)k (Bou-Zo)k

(Ni-Pa = 12, Pa-Bou = 13, Bou-Ga = 11, Ga-Di = 4) / 68(Ni-Bou)k (Pa-Ga)k (Bou-Di)k (Ga-Ke)k diphonic Bou-Di 4th 441.176 423.529 264.706 282.353 476.471 (Ni-Ga)k (Pa-Di)k (Bou-Ke)k (Ga-Zo)k 635.294 494.118 476.471 441.176 (Ni-Di)k (Pa-Ke)k (Bou-Zo)k

diphonic Bou-Di 5th

529.412



PentaphoniesCENTS FRACTION CENTS

-Ga = 27/25, Ni-Ga=4/3)701.955 701.955 701.955 (Ni-Ke)d 1 11/16 (Pa-Zo)d 1 157/243 (Bou-Ni)d 1 1/2 905.865 862.852 701.955

700.000 700.000 700.000

(Ni-Ke)d (Pa-Zo)d (Bou-Ni)d

900.000 866.667 833.333

/15, Ni-Ga=4/3)701.955 701.955 701.955 (Ni-Ke)d 1 11/16 (Pa-Zo)d 1 2/3 (Bou-Ni)d 1 1/2 905.865 884.359 701.955

8/81, Ni-Ga=4/3)701.955 648.682 641.543 (Ni-Ke)d 1 7/11 (Pa-Zo)d 1 47/81 (Bou-Ni)d 1 1/2 852.592 792.180 701.955

8/81, Ni-Ga=4/3)701.955 701.955 701.955 (Ni-Ke)d 1 11/16 (Pa-Zo)d 1 7/11 (Bou-Ni)d 1 1/2 905.865 852.592 701.955

1, Di-Ni=4/3)701.955

705.882 705.882 705.882


917.647 864.706 829.412

709.091 709.091 709.091


927.273 854.545 836.364

700.000 700.000

(Ni-Ke)d (Pa-Zo)d

900.000 850.000

700.000

(Bou-Ni)d

850.000

= 16/15, Ni-Ga=4/3)701.955 701.955 701.955 (Ni-Ke)cM 1 3/5 (Pa-Zo)cM 1 97/128 (Bou-Ni)cM 1 1/2 813.686 976.537 701.955

Ga = 16/15, Ga-Di = 9/8, Ni-Ga=4/3)701.955 701.955 701.955 (Ni-Ke)cM 1 31/50 (Pa-Zo)cM 1 53/72 (Bou-Ni)cM 1 1/2 835.193 955.031 701.955

Di=12) / 72700.000 700.000 700.000 (Ni-Ke)cM (Pa-Zo)cM (Bou-Ni)cM 833.333 933.333 833.333

Di=12) / 64712.500 712.500 712.500 (Ni-Ke)cM (Pa-Zo)cM (Bou-Ni)cM 843.750 937.500 843.750

Di=12) / 68705.882 705.882 705.882 (Ni-Ke)d (Pa-Zo)d (Bou-Ni)d 829.412 952.941 829.412

, Ga-Di=12) / 68670.588 670.588 705.882 (Ni-Ke)d (Pa-Zo)d (Bou-Ni)d 794.118 917.647 829.412

= 16/15, Ni-Ga=4/3)701.955 701.955 701.955 (Ni-Ke)cS 1 3/5 (Pa-Zo)cS 1 97/128 (Bou-Ni)cS 1 3/5 813.686 976.537 813.686

= 243/200, Ga-Di = 25/24)701.955 701.955 701.955 (Pa-Zo)cS 1 47/81 792.180 (Bou-Ni)cS 1 329/400 1,039.103 (Ga-Pa)cS 1 9/16 772.627

-Di=4) / 72700.000 700.000 700.000 (Pa-Zo)cS (Bou-Ni)cS 800.000 1,033.333

-Di=3) / 68705.882 705.882 705.882 (Pa-Zo)cS (Bou-Ni)cS 829.412 1,023.529

i = 2) / 24700.000 700.000 700.000 (Pa-Zo)c (Bou-Ni)c x 800.000 1,000.000

701.955 701.955 701.955 701.955

(Ga-Pa)e 1 11/16 (Di-Bou)e 1 11/16 (Ke-Ga)e 1 47/81

905.865 905.865 792.180

700.000 700.000 700.000 700.000

(Ga-Pa)e (Di-Bou)e (Ke-Ga)e

900.000 900.000

= 13) / 72705.882 705.882 723.529 723.529 (Ga-Pa)e (Di-Bou)e (Ke-Ga)e 917.647 758.824

-Di = 4) / 72566.667 700.000 633.333 (Ni-Ke)s (Pa-Zo)s (Bou-Ni)s 900.000 866.667 766.667

-Di = 6, Di-Ke = 18) / 68600.000 705.882 635.294 (Ni-Ke)s (Pa-Zo)s (Bou-Ni)s 917.647 864.706 758.824

a-Di = 4) / 72700.000 600.000 700.000 (Ni-Ke)z (Pa-Zo)z (Bou-Ni)z 900.000 766.667 833.333

-Di = 5) / 68705.882 670.588 705.882 (Ni-Ke)z (Pa-Zo)z (Bou-Ni)z 917.647 829.412 829.412

a-Di = 4) / 72700.000 600.000 633.333 (Ni-Ke)k (Pa-Zo)k (Bou-Ni)k 900.000 766.667 766.667




68.000 12 9 7 12 12 9 7

Ni Pa Bou Ga Di Ke Zo Ni

1 8/9 22/27 3/4 2/3 16/27 44/81 1/2

0.000 203.910 354.547 498.045 701.955 905.865 1,056.502 1,200.000

0 12 21 28 40 52 61 68

211.765 370.588 494.118 705.882 917.647 1,076.471 1,200.000

7.9 16.0 -3.9 3.9 11.8 20.0 0.0

CHRYSANTHOS

0.908.6 Ni Pa Bou Ga Di Ke Zo Ni 1 8/9 81/100 3/4 2/3 16/27 27/50 1/2 0.000 203.910 364.807 498.045 701.955 905.865 1,066.762 1,200.000 0 12 21 28 40 52 61 68 211.765 370.588 494.118 705.882 917.647 1,076.471 1,200.000

7.9 5.8 -3.9 3.9 11.8 9.7 0.0

EPITROPI

0.985.5 Ni Pa Bou Ga Di Ke Zo Ni 1 8/9 4/5 3/4 2/3 16/27 8/15 1/2 0.000 203.910 386.314 498.045 701.955 905.865 1,088.269 1,200.000 0 12 21 28 40 52 61 68 211.765 370.588 494.118 705.882 917.647 1,076.471 1,200.000

7.9 -15.7 -3.9 3.9 11.8 -11.8 0.0

DIDYMOS

0.9411.0 PYTHAGORAS Ni Pa Bou 1 8/9 64/81 0.000 203.910 407.820 0 12 21 211.765 370.588

7.9 -37.2

PYTHAGORAS

Ga Di Ke Zo Ni

3/4 2/3 16/27 128/243 1/2

498.045 701.955 905.865 1,109.775 1,200.000

28 40 52 61 68

494.118 705.882 917.647 1,076.471 1,200.000

-3.9 3.9 11.8 -33.3 0.0

0.4321.4 Ni Pa Bou Ga Di Ke Zo Ni 1 8/9 22/27 3/4 2/3 11/18 9/16 1/2 0.000 203.910 354.547 498.045 701.955 852.592 996.090 1,200.000 0 12 21 28 40 52 61 68 211.765 370.588 494.118 705.882 917.647 1,076.471 1,200.000

7.9 16.0 -3.9 3.9 65.1 80.4 0.0

AL FAHRABI

0.0335.4 Bou Bou Bou Bou Bou 22/27 81/100 4/5 64/81 22/27 354.547 364.807 386.314 407.820 354.547 21 21 21 21 21 370.588 370.588 370.588 370.588 370.588

16.0 5.8 -15.7 -37.2 16.0

THIRD

Ga Ga Ga Ga Ga

3/4 3/4 3/4 3/4 3/4

498.045 498.045 498.045 498.045 498.045

28 28 28 28 28

494.118 494.118 494.118 494.118 494.118

-3.9 -3.9 -3.9 -3.9 -3.9

FOURTH

Di Di Di Di Di

2/3 2/3 2/3 2/3 2/3

701.955 701.955 701.955 701.955 701.955

40 40 40 40 40

705.882 705.882 705.882 705.882 705.882

3.9 3.9 3.9 3.9 3.9

FIFTH

(D-B)+(N-P) (D-B)+(N-P) (D-B)+(N-P) (D-B)+(N-P) (D-B)+(N-P)

8/11 477/652 20/27 3/4 8/11

551.318 541.058 519.551 498.045 551.318

31 31 31 31 31

547.059 547.059 547.059 547.059 547.059

-4.3 6.0 27.5 49.0 -4.3

DIPHONIC FOURTH

DIPHONIC FIFTH

(D-B)+(N-P) (D-B)+(N-P) (D-B)+(N-P) (D-B)+(N-P)

81/121 21/31 25/36 472/663

694.816 674.295 631.283 588.270

38 38 38 38

670.588 670.588 670.588 670.588

-24.2 -3.7 39.3 82.3

DIPHONIC FIFTH

(D-B)+(N-P)

81/121

694.816

38

670.588

-24.2

72.000 12 10 8 12 12 10 8 9 9 4 9 9 9 4

53.000

0 12 22 30 42 54 64 72

200.000 366.667 500.000 700.000 900.000 1,066.667 1,200.000

-3.9 12.1 2.0 -2.0 -5.9 10.2 0.0

0 9 18 22 31 40 49 53

203.774 407.547 498.113 701.887 905.660 1,109.434 1,200.000

-0.1 53.0 0.1 -0.1 -0.2 52.9 0.0

0.997.5 0 12 22 30 42 54 64 72 200.000 366.667 500.000 700.000 900.000 1,066.667 1,200.000

0.0627.4 0 9 18 22 31 40 49 53 203.774 407.547 498.113 701.887 905.660 1,109.434 1,200.000

-3.9 1.9 2.0 -2.0 -5.9 -0.1 0.0

-0.1 42.7 0.1 -0.1 -0.2 42.7 0.0

1.003.2 0 12 22 30 42 54 64 72 200.000 366.667 500.000 700.000 900.000 1,066.667 1,200.000

0.2422.1 0 9 18 22 31 40 49 53 203.774 407.547 498.113 701.887 905.660 1,109.434 1,200.000

-3.9 -19.6 2.0 -2.0 -5.9 -21.6 0.0

-0.1 21.2 0.1 -0.1 -0.2 21.2 0.0

0.919.8 0 12 22 200.000 366.667 0 9 18 203.774 407.547

0.9011.0

-3.9 -41.2

-0.1 -0.3

30 42 54 64 72

500.000 700.000 900.000 1,066.667 1,200.000

2.0 -2.0 -5.9 -43.1 0.0

22 31 40 49 53

498.113 701.887 905.660 1,109.434 1,200.000

0.1 -0.1 -0.2 -0.3 0.0

0.3120.7 0 12 22 30 42 54 64 72 200.000 366.667 500.000 700.000 900.000 1,066.667 1,200.000 0 9 18 22 31 40 49 53 203.774 407.547 498.113 701.887 905.660 1,109.434 1,200.000

1.000.1

-3.9 12.1 2.0 -2.0 47.4 70.6 0.0

-0.1 53.0 0.1 -0.1 53.1 113.3 0.0

0.1530.8 22 22 22 22 22 366.667 366.667 366.667 366.667 366.667

0.0045.7 18 18 18 18 18 407.547 407.547 407.547 407.547 407.547

12.1 1.9 -19.6 -41.2 12.1

53.0 42.7 21.2 -0.3 53.0

30 30 30 30 30

500.000 500.000 500.000 500.000 500.000

2.0 2.0 2.0 2.0 2.0

22 22 22 22 22

498.113 498.113 498.113 498.113 498.113

0.1 0.1 0.1 0.1 0.1

42 42 42 42 42

700.000 700.000 700.000 700.000 700.000

-2.0 -2.0 -2.0 -2.0 -2.0

31 31 31 31 31

701.887 701.887 701.887 701.887 701.887

-0.1 -0.1 -0.1 -0.1 -0.1

32 32 32 32 32

533.333 533.333 533.333 533.333 533.333

-18.0 -7.7 13.8 35.3 -18.0

22 22 22 22 22

498.113 498.113 498.113 498.113 498.113

-53.2 -42.9 -21.4 0.1 -53.2

40 40 40 40

666.667 666.667 666.667 666.667

-28.1 -7.6 35.4 78.4

26 26 26 26

588.679 588.679 588.679 588.679

-106.1 -85.6 -42.6 0.4

40

666.667

-28.1

26

588.679

-106.1

Units within Octave of First Scale

72Initial Frequency Final Frequency

First scale conversions

Chord fraction

Second Scale's Total number of Units within Octave

Second scale conversions

Interval using Scale 1 Equivalent interval in Scale 2

CENTS

440.00 880.00

72.00Chord fraction

1,200.0

2

1 1/8

12.23

203.9

68

12.00

alent interval in Scale 2

11.33 1 6/49

200.0

Chord fraction

Units within Octave of First Scale

68Initial Frequency Final Frequency

440.00 880.00Chord fractionChord fraction

First scale conversions

1 1/8

Second Scale's Total number of Units within Octave

Second scale conversions

Interval using Scale 1

72

12.00

Equivalent interval in Scale 2

Chord fraction

CENTS

68.002

1,200.0

11.55

203.9

12.71 1 124/953

211.8

ET conversion table (from one ET system to the 1200 cent system

Epitropis Chrysanthou Chrysanthou 66 72 68

Kinezon 53

Roumanon (Pangratiou) 24 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0 600.0 650.0 700.0 750.0 800.0 850.0 900.0 950.0 1,000.0 1,050.0 1,100.0 1,150.0 1,200.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

16.7 33.3 50.0 66.7 83.3 100.0 116.7 133.3 150.0 166.7 183.3 200.0 216.7 233.3 250.0 266.7 283.3 300.0 316.7 333.3 350.0 366.7 383.3 400.0 416.7 433.3 450.0 466.7 483.3 500.0 516.7

17.6 35.3 52.9 70.6 88.2 105.9 123.5 141.2 158.8 176.5 194.1 211.8 229.4 247.1 264.7 282.4 300.0 317.6 335.3 352.9 370.6 388.2 405.9 423.5 441.2 458.8 476.5 494.1 511.8 529.4 547.1

18.2 36.4 54.5 72.7 90.9 109.1 127.3 145.5 163.6 181.8 200.0 218.2 236.4 254.5 272.7 290.9 309.1 327.3 345.5 363.6 381.8 400.0 418.2 436.4 454.5 472.7 490.9 509.1 527.3 545.5 563.6

22.6 45.3 67.9 90.6 113.2 135.8 158.5 181.1 203.8 226.4 249.1 271.7 294.3 317.0 339.6 362.3 384.9 407.5 430.2 452.8 475.5 498.1 520.8 543.4 566.0 588.7 611.3 634.0 656.6 679.2 701.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

533.3 550.0 566.7 583.3 600.0 616.7 633.3 650.0 666.7 683.3 700.0 716.7 733.3 750.0 766.7 783.3 800.0 816.7 833.3 850.0 866.7 883.3 900.0 916.7 933.3 950.0 966.7 983.3 1,000.0 1,016.7 1,033.3 1,050.0 1,066.7 1,083.3 1,100.0 1,116.7 1,133.3

564.7 582.4 600.0 617.6 635.3 652.9 670.6 688.2 705.9 723.5 741.2 758.8 776.5 794.1 811.8 829.4 847.1 864.7 882.4 900.0 917.6 935.3 952.9 970.6 988.2 1,005.9 1,023.5 1,041.2 1,058.8 1,076.5 1,094.1 1,111.8 1,129.4 1,147.1 1,164.7 1,182.4 1,200.0

581.8 600.0 618.2 636.4 654.5 672.7 690.9 709.1 727.3 745.5 763.6 781.8 800.0 818.2 836.4 854.5 872.7 890.9 909.1 927.3 945.5 963.6 981.8 1,000.0 1,018.2 1,036.4 1,054.5 1,072.7 1,090.9 1,109.1 1,127.3 1,145.5 1,163.6 1,181.8 1,200.0

724.5 747.2 769.8 792.5 815.1 837.7 860.4 883.0 905.7 928.3 950.9 973.6 996.2 1,018.9 1,041.5 1,064.2 1,086.8 1,109.4 1,132.1 1,154.7 1,177.4 1,200.0

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

69 70 71 72

1,150.0 1,166.7 1,183.3 1,200.0

69 70 71 72

Epitropis Chrysanthou Chrysanthou Kinezon 53 Roumanon 24 66 72 6816.666667 33.333333 50.000000 66.666667 83.333333 100.000000 116.666667 133.333333 150.000000 166.666667 183.333333 200.000000 216.666667 233.333333 250.000000 266.666667 283.333333 300.000000 316.666667 333.333333 350.000000 366.666667 383.333333 400.000000 416.666667 433.333333 450.000000 466.666667 483.333333 500.000000 516.666667 17.647059 35.294118 52.941176 70.588235 88.235294 105.882353 123.529412 141.176471 158.823529 176.470588 194.117647 211.764706 229.411765 247.058824 264.705882 282.352941 300.000000 317.647059 335.294118 352.941176 370.588235 388.235294 405.882353 423.529412 441.176471 458.823529 476.470588 494.117647 511.764706 529.411765 547.058824 18.181818 36.363636 54.545455 72.727273 90.909091 109.090909 127.272727 145.454545 163.636364 181.818182 200.000000 218.181818 236.363636 254.545455 272.727273 290.909091 309.090909 327.272727 345.454545 363.636364 381.818182 400.000000 418.181818 436.363636 454.545455 472.727273 490.909091 509.090909 527.272727 545.454545 563.636364 22.641509 45.283019 67.924528 90.566038 113.207547 135.849057 158.490566 181.132075 203.773585 226.415094 249.056604 271.698113 294.339623 316.981132 339.622642 362.264151 384.905660 407.547170 430.188679 452.830189 475.471698 498.113208 520.754717 543.396226 566.037736 588.679245 611.320755 633.962264 656.603774 679.245283 701.886792 50.000000 100.000000 150.000000 200.000000 250.000000 300.000000 350.000000 400.000000 450.000000 500.000000 550.000000 600.000000 650.000000 700.000000 750.000000 800.000000 850.000000 900.000000 950.000000 1,000.000000 1,050.000000 1,100.000000 1,150.000000 1,200.000000

533.333333 550.000000 566.666667 583.333333 600.000000 616.666667 633.333333 650.000000 666.666667 683.333333 700.000000 716.666667 733.333333 750.000000 766.666667 783.333333 800.000000 816.666667 833.333333 850.000000 866.666667 883.333333 900.000000 916.666667 933.333333 950.000000 966.666667 983.333333 1,000.000000 1,016.666667 1,033.333333 1,050.000000 1,066.666667 1,083.333333 1,100.000000 1,116.666667 1,133.333333

564.705882 582.352941 600.000000 617.647059 635.294118 652.941176 670.588235 688.235294 705.882353 723.529412 741.176471 758.823529 776.470588 794.117647 811.764706 829.411765 847.058824 864.705882 882.352941 900.000000 917.647059 935.294118 952.941176

581.818182 600.000000 618.181818 636.363636 654.545455 672.727273 690.909091 709.090909 727.272727 745.454545 763.636364 781.818182 800.000000

724.528302 747.169811 769.811321 792.452830 815.094340 837.735849 860.377358 883.018868 905.660377 928.301887 950.943396 973.584906 996.226415

818.181818 1,018.867925 836.363636 1,041.509434 854.545455 1,064.150943 872.727273 1,086.792453 890.909091 1,109.433962 909.090909 1,132.075472 927.272727 1,154.716981 945.454545 1,177.358491 963.636364 1,200.000000 981.818182

970.588235 1,000.000000 988.235294 1,018.181818 1,005.882353 1,036.363636 1,023.529412 1,054.545455 1,041.176471 1,072.727273 1,058.823529 1,090.909091 1,076.470588 1,109.090909 1,094.117647 1,127.272727 1,111.764706 1,145.454545 1,129.411765 1,163.636364 1,147.058824 1,181.818182 1,164.705882 1,200.000000 1,182.352941 1,200.000000

1,150.000000 1,166.666667 1,183.333333 1,200.000000

interval ratio

cents equivalent

72 ET 68 ET equivale equivale nt nt

interval name(s) (if any)

(1:1)

0.000

0.00

0.00

tonic, unison, 1st harmonic (fundamental of the harmonic series), normalised 2nd harmonic ragisma breedsma cent millioctave skhisma, schisma ((3 to the 8th/2 to the 12th) x 5/8) (the difference between five octaves and eight justly tuned tuned fifths plus one justly tuned major third; the difference between the Phythagorean and syntonic commas) Mercator's comma, the ratio of 53 pure thirds to 31 octaves Savart

(4375:4374) (2401:2400) (21/1200:1) (21/1000:1) (32805:3276 385 : 215 8)

0.400 0.720 1.000 1.200 1.954

0.02 0.04 0.06 0.07 0.12

0.02 0.04 0.06 0.07 0.11

(353:284)

3.600 3.990 5.292 7.712 8.107 11.445 13.795 14.367 17.399 17.576 19.553

0.22 0.24 0.32 0.46 0.49 0.69 0.83 0.86 1.04 1.05 1.17

0.20 0.23 0.30 0.44 0.46 0.65 0.78 0.81 0.99 1.00 1.11

((1+(1/100)l og10/(2)):1) (122444006 4:122070312 5) (225:224) (15625:1555 56 : 2635 2) (393216:390 625) (126:125) (121:120) (100:99) (99:98) (2048:2025) 211 : 3452

parakleisma

septimal kleisma or marvel comma kleisma Wrschmidt comma septimal semicomma or starling comma

minor comma, diaschisma

(81:80)

34 : 245

21.506

1.29

1.22

major comma, syntonic comma, comma of Didymus (difference between four justly tuned perfect fifths and two octaves plus a major third). There are 55.79763 syntonic commas in the octave Arabian comma, Holdrian comma or Holder's comma Pythagorean or ditonic comma (312/219) (difference between twelve justly tuned perfect fifths and seven octaves). There are 51.15087 Pythagorean commas in the octave 65th harmonic septimal comma, comma of Archytas (the difference between the 3-limit or Pythagorean seventh and the harmonic seventh) minimal diesis

(21/53:1)

22.640 23.460

1.36 1.41

1.28 1.33

(531441:524 312 : 219 288)

(65:64) (64:63)

2 : 3 7

6

2

26.841 27.264

1.61 1.64

1.52 1.54

(20000:1968 3) (63:62) (3125:3072) (58:57) (57:56) (56:55) (55:54) (52:51) (51:50) (50:49) (49:48) (46:45) (45:44) (128:125)

27.660 27.700 29.614 30.109 30.642 31.194 31.767 33.617 34.283 34.976 35.697 38.051 38.906 41.059

1.66 1.66 1.78 1.81 1.84 1.87 1.91 2.02 2.06 2.10 2.14 2.28 2.33 2.46

1.57 1.57 1.68 1.71 1.74 1.77 1.80 1.90 1.94 1.98 2.02 2.16 2.20 2.33

small diesis

237 : 511

Ptolemy's enharmonic

septimal sixth-tone or jubilisma septimal diesis or slendro diesis inferior quarter-tone (Ptolemy) diminished second (16/15 x 24/25), enharmonic diesis (the difference between three justly tuned major thirds and one octave), great diesis, enharmonic diesis (Vincentino), 5-limit diesis, limma enharmonic diesis (Avicenna) difference between major and minor semitones

2 :5

7

3

(525:512) (40:39)

43.408 43.831

2.60 2.63

2.46 2.48

(39:38) (77:75) (36:35) (250:243) (21/24:1)

44.970 45.561 48.770 49.166

2.70 2.73 2.93 2.95 3.00

2.55 2.58 2.76 2.79 2.83

superior quarter-tone (Eratosthenes) superior or septimal quartertone maximal diesis equal-tempered quarter tone

2 3 : 57

2

2

50.000

(35:34) (34:33) (50331648:4 8828125) (33:32) (32:31) (125:121) (31:30) (30:29) (29:28) (57:55) (648:625) (28:27) (80:77) (27:26) (26:25) (20480:1968 3)

50.184 51.682 52.504 53.273 54.964 56.305 56.767 58.692 60.751 61.836 62.565 62.961 66.170 65.337 67.900 68.719

3.01 3.10 3.15 3.20 3.30 3.38 3.41 3.52 3.65 3.71 3.75 3.78 3.97 3.92 4.07 4.12

2.84 2.93 2.98 3.02 3.11 3.19 3.22 3.33 3.44 3.50 3.55 3.57 3.75 3.70 3.85 3.89

equal temperament (ET) 1/4tone approximation

33rd harmonic inferior quarter-tone (Didymus) superior quarter-tone (Didymus)

major diesis inferior quarter-tone (Archytas)

1/3-tone (Avicenna) comma that is associated with super-Pythagorean temperament

(51:49) (126:121) (25:24)

52 : 233

69.259 70.100 70.672

4.16 4.21 4.24

3.92 3.97 4.00

minor 5-limit semitone (halfstep), chromatic diesis, semitone minimus, lesser or just chromatic semitone, minor chroma

(24:23) (117:112) (23:22) (67:64) (22:21)

73.681 75.612 76.956 79.307 80.537 37 : 225 84.467 87.676 88.801

4.42 4.54 4.62 4.76 4.83 5.07 5.26 5.33

4.18 4.28 4.36 4.49 4.56 4.79 4.97 5.03

67th harmonic hard semitone (1/2-step) (Ptolemy, Avicenna, Safiud) major diesis, septimal chromatic semitone

(21:20) (81:77) (20:19)

(256:243)

28 : 35

90.225

5.41

5.11

(58:55) (135:128)

Pythagorean diatonic semitone (half-step), minor or Pythagorean limma, minor semitone limma ascendant, greater chromatic semitone, semitone medius, chromatic semitone, major chroma

3 5 : 2

3

7

91.946 92.179

5.52 5.53

5.21 5.22

(96:91) (644245094 4:610351562 5) (19:18) (55:52) (128:121) (18:17) (21/12:1)

92.601 93.563 93.603 97.104 97.364 98.955

5.56 5.61 5.62 5.83 5.84 5.94 6.00

5.25 5.30 5.30 5.50 5.52 5.61 5.67

100.000

equal temperament (ET) semitone (half-step) equal temperament (ET) semitone (half-step), equaltempered minor second, exact

(35:33) (52:49) (86:81) (17:16)

101.867 102.876 103.698 104.955 105.882

6.11 6.17 6.22 6.30 6.35

5.77 5.83 5.88 5.95 6.00

overtone semitone (halfstep)

(33:31) (49:46) (16:15)

24 : 35

108.237 109.377 111.731

6.49 6.56 6.70

6.13 6.20 6.33

major 5-limit semitone (halfstep), just diatonic semitone

(2187:2048) 37 : 211

113.685

6.82

6.44

(31:29) (77:72) (15:14)

35 : 27

115.458 116.234 119.443

6.93 6.97 7.17

6.54 6.59 6.77

apotome or apotome Pythagorica, Pythagorean major semitone, Pythagorean chromatic semitone

Cowell just semitone (halfstep), septimal diatonic semitone

123.529

7.41

7.00

(29:27) (14:13) (69:64) (55:51)

123.712 128.298 130.229 130.721 131.250

7.42 7.70 7.81 7.84 7.88

7.01 7.27 7.38 7.41 7.44

major semitone 69th harmonic

(27:25)

33 : 52

133.238

7.99

7.55

alternate Renaissance semitone (half-step), semitone maximus, minor second, large limma

133.333

8.00

7.56

(121:112) (13:12) (64:59) (38:35) (63:58) (88:81)

133.810 138.573 140.828 142.373 143.159 143.498

8.03 8.31 8.45 8.54 8.59 8.61

7.58 7.85 7.98 8.07 8.11 8.13

3/4-tone (Avicenna), minor tone

(25:23) (62:57) (135:124) (49:45)

144.353 145.568 147.143 147.428

150.000

8.66 8.73 8.83 8.85 9.00

8.18 8.25 8.34 8.35 8.50

(12:11)

223 : 11

150.637

9.04

8.54

undecimal "median" semitone (1/2-step), lesser undecimal neutral second

(59:54)

153.307

9.20

8.69

(35:32) (23:21)

155.140 157.493 158.824

9.31 9.45 9.53

8.79 8.92 9.00

35th harmonic

(57:52) (34:31) (800:729)

158.940 159.920 160.897

9.54 9.60 9.65

9.01 9.06 9.12

(56:51) (11:10)

11 : 25

161.915 165.004 166.667

9.71 9.90 10.00

9.18 9.35 9.44

greater undemical neutral second

(54:49) (32:29) (21:19) (31:28) (567:512) (51:46) (71:64) 65536 : 5904916 10

2

:3

168.213 170.423 173.268 176.210 176.646 178.636 179.697 180.450 182.404

10.09 10.23 10.40 10.57 10.60 10.72 10.78 10.83 10.94

9.53 9.66 9.82 9.99 10.01 10.12 10.18 10.23 10.34

71st harmonic Pythagorean diminished third, Pythagorean minor tone minor whole-tone, just minor tone, smaller step

(10:9)

25 : 32

(49:40) (39:35) (29:26) (125:112) (48:43) (19:17) (160:143) (28:25) (121:108) (55:49)

186.338 187.343 189.050 190.115 190.437 192.558 194.468 196.198 196.771 199.980

11.18 11.24 11.34 11.41 11.43 11.55 11.67 11.77 11.81 12.00

10.56 10.62 10.71 10.77 10.79 10.91 11.02 11.12 11.15 11.33

(21/6:1)

200.000

12.00

11.33

equal-tempered whole-tone, exact, equal-tempered major second

(64:57) (9:8)

32 : 23

200.532 203.910

12.03 12.23

11.36 11.55

major whole-tone, major tone, greater step, just major second, sesquioctave, tonus , 5th harmonic (normalised)

(62:55) (44:39) (35:31)

207.404 208.835 210.104 211.765

12.44 12.53 12.61 12.71

11.75 11.83 11.91 12.00

(26:23) (112:99) (17:15) (25:22) (58:51) (256:225) (33:29)

212.253 213.598 216.687 221.309 222.667 222.463 223.696 225.000

12.74 12.82 13.00 13.28 13.36 13.35 13.42 13.50

12.03 12.10 12.28 12.54 12.62 12.61 12.68 12.75

(729:640) (57:50) (73:64)

225.416 226.841 227.789 229.412

13.52 13.61 13.67 13.76

12.77 12.85 12.91 13.00

73rd harmonic

(8:7)

23 : 7

231.174 233.333

13.87 14.00

13.10 13.22

septimal whole-tone, septimal major second

(63:55)

235.104

14.11

13.32

(55:48) (39:34) (225:196) (31:27) (147:128) (169:147) (23:20) (2187:1900) (38:33) (144:125) (121:105)

235.677 237.527 238.886 239.171 239.607 241.449 241.961 243.545 244.240 244.969 245.541 247.059

14.14 14.25 14.33 14.35 14.38 14.49 14.52 14.61 14.65 14.70 14.73 14.82

13.36 13.46 13.54 13.55 13.58 13.68 13.71 13.80 13.84 13.88 13.91 14.00

diminished third (6/5 x 24/25)

(15:13) (52:45) (37:32) (81:70) (125:108)

247.741 250.304 251.344 252.680 253.076

14.86 15.02 15.08 15.16 15.18

14.04 14.18 14.24 14.32 14.34

37th harmonic

(22:19) (51:44) (196:169) (29:25) (36:31) (93:80) (57:49) (64:55)

253.805 255.592 256.596 256.950 258.874 260.677 261.816 262.368 264.706

15.23 15.34 15.40 15.42 15.53 15.64 15.71 15.74 15.88

14.38 14.48 14.54 14.56 14.67 14.77 14.84 14.87 15.00

consonant interval (Avicenna)

266.667

16.00

15.11

(7:6)

7 : 23

266.871

16.01

15.12

septimal minor third, subminor third. the named interval is only approximately equal to 7:6 frequency ratio. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a 5-limit just minor third of 6:5. In the meantone era the interval m

(90:77)

270.080 272.727

16.20 16.36

15.30 15.45

(75:64)

274.582

16.47

15.56

augmented second (9/8 x 25/24)

(34:29) (88:75) (27:23) (20:17)

275.378 276.736 277.591 281.358 282.353

16.52 16.60 16.66 16.88 16.94

15.60 15.68 15.73 15.94 16.00

(33:28) (46:39) (13:11) (58:49) (45:38)

284.447 285.792 289.210 291.925 292.711

17.07 17.15 17.35 17.52 17.56

16.12 16.19 16.39 16.54 16.59

(32:27)

25 : 33

294.135

17.65

16.67

Pythagorean minor third, Pythagorean semiditone

(19:16) (21/4:1)

297.513 300.000

17.85 18.00

16.86 17.00

overtone minor third equal-tempered minor third, exact

(25:21) (31:26) (105:88) (55:46) (6:5)

23 : 5

301.847 304.508 305.777 309.357 315.641

18.11 18.27 18.35 18.56 18.94

17.10 17.26 17.33 17.53 17.89

5-limit just minor third, sesquiquintan, semiditonus

(19683:1638 4)

317.595 317.647

19.06 19.06

18.00 18.00

(77:64) (35:29) (29:24) (75:62) (98:81) (121:100) (23:19) (63:52)

320.144 325.562 327.622 329.547 329.832 330.008 330.761 332.208

19.21 19.53 19.66 19.77 19.79 19.80 19.85 19.93

18.14 18.45 18.57 18.67 18.69 18.70 18.74 18.83

77th harmonic

(40:33)

333.041 333.333

19.98 20.00

18.87 18.89

335.294

20.12

19.00

(17:14) (2:3):(81:10 0)

336.130 337.146

20.17 20.23

19.05 19.10

Epitropi FRACTIONAL diatonic diphonia Di-Bou

(243:200)

337.148

20.23

19.11

(62:51) (28:23) (39:32)

338.125 340.552 342.483 342.905 343.301 345.455

20.29 20.43 20.55 20.57 20.60 20.73

19.16 19.30 19.41 19.43 19.45 19.58

39th harmonic (major third minus a minor diesis)

(128:105) (8000:6561)

(11:9)

11 : 32

347.408 350.000

20.84 21.00

19.69 19.83

undecimal "median" third, undecimal neutral third

(60:49)

350.617

21.04

19.87

(49:40) (38:31) (27:22)

351.338 352.477 354.547

21.08 21.15 21.27

19.91 19.97 20.09

(16:13)

359.472 356.250

21.57 21.38

20.37 20.19

24 : 13(79:64) (100:81)

359.470 364.537 364.807

21.57 21.87 21.89

20.37 20.66 20.67

Tridecimal neutral third 79th harmonic

(121:98) (21:17)

364.984 365.825 366.667

21.90 21.95 22.00

20.68 20.73 20.78

(99:80) (26:21)

368.914 369.747

22.13 22.18

20.91 20.95

370.588

22.24

21.00

(57:46) (31:25) (36:29) (56:45) (96:77) (8192:6561) (5:4)

371.194 372.408 374.333 378.602 381.811 384.360 5 : 22 386.314

22.27 22.34 22.46 22.72 22.91 23.06 23.18

21.03 21.10 21.21 21.45 21.64 21.78 21.89

Pythagorean "schismatic" third 5-limit just major third, sesquiquartan, ditonus

388.235

23.29

22.00

(64:51) (49:39) (44:35) (39:31) (34:27)

393.090 395.169 396.178 397.447 399.090

23.59 23.71 23.77 23.85 23.95

22.28 22.39 22.45 22.52 22.62

(21/3:1)

400.000

24.00

22.67

equal-tempered major third, exact

(63:50) (121:96) (29:23) (125:99) (24:19) (512:405) (62:49) (81:64)

34 : 26

400.108 400.681 401.303 403.713 404.442 405.866 407.384 407.820

24.01 24.04 24.08 24.22 24.27 24.35 24.44 24.47

22.67 22.71 22.74 22.88 22.92 23.00 23.09 23.11

Pythagorean major third, Pythagorean ditone (that is two 9:8 tones)

(19:15) (33:26) (80:63) (14:11) (51:40) (125:98)

27 : 11

409.244 412.745 413.578 417.508 420.597 421.289 423.529

24.55 24.76 24.81 25.05 25.24 25.28 25.41

23.19 23.39 23.44 23.66 23.83 23.87 24.00

undecimal major third

(23:18)

424.364

25.46

24.05

(32:25)

427.373

25.64

24.22

diminished fourth

(41:32) (50:39) (77:60)

429.062 430.145 431.875 433.333

25.74 25.81 25.91 26.00

24.31 24.37 24.47 24.56

41st harmonic major third plus a minor diesis

(9:7)

32 : 7

435.084

26.11

24.65

septimal major third, supermajor third; the named interval interval is exactly or approximately equal to a 9:7 frequency ratio. In terms of cents, it is 435 cents, a quartertone of size 36/35 sharper than a just major third of 5/4. In the early meantone e

436.364

26.18

24.73

(58:45) (49:38)

439.353 440.139

26.36 26.41

24.90 24.94

441.176

26.47

25.00

(40:31) (31:24) (1323:1024) (128:99) (22:17) (57:44) (162:125)

441.278 443.081 443.517 444.772 446.363 448.150 448.879

26.48 26.58 26.61 26.69 26.78 26.89 26.93

25.01 25.11 25.13 25.20 25.29 25.40 25.44

(35:27) (83:64) (100:77) (13:10) (125:96)

449.275 450.047 452.484 454.214 456.986 458.824

26.96 27.00 27.15 27.25 27.42 27.53

25.46 25.50 25.64 25.74 25.90 26.00

83rd harmonic perfect fourth minus a minor diesis augmented third (5/4 x 25/24)

(30:23) (64:49) (98:75) (17:13)

459.994 462.348 463.069 464.428

27.60 27.74 27.78 27.87

26.07 26.20 26.24 26.32

(72:55)

466.278

27.98

26.42

(55:42) (38:29) (21:16) (46:35) (25:19)

466.851 467.936 470.781 473.135 475.114 476.471

28.01 28.08 28.25 28.39 28.51 28.59

26.45 26.52 26.68 26.81 26.92 27.00

septimal fourth

(320:243) (29:22) (675:512) (33:25) (45:34)

476.539 478.259 478.492 480.646 485.286 487.500

28.59 28.70 28.71 28.84 29.12 29.25

27.00 27.10 27.11 27.24 27.50 27.63

490.909

29.45

27.82

(85:64)

491.269

29.48

27.84

85th harmonic

494.118

29.65

28.00

(4:3)

22 : 3

498.045

29.88

28.22

harmonic fourth, just perfect fourth, sesquitertan, sesquitertium, diatessaron

(25/12:1)

500.000

30.00

28.33

equal-tempered perfect fourth, exact

(75:56) (51:38) (43:32) (121:90) (39:29) (35:26) (66:49) (31:23) (27:20)

33 : 225

505.757 509.397 511.518 512.412 512.905 514.612 515.621 516.761 519.551

30.35 30.56 30.69 30.74 30.77 30.88 30.94 31.01 31.17

28.66 28.87 28.99 29.04 29.06 29.16 29.22 29.28 29.44

43rd harmonic

5-limit wolf fourth

(177147:131 072) (23:17) (42:31) (19:14)

521.505 523.319 525.745 528.687 529.412

31.29 31.40 31.54 31.72 31.76

29.55 29.65 29.79 29.96 30.00

(110:81) (87:64) (34:25)

529.812 531.532 532.328 533.333

31.79 31.89 31.94 32.00

30.02 30.12 30.17 30.22

87th harmonic

(49:36) (15:11) (512:375) (27:16)/(10 0:81)

533.742 536.951 539.104 541.058

32.02 32.22 32.35 32.46

30.25 30.43 30.55 30.66

Epitropi FRACTIONAL triphonia Bou-Ke

(160/117) (26:19) (63:46) (48:35)

541.900 543.015 544.462 546.815 547.059

32.51 32.58 32.67 32.81 32.82

30.71 30.77 30.85 30.99 31.00

perfect fourth plus a minor diesis

(1000:729)

547.211

32.83

31.01

550.000

33.00

31.17

(11:8)

11 : 23

551.318

33.08

31.24

undecimal tritone (11th harmonic)

(62:45) (40:29)

554.812 556.737 558.457

33.29 33.40 33.51

31.44 31.55 31.65

(29:21) (112:81) (18:13)

558.796 561.006 563.382 563.636

33.53 33.66 33.80 33.82

31.67 31.79 31.92 31.94

566.667

34.00

32.11

(25:18)

568.717

34.12

32.23

augmented fourth (4/3 x 25/24)

(89:64) (32:23) (39:28) (46:33) (88:63)

570.880 571.726 573.657 575.001 578.582 581.250

34.25 34.30 34.42 34.50 34.71 34.88

32.35 32.40 32.51 32.58 32.79 32.94

89th harmonic

581.818

34.91

32.97

582.353

34.94

33.00

(7:5) (108:77) (1024:729)

7:5

582.512 585.721 588.270

34.95 35.14 35.30

33.01 33.19 33.34

lesser septimal tritone low Pythagorean tritone

(45:32)

590.224

35.41

33.45

high 5-limit tritone (major third plus 9/8 whole tone)

(38:27) (31:22) (55:39) (24:17) (21/2:1)

591.648 593.718 595.149 597.000 600.000

35.50 35.62 35.71 35.82 36.00

33.53 33.64 33.73 33.83 34.00

equal-tempered tritone, exact

(99:70) (17:12) (44:31) (125:88) (27:19) (91:64) (64:45) (729:512)

600.088 603.000 606.282 607.623 608.352 609.354 609.776 611.730

36.01 36.18 36.38 36.46 36.50 36.56 36.59 36.70

34.00 34.17 34.36 34.43 34.47 34.53 34.55 34.66

91st harmonic low 5-limit tritone high Pythagorean tritone, Tritonus (German)

(57:40) (77:54) (10:7)

25 : 7

613.154 614.279 617.488 617.647

36.79 36.86 37.05 37.06

34.75 34.81 34.99 35.00

greater septimal tritone

(63:44) (33:23) (56:39)

621.418 624.999 626.343

37.29 37.50 37.58

35.21 35.42 35.49

(23:16) (36:25)

628.274 631.283

37.70 37.88

35.60 35.77

23rd harmonic diminished fifth (3/2 x 24/25)

(121:84) (49:34)

631.855 632.696 633.333

37.91 37.96 38.00

35.81 35.85 35.89

(75:52)

634.100 635.294

38.05 38.12

35.93 36.00

diminished fifth (tritone plus a minor diesis)

(13:9) (81:56) (55:38) (42:29)

636.618 638.994 640.119 641.204 641.543

38.20 38.34 38.41 38.47 38.49

36.08 36.21 36.27 36.33 36.35

(29:20) (45:31) (93:64)

643.263 645.188 646.991

38.60 38.71 38.82

36.45 36.56 36.66

93rd harmonic

(16:11)

24 : 11

648.682

38.92

36.76

inversion of eleventh harmonic

(51:35) (729:500)

651.771 652.789 652.941

39.11 39.17 39.18

36.93 36.99 37.00

(35:24) (19:13) (117:80) (375:256) (22:15) (47:32) (72:49)

653.185 656.985 658.100 660.896 663.049 665.507 666.258 666.667

39.19 39.42 39.49 39.65 39.78 39.93 39.98 40.00

37.01 37.23 37.29 37.45 37.57 37.71 37.75 37.78

perfect fifth minus a minor diesis

47th harmonic

(25:17) (81:55)

667.672 670.188 670.588

40.06 40.21 40.24

37.83 37.98 38.00

(28:19) (31:21)

671.313 674.295

40.28 40.46

38.04 38.21

(189:128) (34:23)

674.691 676.681

40.48 40.60

38.23 38.35

(262144:177 147)

678.495

40.71

38.45

wolf fifth, the interval between G# and Ed in early Pythagorean chromatic tuning, a fifth containing (i.e. narrowed by) the whole of a Pythagorean or ditonic comma (531441:524288) (dissonant) 5-limit wolf fifth or diminished sixth 95th harmonic

(40:27) (46:31) (95:64) (49:33) (52:35) (58:39)

235 : 33

680.449 683.239 683.827 684.379 685.388 687.095

40.83 40.99 41.03 41.06 41.12 41.23 0.00

38.56 38.72 38.75 38.78 38.84 38.94 0.00

(125:84) (112:75) (121:81)

688.160 694.243 694.816

41.29 41.65 41.69

39.00 39.34 39.37

(27/12:1)

700.000

42.00

39.67

equal-tempered perfect fifth, exact

(3:2)

3:2

701.955

42.12

39.78

harmonic fifth, just perfect fifth, trihemitone, Pythagorean perfect fifth or diapente, sesquialterum, 3rd harmonic,

705.882

0.00 0.00 42.35

0.00 0.00 40.00

712.500

42.75

40.38

(121:80) (50:33) (97:64) (1024:675) (44:29) (243:160)

716.322 719.354 719.895 721.508 721.741 723.461 723.529

42.98 43.16 43.19 43.29 43.30 43.41 43.41

40.59 40.76 40.79 40.89 40.90 41.00 41.00

97th harmonic

(38:25) (35:23) (32:21) (29:19) (84:55) (55:36) (26:17) (75:49) (49:32) (23:15) (192:125) (20:13) (77:50) (54:35) (125:81) (17:11) (99:64) (48:31) (31:20) (45:29) (14:9) (120:77) (39:25)

724.886 726.865 729.219 732.064 733.149 733.722 735.572 736.931 737.652 740.006 743.014 745.786 747.516 750.725 751.121 753.637 755.228 756.919 758.722 760.647 764.916 768.125 769.855

43.49 43.61 43.75 43.92 43.99 44.02 44.13 44.22 44.26 44.40 44.58 44.75 44.85 45.04 45.07 45.22 45.31 45.42 45.52 45.64 45.89 46.09 46.19

41.08 41.19 41.32 41.48 41.55 41.58 41.68 41.76 41.80 41.93 42.10 42.26 42.36 42.54 42.56 42.71 42.80 42.89 42.99 43.10 43.35 43.53 43.63

49th harmonic diminished sixth (8/5 x 24/25) perfect fifth plus a minor diesis

99th harmonic

27 : 32

septimal minor sixth minor sixth minus a minor diesis

(25:16)

772.627

46.36

43.78

augmented fifth

(36:23) (11:7) (63:40) (52:33) (101:64) (30:19) (128:81) (49:31) (405:256) (19:12) (46:29) (100:63) (22/3:1)

11 : 7

27 : 34

775.636 782.492 786.422 787.255 789.854 790.756 792.180 792.616 794.134 795.558 798.697 799.892 800.000

46.54 46.95 47.19 47.24 47.39 47.45 47.53 47.56 47.65 47.73 47.92 47.99 48.00

43.95 44.34 44.56 44.61 44.76 44.81 44.89 44.91 45.00 45.08 45.26 45.33 45.33

undecimal minor sixth

101st harmonic Pythagorean minor sixth

equal-tempered minor sixth, exact

(27:17) (62:39) (35:22) (51:32) (8:5)3

2 :5

800.910 802.553 803.822 806.910 813.686 815.640

48.05 48.15 48.23 48.41 48.82 48.94

45.38 45.48 45.55 45.72 46.11 46.22

51st harmonic 5-limit just minor sixth (inversion of 5-limit just major third) Pythagorean "schismatic" sixth

(6561:4096)

(77:48) (45:28) (103:64) (29:18) (50:31) (121:75) (21:13) (55:34) (34:21) (81:50) (125:77) (13:8) (57:35) (44:27) (31:19) (80:49) (49:30) (18:11)

13 : 23

818.189 821.398 823.801 825.667 827.592 828.053 830.253 832.676 834.175 835.193 838.797 840.528 844.328 845.453 847.523 848.662 849.383 852.592

49.09 49.28 49.43 49.54 49.66 49.68 49.82 49.96 50.05 50.11 50.33 50.43 50.66 50.73 50.85 50.92 50.96 51.16

46.36 46.55 46.68 46.79 46.90 46.92 47.05 47.18 47.27 47.33 47.53 47.63 47.85 47.91 48.03 48.09 48.13 48.31

103rd harmonic

overtone sixth

232 : 11

undecimal "median" sixth, undecimal neutral sixth

854.745

51.28

48.44

(105:64) (64:39) (23:14) (51:31) (400:243) (28:17)

857.095 857.517 859.448 861.875 862.852 863.870 866.667

51.43 51.45 51.57 51.71 51.77 51.83 52.00

48.57 48.59 48.70 48.84 48.89 48.95 49.11

105th harmonic minor sixth plus a minor diesis

(33:20) (38:23) (81:49) (48:29) (53:32) (58:35) (63:38) (128:77)

866.959 869.239 870.168 872.378 873.505 874.438 875.223 879.856 882.353

52.02 52.15 52.21 52.34 52.41 52.47 52.51 52.79 52.94

49.13 49.26 49.31 49.43 49.50 49.55 49.60 49.86 50.00

53rd harmonic

(5:3)

5:3

884.359 889.760 894.513 895.492 898.153 898.726 900.000 902.487 905.865

53.06 53.39 53.67 53.73 53.89 53.92 54.00 54.15 54.35

50.11 50.42 50.69 50.74 50.90 50.93 51.00 51.14 51.33

(107:64) (57:34) (52:31) (42:25) (121:72) (23/4:1) (32:19) (27:16)

5-limit just major sixth (inversion of a 5-limit just minor third) 107th harmonic

equal-tempered major sixth, exact Pythagorean major sixth

33 : 24

(49:29) (22:13) (39:23) (56:33) (17:10) (109:64) (46:27) (75:44) (29:17) (128:75)

908.075 910.790 914.208 915.553 918.642 921.821 922.409 923.264 924.622 925.418

54.48 54.65 54.85 54.93 55.12 55.31 55.34 55.40 55.48 55.53

51.46 51.61 51.81 51.88 52.06 52.24 52.27 52.32 52.40 52.44

109th harmonic

diminished seventh (16/9 x 24/25)

(77:45) (12:7)

2 3 : 7

2

929.920 933.129 937.632 941.126 941.562 943.050 946.195 946.924 947.496 949.696 952.259 953.299 955.031 955.760 958.039 960.829 964.323 964.896 968.826

55.80 55.99 56.26 56.47 56.49 56.58 56.77 56.82 56.85 56.98 57.14 57.20 57.30 57.35 57.48 57.65 57.86 57.89 58.13

52.70 52.88 53.13 53.33 53.36 53.44 53.62 53.66 53.69 53.82 53.96 54.02 54.12 54.16 54.29 54.45 54.64 54.68 54.90

(55:32) (31:18) (441:256) (50:29) (19:11) (216:125) (121:70) (45:26) (26:15) (111:64) (125:72) (33:19) (40:23) (54:31) (96:55) (110:63) (7:4)

septimal major sixth (inversion of a minimal third) 55th harmonic

111th harmonic augmented sixth (5/3 x 25/24)

7 : 22

septimal minor seventh, harmonic seventh, (inversion of major tone). [some assert that 7:4 is one of the blue notes used in jazz]

(58:33) (225:128) (51:29) (44:25) (30:17) (113:64) (99:56) (23:13) (62:35) (39:22) (55:31) (16:9)

24 : 32

976.304 976.537 977.333 978.691 983.313 984.215 986.402 987.747 989.896 991.165 992.596 996.090

58.58 58.59 58.64 58.72 59.00 59.05 59.18 59.26 59.39 59.47 59.56 59.77

55.32 55.34 55.38 55.46 55.72 55.77 55.90 55.97 56.09 56.17 56.25 56.45

225th harmonic

113th harmonic

Pythagorean small minor seventh, lesser just minor seventh, dominant seventh 57th harmonic equal-tempered minor seventh

(57:32) (25/6

:1)

999.468 1,000.000 1,000.020 1,003.802 1,007.442 1,010.950 1,013.666 1,014.588

59.97 60.00 60.00 60.23 60.45 60.66 60.82 60.88

56.64 56.67 56.67 56.88 57.09 57.29 57.44 57.49

(98:55) (25:14) (34:19) (52:29) (88:49) (115:64)

115th harmonic

(9:5)

32 : 5

1,017.596

61.06

57.66

5-limit large minor seventh, greater just minor seventh, tonic seventh

(59049:3276 8) (56:31) (38:21) (29:16) (49:27) (20:11) (51:28) (729:400) (31:17) (42:23) (117:64) (64:35) (4000:2187) (11:6)

1,019.550 1,023.790 1,026.732 1,029.577 1,031.787 1,034.996 1,038.085 1,039.103 1,040.080 1,042.507 1,044.438 1,044.860 1,045.256 11 : 23 1,049.363

61.17 61.43 61.60 61.77 61.91 62.10 62.29 62.35 62.40 62.55 62.67 62.69 62.72 62.96

57.77 58.01 58.18 58.34 58.47 58.65 58.82 58.88 58.94 59.08 59.18 59.21 59.23 59.46

29th harmonic lesser undecimal neutral seventh

225 : 11

117th harmonic

undecimal "median" seventh, greater undecimal neutral seventh

(90:49) (57:31) (46:25) (81:44) (35:19) (59:32) (24:13) (50:27) (63:34) (13:7) (119:64) (54:29) (28:15) (58:31) (15:8)

1,052.572 1,054.432 1,055.647 1,056.502 1,057.627 1,059.172 1,061.427 1,066.762 1,067.780 1,071.702 1,073.781 1,076.288 1,080.557 1,084.542 1,088.269

63.15 63.27 63.34 63.39 63.46 63.55 63.69 64.01 64.07 64.30 64.43 64.58 64.83 65.07 65.30

59.65 59.75 59.82 59.87 59.93 60.02 60.15 60.45 60.51 60.73 60.85 60.99 61.23 61.46 61.67

59th harmonic inversion of minor tone

inversion of major semitone 119th harmonic

35 : 23

5-limit just major seventh

(62:33) (32:17) (49:26) (66:35) (211/12:1) (17:9) (121:64) (125:66) (36:19) (256:135)

1,091.763 1,095.045 1,097.124 1,098.133 1,100.000 1,101.045 1,102.636 1,105.668 1,106.397 1,107.821

65.51 65.70 65.83 65.89 66.00 66.06 66.16 66.34 66.38 66.47

61.87 62.05 62.17 62.23 62.33 62.39 62.48 62.65 62.70 62.78

equal-tempered major seventh, exact 121st harmonic

(55:29) (243:128) (19:10) (40:21) (61:32) (21:11) (44:23) (23:12) (48:25) (121:63) (123:64) (25:13) (77:40) (52:27) (27:14) (56:29) (29:15) (60:31) (31:16) (64:33) (33:17) (243:125) (39:20) (125:64) (88:45) (45:23) (96:49) (49:25) (51:26) (108:55) (55:28) (57:29) (63:32) (160:81) (99:50) (125:63) (127:64) (2:1)

3 :2

5

7

1,108.054 1,109.775 1,111.199 1,115.533 1,116.885 1,119.463 1,123.044 1,126.319 1,129.328 1,129.900 1,131.017 1,132.100 1,133.830 1,134.663 1,137.039 1,139.249 1,141.308 1,143.233 1,145.036 1,146.727 1,148.318 1,150.834 1,156.169 1,158.941 1,161.094 1,161.991 1,164.303 1,165.024 1,166.383 1,168.233 1,168.806 1,169.891 1,172.736 1,178.494 1,182.601 1,186.205 1,186.422 1,200.000

66.48 66.59 66.67 66.93 67.01 67.17 67.38 67.58 67.76 67.79 67.86 67.93 68.03 68.08 68.22 68.35 68.48 68.59 68.70 68.80 68.90 69.05 69.37 69.54 69.67 69.72 69.86 69.90 69.98 70.09 70.13 70.19 70.36 70.71 70.96 71.17 71.19 72.00

62.79 62.89 62.97 63.21 63.29 63.44 63.64 63.82 64.00 64.03 64.09 64.15 64.25 64.30 64.43 64.56 64.67 64.78 64.89 64.98 65.07 65.21 65.52 65.67 65.80 65.85 65.98 66.02 66.10 66.20 66.23 66.29 66.46 66.78 67.01 67.22 67.23 68.00

Pythagorean major seventh

inversion of minor semitone/major diesis 61st harmonic

diminished octave 123rd harmonic

septimal major seventh

31st harmonic

inversion of minor diesis augmented seventh (15/8 x 25/24)

63rd harmonic

127th harmonic octave, diapason, 2nd harmonic (which normalizes to 1:1, i.e. the 1st harmonic or fundamental)

2:1

ABOVE octave (32:15) (9:4) (7:3) (12:5) (5:2)

0.00 1,311.731 1,403.910 1,466.871 1,515.641 1,586.314 78.70 84.23 88.01 90.94 95.18

0.00 74.33 79.55 83.12 85.89 89.89minor ninth major ninth harmonic minor tenth minor ninth major ninth

(8:3) (11:4) (45:16) (3:1) (25:8) (16:5) (13:4) (10:3) (7:2) (32:9) (18:5) (15:4) (4:1)

1,698.045 1,751.318 1,790.224 1,901.955 1,972.627 2,013.686 2,040.528 2,084.359 2,168.826 2,196.090 2,217.596 2,288.269 2,400.000

101.88 105.08 107.41 114.12 118.36 120.82 122.43 125.06 130.13 131.77 133.06 137.30 144.00

96.22 99.24 101.45 107.78 111.78 114.11 115.63 118.11 122.90 124.45 125.66 129.67 136.00

perfect eleventh harmonic eleventh augmented eleventh just perfect twelfth, tritave augmented twelfth minor thirteenth harmonic thirteenth major thirteenth harmonic fourteenth dominant thirteenth tonic fourteenth major fourteenth double octave, fifteenth

Chrysanthos Chrysanth Chrysant Epitropi FRACTIONS os (68 ET) hos 1881 R FRACTION (correcte S d 66 ET, UNcorrec ted 64 or 68 ET chromati c)

Epitropi 1881 (72 ET)

Didymos EL OTHER Pangratio FRACTIO FARABI (Euthemi s (24 ET) NS FRACTIO ades, NS Debrelis, Papadimi triou)

Pangratio s ET quarter tone

Epitropi enharmonic Ke Zo (6/72)

Pangratio s 24 ET semi tone = 2/24

Chrysanth os Chroa 6/68

Didymos FR just diatonic semi TONE = (16/15)

Euthem/D ebr/Papa dim chromati c Neanes

Chrysanth os 68 ET chromatic neanes 7/68

Chrysant hos corrected 66 ET chromati c neanes 7/66 Epitropi FR (27/25) Papadim. chromati c Neanes

Epitropi 72 ET diatonic (8/72), chroa spathi diphonia

Chrysanthos FR diatonic 88/81

El Farabi 88/81

Pangratio s ET 3/4 tone = 3/24 Chysanthos FR diatonic 12/11 El Farabi diatonic 12/11

Chrysanth os 68 ET diatonic (9/68), chroa (S,Z,K) diphonia

Epitropi FR minor TONE = (800/729)

Epitropi 72 ET 10/72

Didymos FR minor TONE = (10/9)

Epitropi 72 ET TONE = 12/72

Pangratio s ET whole TONE = 4/24 Didymos EL Farabi Euthem/D FR TONE FR TONE ebr/Papa dim = = = (9/8) (9/8) (9/8)

Chrysanthos FR TONE = (9/8)

Epitropi FR TONE = (9/8)

Chrysanth Chrysant os 68 ET hos TONE UNcorrec = ted (12/68) chromati c = (12/68)

Chrysant hos UNcorrec ted 64 ET chromati c 7/64

Chrysanth os 68 ET enharmoni c tone 13/68 Epitropi 72 ET chromatic 14/72

Chrysanth os 68 ET corrected chromatic 14/68

Papadim. chromati c neanes (125/108)

Chrysanth os ET diphonia chromatic necheanes , enharmoni c

Epitropi 72 ET Zygos (16/72), diphonia diatonic, chromatic necheanes, chroa Zygos, Kliton

Chrysant hos corrected 66 ET diphonia Epitropi FR chromatic necheanes diphonia Euthem/D ebr Chromati c neanes (75/64)

Chrysanth os 68 ETdiphoni a diatonic, enharmoni c chroa spatheion

Chrysanthos FR diphonia diatonic

Epitropi FR diphonia diatonic, enharmonic , chromatic necheanes

Didymos El Farabi FR FR diphonia diphonia diatonic diatonic

Epitropi 72 ET chroa zygos kliton (18/72), diphonia [diatonic, chromatic necheanes, enharmonic ]

Pangratio s 24 ET [diphonia chromati c]

Didymos FR diphonia (diatonic)

Euthem/D ebrel/Pap adim Diphonia (chromati c neanes, necheane s)

Chrysanth os 68 ET chroa spatheion 18/68, (diphonic Bou-Di 5th)

Chrysanth os 68 ET [diphonia diatonic, chromatic neanes] Epitropi FR diphonia [diatonic (2/3):(81/10 0)] Epitropi FR chromatic necheanes (243/200)

Epitropi 72 ET chromatic Necheanes 20/72, diphonia [diatonic, chromatic neanes, chroa Kliton, Zygos]

Chrysant hos 66 ET corrected diphonia

Pangratio s 24 ET [diphonia diatonic}

Chrysanthos FR [diphonia diatonic]

El Farabi FR [diphonia diatonic]

Chrysant hos 64 ET UNcorrec ted diphonia

Epitropi FR [diphonia diatonic], [triphonia chromatic neanes]

Papadim FR [diphonia chromati c neanes]

Epitropi 72 ET [diphonia diatonic, chromatic neanes, chroa zygos],[trip honia diatonic, chromatic neanes]

Chrysanth os 68 ET [diphonia diatonic, chromatic neanes, necheanes , chroa zygos, kliton], diaphonic 4th spatheion

Didymos FR [diphonia diatonic 5/4 ]

Euthem, Deber,Pa padim [diphonia chromati c neanes, necheane s 5/4 ]

Chrysanth os 68 ET chroa Zygos (22/68),[tri phonia diatonic, chromatic neanes]

Epitropi [diphonia enharmonic , chromatic neanes, spatheion], [triphonia chroa zygos, kliton]

Chrysanthos FR, [diphonia diatonic]

Epitropi FR, [diphonia diatonic, chromatic neacheanes , enharmonic ]

Chrysanth os 68 ET [diphonai diatonic, enarmonic , spatheion]

Epitropi FR [diphonia chromatic necheanes]

Euthem, Deber, [diphonic 4th], [triphonia chromati c neanes, necheane s]

Epitropi 72 ET [diphonia chromatic necheanes, chroa spatheion, kliton], [triphonia chroa zygos, kliton]

Chrysant hos 66 ET corrected [diphonia ]

Chrysanth os 68 ET [diphonia chromatic necheanes , enharmoni c, chroa spatheion, zygos, kliton], [triphonia chroa kliton]

Euthem Debrelis, Papadim diphonic 4th, [triphonia chromati c neanes]

Chrysanth os 68 ET diphonic 4th, [triphonia cromatic neanes, chroa zygos]

Epitropi 72 ET diphonic 4th [triphonia chromatic neanes, chroa spatheion, kliton]

Chrysanth os 68 ET diphonic 4th, [triphonia enharmoni c, chroa spatheion, kliton]

Chrysant hos 66 corrected diphonic 4th, [triphonia chromati c neanes]

Chrysant hos 66 corrected [triphonia diatonic]

Chrysanthos FR [triphonia diatonic]

Chrysanth os 68 ET [triphonia diatonic, chromatic neanes, necheanes , enharmoni c, chroa spatheion, kliton]

Epitropi FR diatonic diphonic 4th 4/3 [triphonia diatonic, chromatic neanes, enharmonic ]

Didymos El Farabi FR FR diatonic [triphonia diphonic diatonic] 4th 4/3, [triphonia diatonic]

Euthem, Debrelis, Papadim, FR [triphonia diatonic]

Eptiropi 72 ET diphonic 4th, [triphonia diatonic, chromatic neanes, necheanes, enharmonic , chroa spatheion]

Pangratio s 24 ET [diphonic 4th, triphonia chromati c}

Didymos FR diphonic 4th, [triphonia diatonic]

Euthem, Debrel, Papadim, diphonic 4th

Chrysanth os 68 ET diphonic 5th

Epitropi 72 ET diphonic diatonic 4th, diphonic Kliton 5th, [triphonia diatonic, chroa zygos]

Epitropi FR diphonic 4th (27:16)/(100 :81), [triphonia diatonic]

Chrysanth os 68 ET diphonic 4th, [triphonia diatonic, chromatic neanes, chroa zygos]

Chrysant hos corrected 66 ET[tripho nia chromati c neanes]

Pangratio s 24 ET [diphonic 4th}, [triphonia diatonic] Chrysanthos FR diphonic 4th, [triphonia diatonic] El Farabi FR diphonic 4th

Chrysanthos FR [triphonia diatonic]

Chrysant hos corrected 66 ET diphonic 4th, [triphonia diatonic] Epitropi 72 ET diphonic 4th, [triphonia diatonic, chromatic neanes, chroa spatheion], [tetraphonia chroa spatheion, zygos, kliton]

Epitropi FR [triphonia diatonic]

Euthemia d, Debrel, Papadim FR [triphonia diatonic]

Chrysant hos corrected 66 ET [triphonia chromati c neanes]

Chrysant hos corrected 66 ET [triphonia diatonic] Chrysanth os 68 ET [triphonia diatonic, chromatic neanes, necheanes , chroa spatheion] Chrysant hos corrected 66 ET [triphonia chromati c neanes]

Epitropi FR diphonic 5th

Didymos FR [triphonia chromati c neanes, necheane s]

Euthem, Debr, Papadim diphonic 4th, [triphonia chromati c neanes, necheane s]

Chrysanth os 68 ET [tetraphoni a chroa spatheion]

Epitropi 72 ET diphonic 5th, [triphonia chromatic neanes, enharmonic ]

Epitropi FR diphonic 4th, [triphonia chromatic neanes, enharmonic ]

Chrysanth os 68 ET (triphonia chroa zygos]

Epitropi FR diphonic 4th, [triphonia chromatic neanes]

Didymos FR diphonic 5th

Euthem, Debr, Papadim diphonic 5th

Epitropi 72 ET diphonic 4th, [triphonia chromatic neanes, chroa zygos, kliton], [tetraphonia chroa spatheion, kliton]

Chrysanth os 68 ET [triphonia chroa kliton], [tetraphoni a chroa spatheion, kliton]

EL Farabi FR [tetrapho nia diatonic]

EL Farabi FR [tetrapho nia diatonic]

Chrysanth os 68 ET diphonic 4th, [triphonia chromatic necheanes ]

Epitropi 72 ET diphonic 5th

Chrysanth os 68 ET diphonic 5th, [tetraphoni a chroa zygos]

Chrysant hos corrected 66 ET [tetrapho nia chromati c neanes]


Chrysant hos corrected diphonic 5th

Chrysanthos FR diphonic 5th Epitropi 72 ET [tetraphonia diatonic, chromatic, enharmonic, chroa] Chrysanthos FR {tetraphonia diatonic] Epitropi FR 3/2 (Ni-Di)d, (Bou-Zo)d, [tetraphoni a diatonic, chromatic, enharmonic , chroa spatheion]

El Farabi FR diphonic 5th

Didymos El Farabi FR 3/2 (NiFR Di)d, {tetrapho {tetrapho nia nia diatonic] diatonic]

Euthemia d, Debrel Papadim FR [tetrapho nia chromati c neanes, necheane s]

Chrysanth os 68 ET [tetraphoni a ALL]

Chrysant hos corrected 66 ET diphonic 5th, [tetrapho nia chromati c neanes]

Chrysanth os 68 ET, [tetraphoni a enharmoni c]

Euthem, Debrel, Papadim diphonic 5th





Chrysanth os 68 ET diphonic 5th

Epitropi FR 27/16 (NiKe)d

Didymos FR 27/16 (Ni-Ke)d

Didymos FR 27/16 (Ni-Zo)d

68 ET

RATIO

Chord fraction equivalent of ET

68

Chord distances

via 68 ET 3-digit fraction Ni 2-digit fraction 1-digit fraction 1000 mm

Pa

Bou

Ga

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

68

1 1488/493 437/446 161/166 817/851 153/161 111/118 649/697 153/166 510/559 643/712 59/66 0.97982

198/99 97/99 32/33 24/25 19/20 16/17 27/29 47/51 52/57 28/31 59/66

11 1 1 1 1 1 1 1 1 1

1000989.8584 979.8197 969.8828 960.0467 950.3103 940.6727 931.1328 921.6896 912.3423 903.0897 8/9 893.9309

68 0.989858 68

68 0.969883 68 0.960047 68 0.95031

68 0.940673 68 0.931133 68 0.92169

68 0.912342 68 0.90309

68 0.893931 68

0.88487

684/773614/701 339/391 115/134 175/206 37/44 144/173 365/443 199/244

23/267/8 13/15 6/7 79/93 37/44 5/6 14/17 31/38

8/9 884.8657/8 875.8912 7/8 867.0083 6/7 858.2154 6/7 849.5118 5/6 840.8964 5/6 832.3684 5/6 823.9269 4/5 815.571

68 0.875891 68 0.867008 68 0.858215 68 0.849512 68 0.840896 68 0.832368 68 0.823927 68 0.815571 68

0.8073

553/685720/901 651/823 451/576 441/569 346/451 101/133

67/834/5 53/67 18/23 31/40 56/73 60/79

4/5

807.3

68 0.799113 68 0.791008 68 0.782986 68 0.775046 68 0.767185 68 0.759405 68 68

4/5 799.1125 4/5 791.0083 7/9 782.9862 7/9 775.0455 3/4 767.1853 3/4 759.4049

0.75170.74408

221/294157/211 629/854

3/432/43 14/19

3/4 751.7033/4 744.0799 3/4 736.5337

68 0.736534

Di

Ke

Zo

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

68 0.729064 68 0.72167

148/203 363/503 5/7 408/577 7/10 503/726 203/296 427/629 549/817

35/48 70/97 5/7 70/99 7/10 9/13 24/35 19/28 43/64

3/4 729.0641 5/7 721.6703 5/7 714.3514 5/7 707.1068 2/3 699.9356 2/3 692.8372 2/3 685.8107 2/3 678.8555 2/3 671.9709

68 0.714351 68 0.707107 68 0.699936 68 0.692837 68 0.685811 68 0.678856 68 0.671971 68 68

0.665160.65841

147/221638/969 451/692 549/851 288/451 189/299 112/179 589/951 122/199 443/730 173/288 22/37

2/327/41 58/89 20/31 53/83 55/87 5/8 13/21 19/31 17/28 3/5 22/37

2/3 665.1562/3 658.4103 2/3 651.733

68 0.651733 68 0.645123 68 0.638581 68 0.632105 68 0.625694 68 0.619349 68 0.613067 68 0.60685

2/3 645.1234 2/3 638.5808 5/8 632.1046 5/8 625.6941 5/8 619.3486 3/5 613.0674 3/5 606.85

68 0.600696 68 0.594604 68

3/5 600.6956 3/5 594.6036

0.58857

103/17567/115 391/678 141/247 152/269 33/59 356/643 154/281 83/153

10/177/12 15/26 4/7 13/23 33/59 31/56 40/73 32/59

3/5 588.5734/7 582.6043 4/7 576.6958 4/7 570.8472 4/7 565.0579 5/9 559.3273 5/9 553.6549 5/9 548.0399 1/2 542.482

68 0.582604 68 0.576696 68 0.570847 68 0.565058 68 0.559327 68 0.553655 68 0.54804

68 0.542482 68

0.53698

530/98759/111 161/306 438/841 83/161 223/437

29/5442/79 10/19 25/48 50/97 25/49

1/2

536.98

68 0.531535 68 0.526144 68 0.520808 68 0.515526 68 0.510298

1/2 531.5345 1/2 526.1439 1/2 520.808

1/2 515.5262 1/2 510.298

Ni

67 68

68 0.505123 68

493/976

49/97

1/2 505.1227

0.5

1/2

1/2

1/2

500

Chord distances

Chord fractions

Chord distances

EPITROPI Chord distances via 72 ET 1000 mm 650 mm

via 68 ET 650 mm

(via Chrysanthos' fractions) 1000 mm 650 mm

650643.408 636.8828 630.4238 624.0303 617.7017 611.4372 605.2363 599.0983 593.0225 587.0083 581.0551

1

1,000.00

650.00

1000 990.419 980.93 971.532 962.224 953.005 943.874 934.831 925.875 917.004 908.218 899.517 882.363 873.909 865.537 857.244 849.031 840.896 832.84 824.861

650 643.772 637.605 631.496 625.445 619.453 613.518 607.64 601.819 596.053 590.342 584.686 573.536 568.041 562.599 557.209 551.87 546.583 541.346 536.159

575.162569.3293 563.5554 557.84 552.1827 546.5827 541.0395 535.5525 530.1211

8/9

888.89

577.78

890.899 579.084

524.745519.4231 514.1554 508.941 503.7796 498.6705 493.6132

22/27

814.81

529.63

816.958 531.023 809.131 525.935 801.378 793.701 786.096 778.565 771.105 520.896 515.905 510.963 506.067 501.219 491.66 486.95

488.607483.6519 478.7469

3/4

750.00

487.50

763.718 496.416 756.401 749.154

473.8917 469.0857 464.3284 459.6194 454.9581 450.3442 445.777 441.2561 436.7811

741.976 482.284 734.867 477.664 727.827 720.853 713.947 707.107 700.332 693.622 686.977 2/3 666.67 433.33 473.087 468.555 464.066 459.619 455.216 450.855 446.535

432.351427.9667 423.6264 419.3302 415.0775 410.868 406.7012 402.5766 398.4938 394.4525 390.4521 386.4923

680.395 442.257 673.876 438.02 667.42 433.823 661.025 429.667 654.692 648.42 642.207 636.054 629.961 623.925 617.947 612.027 425.55 421.473 417.435 413.435 409.474 405.551 401.666 397.817

382.573378.6928 374.8523 371.0507 367.2876 363.5628 359.8757 356.226 352.6133

16/27

592.59

385.19

606.163 394.006 600.355 390.231 594.604 386.492 588.907 382.789 583.265 379.122 577.676 375.49 572.142 371.892 566.66 368.329 561.231 364.8

349.037345.4974 341.9936 338.5252 335.092 331.6937

44/81

543.21

353.09

555.854 361.305 550.528 545.254 540.03 534.856 357.843 354.415 351.019 347.656

529.732 344.326

328.3298

524.656 341.027 1/2 500.00 325.00 519.63 337.759 514.651 334.523 509.72 331.318 504.837 328.144 500 325

325

RATIO

Epitropi fraction equivalent of 72 ET

72 ET

3-digit fraction

2-digit fraction

1-digit fraction

1 0.99042 0.98093 0.97153 0.96222 0.953 0.94387 0.93483 0.92587 0.917 0.90822 0.89952 0.8909 0.88236 0.87391 0.86554 0.85724 0.84903 0.8409 0.83284 0.82486

1 0.99042 0.98093 0.97153 0.96222 0.953 0.94387 0.93483 0.92587 0.917 0.90822 0.89952 0.8909 0.88236 0.87391 0.86554 0.85724 0.84903 0.8409 0.83284 0.82486

1 0.99042 0.98093 0.97153 0.96222 0.953 0.94387 0.93483 0.92587 0.917 0.90822 0.89952 0.8909 0.88236 0.87391 0.86554 0.85724 0.84903 0.8409 0.83284 0.82486

1 0.99042 0.98093 0.97153 0.96222 0.953 0.94387 0.93483 0.92587 0.917 0.90822 0.89952 0.8909 0.88236 0.87391 0.86554 0.85724 0.84903 0.8409 0.83284 0.82486

0.81696 0.81696 0.81696 0.81696 0.80913 0.80913 0.80913 0.80913 0.80138 0.80138 0.80138 0.80138 0.7937 0.7937 0.7937 0.7937 0.7861 0.7861 0.7861 0.7861 0.77856 0.77856 0.77856 0.77856 0.77111 0.77111 0.77111 0.77111 0.76372 0.76372 0.76372 0.76372 0.7564 0.7564 0.7564 0.7564 0.74915 0.74915 0.74915 0.74915

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72

0.74198 0.74198 0.74198 0.74198 0.73487 0.73487 0.73487 0.73487 0.72783 0.72085 0.71395 0.70711 0.70033 0.69362 0.68698 0.6804 0.72783 0.72085 0.71395 0.70711 0.70033 0.69362 0.68698 0.6804 0.72783 0.72085 0.71395 0.70711 0.70033 0.69362 0.68698 0.6804 0.72783 0.72085 0.71395 0.70711 0.70033 0.69362 0.68698 0.6804

0.67388 0.67388 0.67388 0.67388 0.66742 0.66742 0.66742 0.66742 0.66103 0.66103 0.66103 0.66103 0.65469 0.65469 0.65469 0.65469 0.64842 0.64221 0.63605 0.62996 0.62392 0.61795 0.61203 0.64842 0.64221 0.63605 0.62996 0.62392 0.61795 0.61203 0.64842 0.64221 0.63605 0.62996 0.62392 0.61795 0.61203 0.64842 0.64221 0.63605 0.62996 0.62392 0.61795 0.61203

0.60616 0.60616 0.60616 0.60616 0.60036 0.60036 0.60036 0.60036 0.5946 0.5946 0.5946 0.5946 0.58891 0.58891 0.58891 0.58891 0.58326 0.58326 0.58326 0.58326 0.57768 0.57214 0.56666 0.56123 0.55053 0.54525 0.54003 0.53486 0.57768 0.57214 0.56666 0.56123 0.55053 0.54525 0.54003 0.53486 0.57768 0.57214 0.56666 0.56123 0.55053 0.54525 0.54003 0.53486 0.57768 0.57214 0.56666 0.56123 0.55053 0.54525 0.54003 0.53486

0.55585 0.55585 0.55585 0.55585

0.52973 0.52973 0.52973 0.52973

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72

0.52466 0.52466 0.52466 0.52466 0.51963 0.51963 0.51963 0.51963 0.51465 0.51465 0.51465 0.51465 0.50972 0.50972 0.50972 0.50972 0.50484 0.50484 0.50484 0.50484 0.5 0.5 0.5

67 68 69 70 71 72

72 72 72 72 72 72

Epitropi vs Chrysanthos vs Piano (Percent equivalents)100% 90% 80% Percent equivalent 70% 60% 50% 40% 30% 20% 10% 0%Pythagorea n Cents D-H S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- 0 203.91 90.22 203.91 Equally Tempered Cents 0 200 100 200 72 units (diatonic) 0 10 8 12 68 units (diatonic) 0 9 7 12 66 units (diatonic) 0 8 7 12

0

0

0

0

0

203.91 90.22 203.91

200 100 200

10 8 12

9 7 12

8 7 12

6 20 4

0

7 18 3

0

0

0

0

0

0

8 14 8

9 12 9

9 12 7

8 12 8

12 6 12

13

312

203.92203.91 90.22 203.91

200200 100 200

1210 8 12

129 7 12

128 7 12

126 20 472 units (hard chromatic) 0 6 20 4

127 18 368 units (hard chromatic) 0 7 18 3

128 14 872 units (soft chromatic) 0 8 14 8

129 12 972 units (iso soft chromatic DEACON) 0 9 12 9

129 12 768 units (soft chromatic) 0 9 12 7

128 12 868 units (iso soft chromatic) 0 8 12 8

126 12 1272 units (enharmoni c) 0 12 6 12

123 13 1268 units (enharmoni

203.92203.91 90.22 203.91

200200 100 200

1210 8 12

129 7 12

128 7 12

126 20 4

127 18 3

128 14 8

129 12 9

129 12 7

128 12 8

126 12 12

ScaleDH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- D-H S-

Chrysanthos vs Epitropi vs Piano (cents)1400 1200

203.911000 Cents 800 600 400 200 0

200100 200200 200 100

90.22 203.91203.92 203.91 90.22

52.94117647 66.66666667 133.3333333 211.7647059 218.1818182 317.6470588 333.3333333 233.3333333 133.3333333 123.5294118 127.2727273

200

150 200 150200 150 200

123.5294118 141.1764706 211.7647059 211.7647059 158.8235294 141.1764706211.7647059 211.7647059

200 200100200 200 100

211.7647059 229.4117647 52.94117647211.7647059 211.7647059 52.94117647

158.8235294 166.6666667 145.4545455200 200 211.7647059 218.1818182

100200

123.5294118 133.3333333211.7647059 200

52.94117647 66.66666667 133.3333333 211.7647059 218.1818182 317.6470588 333.3333333 233.3333333 133.3333333 123.5294118 127.2727273

123.5294118 141.1764706211.7647059 211.7647059

203.910 Pythagore an Cents DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- D-H S- 203.91 90.22 203.91 203.92

2000 Equally Tempered Cents 200 100 200 200

166.6666667 158.8235294 145.45454550 72 units (diatonic) 200 0 68 units (diatonic) 0 66 units (diatonic)

0 0 0 72 units 68 units 72 units (hard (hard (soft chromatic) chromatic) chromatic)

100

133.3333333 123.5294118

1500 72 units (iso soft chromatic DEACON) 150 200 150 200

158.8235294 141.1764706

200

229.4117647

0 0 0 68 units 68 units 72 units 68 units (soft (iso soft (enharmon (enharmon chromatic) chromatic) ic) 123.529412 141.176471 211.764706 211.764706 158.823529 141.176471 211.764706 211.764706 200 200 100 200 211.764706 229.411765 52.9411765 211.764706

211.764706 218.181818 66.6666667 52.9411765 133.333333 100 200 123.529412 133.333333 211.764706 200

133.333333 123.529412 127.272727 333.333333 317.647059 233.333333 166.666667 158.823529 145.454545 200 211.764706 218.181818

203.9190.22 203.91 0

200100 200 0

200

211.764706 218.181818 66.6666667 52.9411765 133.333333100 0 123.529412 133.333333 0 0

150200 150 0

123.529412 141.176471211.764706 211.764706 158.823529 141.176471 0 0

200100 200 0

211.76470652.9411765 229.411765

133.333333 123.529412 127.272727 333.333333 317.647059 233.333333 166.666667 158.823529 145.454545 0 0 0

ScaleD-H S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S-

Chrysanthos vs Epitropi vs Piano100% 90%203.91 200 100 200 12 8 10 12 7 9 12 7 8 20 18

4

3

8

9 12 9 12 9 12 9 0 72 units (iso soft chromatic DEACON) 9 12 9 12

7 12

8 12

12

80% 90.22 Relative percent 70% 60% 50% 40% 30% 20% 10% 0%203.91 90.22 203.91 0 Pythagore an Cents DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- D-H S- 203.91 90.22 203.91 203.92 203.92 203.91

14

12 6 12

712 3

8 12 8

9

8 12

6 12

200

12

12

12

12 7 12

4200 100 200 0 Equally Tempered Cents 200 100 200 200 12 8 10 0 72 units (diatonic) 12 8 10 12 12 7

1220 7 8 0 66 units (diatonic) 12 7 8 12 18 14

812

12 6

9 068 units (diatonic) 12 7 9 12

8 7 6 0 0 0 72 units 68 units 72 units (hard (hard (soft chromatic) chromatic) chromatic) 4 20 6 12 3 18 7 12 8 14 8 12

12 9 8 0 0 0 68 units 68 units 72 units 68 units (soft (iso soft (enharmon (enharmon chromatic) chromatic) ic) 7 12 9 12 8 12 8 12 12 12 6 12

203.9190.22 203.91 0

200100 200 0

128 10 0

127 9 0

127 8 0

420 6 0

318 7 0

814 8 0

912 9 0

712 9 0

812 8 0

126 12 0

ScaleD-H S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S-

Piano vs Epitropi vs Chrysanthos (1200 cent equivalents)1400 1200 1000 Cents 800 600 400 200 0Pythagorea n Cents D-H S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- DH- S- 0 203.91 90.22 203.91

0

203.91 90.22 203.91 203.92 203.91 90.22 203.91

133.3333333 150 158.8235294 141.1764706 200 145.4545455 100 123.5294118 158.8235294 200 166.6666667 229.4117647 127.2727273 123.5294118 211.7647059 100 52.94117647 100 133.3333333 233.3333333 200 211.7647059 333.3333333 317.6470588 218.1818182 200 200 211.7647059 141.1764706 200 211.7647059 133.3333333 150 123.5294118 66.66666667 52.94117647 200 200 211.7647059 218.1818182 200 211.7647059 200 200 211.7647059 211.7647059 200 211.7647059 52.94117647 133.3333333 150 158.8235294 141.1764706 100 145.4545455 100 123.5294118 158.8235294 200 166.6666667 200 229.4117647 127.2727273 123.5294118 100 133.3333333 233.3333333 200 211.7647059 211.7647059 333.3333333 317.6470588 218.1818182 200 200 211.7647059 141.1764706 200 211.7647059 133.3333333 150 123.5294118 66.66666667 52.9411764

Documents

GKM 2009 Diastemata Intervals Psaltiki ALL 001