GKM 2009 Diastemata Intervals Psaltiki ALL 001 g

PSALTIKI SMonophoniesFRACTION CENTS

Fractions and Equal TempereDIATONIC 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 ## ## ## ## ## ## ## ### #### ### ### ### #### ### 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 ## ## ## ## ## ## ## ### ### ### ### ### ### ### 203.910 182.404 111.731 203.910 203.910 182.404 111.731

Al-FarabiNi 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 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni ## ## ## ## ## ## ## ### ### ### ### ### ### ### 203.910 150.637 143.498 203.910 150.637 143.498 203.910

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 ## ## ## ## ## ## ## ### ### ### ### ### ### ### 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 ## ## ## ## ## ## ## ### ### ### ### ### ### ### 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 Ga Di Ke Zo Ni 0 4 7 10 14 18 21 24 24 200.000 350.000 500.000 700.000 900.000 1,050.000 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 4 3 3 4 4 3 3 200.000 150.000 150.000 200.000 200.000 150.000 150.000

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 ## ## ## ## ## ## ## ### #### ### ### ### #### ### 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 ## ## ## ## ## ## ## ### #### ### ### ### #### ### 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 Ni 0 7 19 26 38 45 57 64 64 131.250 356.250 487.500 712.500 843.750 1,068.750 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 7 12 7 12 7 12 7 131.250 225.000 131.250 225.000 131.250 225.000 131.250

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 Neanes

Ni 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 ## ## ## ## ## ## ## ### #### ### ### ### #### ### 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 ## ## ## ## ## ## ## #### #### ### ### #### #### ### 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 0 7 0.000 123.529 Pa-Bou 7 123.529

Ga Di Ke Zo Ni Pa

25 28 40 47 65 68

441.176 494.118 705.882 829.412 1,147.059 1,200.000

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

18 3 12 7 18 3

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 Pa Bou Ga 1 8/9 64/81 3/4 2/3 16/27 128/243 1/2 0.000 203.910 407.820 498.045 701.955 905.865 1,109.775 1,200.000 Ga-Di Di-Ke Ke-Zo Zo-Ni Ni-Pa Pa-Bou Bou-Ga ## ## ## ## ## ## ## ### ### #### ### ### ### #### 203.910 203.910 90.225 203.910 203.910 203.910 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 0 12 24 27 0.000 211.765 423.529 476.471 Ga-Di Di-Ke Ke-Zo 12 12 3 211.765 211.765 52.941

Ni Pa Bou Ga

40 52 55 68

705.882 917.647 970.588 1,200.000

Zo-Ni Ni-Pa

13 12

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 Ni 0 12 25 36 40 52 61 68 68 211.765 441.176 635.294 705.882 917.647 1,076.471 1,200.000 Ni-Pa Pa-Bou Bou-Ga Ga-Di Di-Ke Ke-Zo Zo-Ni 12 13 11 4 12 9 7 211.765 229.412 194.118 70.588 211.765 158.824 123.529

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

PSALTIKI SCALESDiphoniesFRACTION CENTS

qual 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 1 1 1 19/81 5/27 43/200 17/64 364.807 294.135 337.148 407.820 541.058 674.295 (Ni-Ga)d (Pa-Di)d (Bou-Ke)d (Ga-Zo)d ## ### ## ### ## ##### ## ### 498.045 498.045 541.058 568.717 (Ni-Di)d (Pa-Ke)d (Bou-Zo)d

1 587/1600 1 3285/6898

(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 1 1 1 1 1/4 5/27 1/5 17/64 7/20 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 ## ## ## ## ### ### ### #### 498.045 498.045 519.551 590.224 (Ni-Di)d (Pa-Ke)d (Bou-Zo)d

(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 1 1 1 1 1 5/22 5/27 2/9 5/22 3/8 40/81 354.547 294.135 347.408 354.547 551.318 694.816 (Ni-Ga)d (Pa-Di)d (Bou-Ke)d (Ga-Zo)d ## ## ## ## ### ### ### ### 498.045 498.045 498.045 498.045 (Ni-Di)d (Pa-Ke)d (Bou-Zo)d

(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 1 1 1 1 1 5/22 5/27 2/9 17/64 3/8 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 ## ## ## ## ### ### ### #### 498.045 498.045 551.318 558.457 (Ni-Di)d (Pa-Ke)d (Bou-Zo)d

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

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

1 1

3/8 40/81

551.318 694.816




(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 1 1/4 1/4 1/5 1/5 7/25 11/25 386.314 (Ni-Ga)cM ## 386.314 (Pa-Di)cM ## 315.641 (Bou-Ke)cM ## 315.641 (Ga-Zo)cM ## 427.373 631.283 ### #### ### #### 498.045 (Ni-Di)cM 590.224 (Pa-Ke)cM 427.373 (Bou-Zo)cM 590.224

(Ni-Pa = 27/25, Pa-Bou = 125/108, Bou-Ga = 16/15, Ga-Di = 9(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 1 1/4 19/81 1/5 43/200 37/125 11/25 386.314 (Ni-Ga)cM ## 364.807 (Pa-Di)cM ## 315.641 (Bou-Ke)cM ## 337.148 (Ga-Zo)cM ## 448.879 631.283 ### ### #### #### 498.045 (Ni-Di)cM 568.717 (Pa-Ke)cM 448.879 (Bou-Zo)cM 590.224

(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) / 68

(Ni-Bou)cM (Pa-Ga)cM (Bou-Di)cM (Ga-Ke)cM diphonic Bou-Di 4th diphonic Bou-Di 5th

335.294 (Ni-Ga)cM 335.294 (Pa-Di)cM 335.294 (Bou-Ke)cM 335.294 (Ga-Zo)cM 458.824 670.588

458.824 (Ni-Di)cM 547.059 (Pa-Ke)cM 458.824 (Bou-Zo)cM 582.353

(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 1 1 1 1/4 1/4 1/5 1/5 7/20 11/25 13/32 9/16 386.314 (Ni-Ga)cS ## 386.314 (Pa-Di)cS ## 315.641 (Bou-Ke)cS ## 315.641 (Ga-Zo)cS ## 519.551 631.283 590.224 772.627 ### #### ### #### 498.045 (Ni-Di)cS 590.224 (Pa-Ke)cS 427.373 (Bou-Zo)cS 590.224

(Ni-Pa = 9/8, Pa-Bou = 256/243, Bou-Ga = 243/200, Ga-Di = 2(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 ## ### 407.820 (Bou-Ke)cS ## ##### 274.582 (Ga-Zo)cS ## #### 294.135 (Di-Ni)cS ## #### 498.045 588.270 631.283 854.745 498.045 (Pa-Ke)cS 611.730 (Bou-Zo)cS 364.807 (Ga-Ni)cS 631.283

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 441.176 (Pa-Di)cS 494.118 (Pa-Ke)cS


(Bou-Di)cS (Ga-Ke)cS (Di-Zo)cS Di-Zo 4th Di-Zo 5th Pa-Ga 4th Pa-Ga 5th

370.588 (Bou-Ke)cS 264.706 (Ga-Zo)cS 335.294 (Di-Ni)cS 547.059 670.588 652.941 882.353

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 (Ni-Bou)e diphonic Ni-Bou 4th diphonic Ni-Bou 5th 1 17/64 1 5/27 1 5/27 1 17/64 1 17/64 1 217/512 1 2465/4096 407.820 294.135 294.135 407.820 407.820 611.730 815.640 (Ga-Zo)e (Di-Ni)e (Ke-Pa)e (Zo-Bou)e ## ### ## ### ## ### ## ##### 498.045 498.045 498.045 611.730 (Ga-Ni)e (Di-Pa)e (Ke-Bou)e (Zo-Ga)e

(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 423.529 264.706 282.353 441.176 (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-Bou)e diphonic Ni-Bou 4th diphonic Ni-Bou 5th

264.706 476.471 529.412

(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

pg 115 et 170)(Ni-Bou)z (Pa-Ga)z (Bou-Di)z (Ga-Ke)z diphonic Bou-Di 4th diphonic Bou-Di 5th

(Ni-Pa = 14, Pa-Bou = 7, Bou-Ga = 14, Ga-Di = 5) / 68370.588 370.588 335.294 300.000 582.353 670.588 (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

g 115 et 170)(Ni-Bou)k (Pa-Ga)k (Bou-Di)k (Ga-Ke)k diphonic Bou-Di 4th diphonic Bou-Di 5th

(Ni-Pa = 12, Pa-Bou = 13, Bou-Ga = 11, Ga-Di = 4) / 68441.176 423.529 264.706 282.353 476.471 529.412 (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



TetraphoniesFRACTION CENTS

PentaphoniesFRACTION CENTS

00/729, Bou-Ga = 27/25, Ni-Ga=4/3)## ## ## ### ### ### 701.955 701.955 701.955 (Ni-Ke)d ## #### (Pa-Zo)d ## ##### (Bou-Ni)d ## ### 905.865 862.852 701.955

ou-Ga = 8) / 72700.000 700.000 700.000 (Ni-Ke)d (Pa-Zo)d (Bou-Ni)d 900.000 866.667 833.333

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

1, Bou-Ga = 88/81, Ni-Ga=4/3)## ### ## ### ## ##### 701.955 648.682 641.543 (Ni-Ke)d ## (Pa-Zo)d ## (Bou-Ni)d ## ### #### ### 852.592 792.180 701.955

1, Bou-Ga = 88/81, Ni-Ga=4/3)## ## ## ### ### ### 701.955 701.955 701.955 (Ni-Ke)d ## (Pa-Zo)d ## (Bou-Ni)d ## #### ### ### 905.865 852.592 701.955

Zo-Ni = 88/81, Di-Ni=4/3)## ### 701.955

705.882 705.882 705.882

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

917.647 864.706 829.412

709.091 709.091 709.091


927.273 854.545 836.364

700.000 700.000 700.000


900.000 850.000 850.000

/64, Bou-Ga = 16/15, Ni-Ga=4/3)## ## ## ### ### ### 701.955 701.955 701.955 (Ni-Ke)cM ## (Pa-Zo)cM ## (Bou-Ni)cM ## ### #### ### 813.686 976.537 701.955

25/108, Bou-Ga = 16/15, Ga-Di = 9/8, Ni-Ga=4/3)## ## ## ### ### ### 701.955 701.955 701.955 (Ni-Ke)cM ## (Pa-Zo)cM ## (Bou-Ni)cM ## #### #### ### 835.193 955.031 701.955

u-Ga = 8, Ga-Di=12) / 72700.000 700.000 700.000 (Ni-Ke)cM (Pa-Zo)cM (Bou-Ni)cM 833.333 933.333 833.333

u-Ga = 7, Ga-Di=12) / 64712.500 712.500 712.500 (Ni-Ke)cM (Pa-Zo)cM (Bou-Ni)cM 843.750 937.500 843.750

u-Ga = 7, Ga-Di=12) / 68705.882 705.882 705.882 (Ni-Ke)d (Pa-Zo)d (Bou-Ni)d 829.412 952.941 829.412

Bou-Ga = 7, Ga-Di=12) / 68

670.588 670.588 705.882


794.118 917.647 829.412

/64, Bou-Ga = 16/15, Ni-Ga=4/3)## ## ## ### ### ### 701.955 701.955 701.955 (Ni-Ke)cS ## (Pa-Zo)cS ## (Bou-Ni)cS ## ### #### ### 813.686 976.537 813.686

243, Bou-Ga = 243/200, Ga-Di = 25/24)## ## ## ### ### ### 701.955 701.955 701.955 (Pa-Zo)cS ## #### 792.180 (Bou-Ni)cS ## ##### 1,039.103 (Ga-Pa)cS ## ### 772.627

u-Ga = 20, Ga-Di=4) / 72700.000 700.000 700.000 (Pa-Zo)cS (Bou-Ni)cS 800.000 1,033.333

u-Ga = 18, Ga-Di=3) / 68705.882 (Pa-Zo)cS 829.412

705.882 705.882

(Bou-Ni)cS

1,023.529

-Ga = 8, Ga-Di = 2) / 24700.000 700.000 700.000 (Pa-Zo)c (Bou-Ni)c x 800.000 1,000.000

e-Zo = 256/243)## ## ## ## ### ### ### ### 701.955 701.955 701.955 701.955 (Ga-Pa)e ## (Di-Bou)e ## (Ke-Ga)e ## #### #### #### 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

Zo = 3, Zo-Ni = 13) / 72705.882 705.882 723.529 723.529 (Ga-Pa)e (Di-Bou)e (Ke-Ga)e 917.647 758.824

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

ou-Ga = 3, Ga-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

ou-Ga = 16, Ga-Di = 4) / 72700.000 600.000 700.000 (Ni-Ke)z (Pa-Zo)z (Bou-Ni)z 900.000 766.667 833.333

u-Ga = 14, Ga-Di = 5) / 68705.882 670.588 705.882 (Ni-Ke)z (Pa-Zo)z (Bou-Ni)z 917.647 829.412 829.412

ou-Ga = 16, Ga-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 1600/2187 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 2755/4067 25/36 729/1024

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

108

Total chord length

68Initial Frequency Final Frequency

First scale conversions

Chord fractionChord length M Chord fraction

Second Scale's Total number of Units within Octave

Second scale conversions

Interval using Scale 1

Equivalent interval in Scale 2

CENTS

597.92 676.70

12.141.1318 95.43

214.3

1 679/515312.57

1

7/81

8.13

143.5

68

7.00

alent interval in Scale 2

7.00 1 459/6206

123.5

Chord fraction

Units within Octave of First Scale

108

Total chord length


597.92 676.701.1318 95.43


Chord fractionChord length M Chord fraction

1

7/81

Second Scale's Total number of Units within Octave


Interval using Scale 1

72

12.00


Chord fraction

CENTSUnits within Octave of First Scale

Total chord length

12.861 679/515312.57

214.3


8.61

143.5

Second scale conversions 12.00 1 573/4679

200.0

108


597.92 676.70


11.43Chord fractionChord length M Chord fraction

1.1318 95.43

1 679/515312.57

1

7/81

7.65Second Scale's Total number of Units within Octave Interval using Scale 1


72

12.00


13.50 1 346/2493

Chord fraction

CENTSUnits within Octave of First Scale

108

Total chord length

51

214.3


143.5

Second scale conversions 225.0

Initial Frequency Final Frequency

597.92 676.70

9.11Chord fractionChord length M Chord fraction

1.1318 95.43

1 679/515312.57

1

7/81

6.10Second Scale's Total number of Units within Octave Interval using Scale 1

72

12.00


16.94 1 262/1479

Chord fraction

CENTS

214.3

143.5

282.4

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

Epitropis Chrysanthou Chrysanthou Chrysanthou 66 64 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 32 33 34

16.7 33.3 50.0 66.7 83.3 100.0 116.7 133.3 150.0 166.7 183.3 200.0 216.7 233.3 250.0 266.7 283.3 300.0 316.7 333.3 350.0 366.7 383.3 400.0 416.7 433.3 450.0 466.7 483.3 500.0 516.7 533.3 550.0 566.7

17.6 35.3 52.9 70.6 88.2 105.9 123.5 141.2 158.8 176.5 194.1 211.8 229.4 247.1 264.7 282.4 300.0 317.6 335.3 352.9 370.6 388.2 405.9 423.5 441.2 458.8 476.5 494.1 511.8 529.4 547.1 564.7 582.4 600.0

18.2 36.4 54.5 72.7 90.9 109.1 127.3 145.5 163.6 181.8 200.0 218.2 236.4 254.5 272.7 290.9 309.1 327.3 345.5 363.6 381.8 400.0 418.2 436.4 454.5 472.7 490.9 509.1 527.3 545.5 563.6 581.8 600.0 618.2

18.8 37.5 56.3 75.0 93.8 112.5 131.3 150.0 168.8 187.5 206.3 225.0 243.8 262.5 281.3 300.0 318.8 337.5 356.3 375.0 393.8 412.5 431.3 450.0 468.8 487.5 506.3 525.0 543.8 562.5 581.3 600.0 618.8 637.5

22.6 45.3 67.9 90.6 113.2 135.8 158.5 181.1 203.8 226.4 249.1 271.7 294.3 317.0 339.6 362.3 384.9 407.5 430.2 452.8 475.5 498.1 520.8 543.4 566.0 588.7 611.3 634.0 656.6 679.2 701.9 724.5 747.2 769.8

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

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

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

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

656.3 675.0 693.8 712.5 731.3 750.0 768.8 787.5 806.3 825.0 843.8 862.5 881.3 900.0 918.8 937.5 956.3 975.0 993.8 1,012.5 1,031.3 1,050.0 1,068.8 1,087.5 1,106.3 1,125.0 1,143.8 1,162.5 1,181.3 1,200.0

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

Epitropis Chrysanthou Chrysanthou Chrysanthou Kinezon 53 66 64 72 68

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

16.666667 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 533.333333 550.000000 566.666667

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 564.705882 582.352941 600.000000

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 581.818182 600.000000 618.181818

18.750000 37.500000 56.250000 75.000000 93.750000 112.500000 131.250000 150.000000 168.750000 187.500000 206.250000 225.000000 243.750000 262.500000 281.250000 300.000000 318.750000 337.500000 356.250000 375.000000 393.750000 412.500000 431.250000 450.000000 468.750000 487.500000 506.250000 525.000000 543.750000 562.500000 581.250000 600.000000 618.750000 637.500000

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 724.528302 747.169811 769.811321

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

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

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

636.363636 654.545455 672.727273 690.909091 709.090909 727.272727 745.454545 763.636364 781.818182 800.000000 818.181818 836.363636 854.545455 872.727273 890.909091 909.090909 927.272727 945.454545 963.636364

656.250000 675.000000 693.750000 712.500000 731.250000 750.000000 768.750000 787.500000 806.250000 825.000000

792.452830 815.094340 837.735849 860.377358 883.018868 905.660377 928.301887 950.943396 973.584906 996.226415

843.750000 1,018.867925 862.500000 1,041.509434 881.250000 1,064.150943 900.000000 1,086.792453 918.750000 1,109.433962 937.500000 1,132.075472 956.250000 1,154.716981 975.000000 1,177.358491 993.750000 1,200.000000

981.818182 1,012.500000

970.588235 1,000.000000 1,031.250000 988.235294 1,018.181818 1,050.000000 1,005.882353 1,036.363636 1,068.750000 1,023.529412 1,054.545455 1,087.500000 1,041.176471 1,072.727273 1,106.250000 1,058.823529 1,090.909091 1,125.000000 1,076.470588 1,109.090909 1,143.750000 1,094.117647 1,127.272727 1,162.500000 1,111.764706 1,145.454545 1,181.250000 1,129.411765 1,163.636364 1,200.000000 1,147.058824 1,181.818182 1,164.705882 1,200.000000 1,182.352941 1,200.000000

Roumanon 24

50.000000 70.61939031 100.000000 141.2387806 150.000000 211.8581709 200.000000 282.4775612 250.000000 353.0969515 300.000000 423.7163419 350.000000 494.3357322 400.000000 564.9551225 450.000000 635.5745128 500.000000 706.1939031 550.000000 776.8132934 600.000000 847.4326837 650.000000 918.052074 700.000000 988.6714643 750.000000 1059.290855 800.000000 1129.910245 850.000000 1200.529635 900.000000 1271.149026 950.000000 1341.768416 1,000.000000 1412.387806 1,050.000000 1483.007196 1,100.000000 1553.626587 1,150.000000 1624.245977 1,200.000000 1694.865367 1765.484758 1836.104148 1906.723538 1977.342929 2047.962319 2118.581709 2189.2011 2259.82049 2330.43988 2401.05927

2471.678661 2542.298051 2612.917441 2683.536832 2754.156222 2824.775612 2895.395003 2966.014393 3036.633783 3107.253174 3177.872564 3248.491954 3319.111344 3389.730735 3460.350125 3530.969515 3601.588906 3672.208296 3742.827686 3813.447077 3884.066467 3954.685857 4025.305248 4095.924638 4166.544028 4237.163419 4307.782809 4378.402199 4449.021589 4519.64098 4590.26037 4660.87976 4731.499151 4802.118541 4872.737931 4943.357322 5013.976712 5084.596102

LENGTH DIVISIONS ET output

108 68 72

108

CHORD FRACTION FIN

72

Chord FRACTION68

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1.01024547 1.02059591 1.031052394 1.041616011 1.052287856 1.06306904 1.073960682 1.084963914 1.096079879 1.107309732 1.118654641 1.130115783 1.141694351 1.153391546 1.165208584 1.177146694 1.189207115 1.201391101

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

1.213699917 1.226134843 1.238697171 1.251388206 1.264209266 1.277161684 1.290246806 1.303465991 1.316820612 1.330312058 1.34394173 1.357711045 1.371621433 1.385674339 1.399871224 1.414213562 1.428702845 1.443340577 1.45812828 1.473067489 1.488159758 1.503406654 1.518809762 1.534370681 1.55009103 1.565972441 1.582016565 1.598225068

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 73 74

1.614599635 1.631141967 1.647853783 1.664736819 1.681792831 1.699023588 1.716430884 1.734016525 1.751782339 1.769730172 1.78786189 1.806179375 1.824684531 1.843379282 1.862265569 1.881345355 1.900620622 1.920093374 1.939765633 1.959639443 1.97971687 2 2.02049094 2.041191819 2.062104789 2.083232021 2.104575712 2.12613808

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

2.147921363 2.169927827 2.192159758 2.214619465 2.237309282 2.260231567 2.283388701 2.306783092 2.330417169 2.354293388 2.37841423 2.402782202 2.427399834 2.452269687 2.477394342 2.502776411 2.528418532 2.554323368 2.580493611 2.606931981 2.633641224 2.660624116 2.687883461 2.71542209 2.743242866 2.771348678 2.799742448 2.828427125

103 104 105 106 107 108

2.85740569 2.886681154 2.916256559 2.946134978 2.976319516 3.006813308

FRACTION FRACTION

12/68 8/68 9/68 7/68 12/72 10/72 8/72

1 8/9 ## 8 9 7 ## ## 8 ##

1.000

1.125

1.125

1.080

1.091

1.069

1.125

1.102

1.080

#DIV/0!

FRACTION FRACTION FRACTION

8/9 22/27 81/100

1.125

1.227272727

1.234567901

FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION

4/5 3/4 2/3 11/18 16/27 9/16 27/50 8/15 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

1.25

1.333333333

1.5

1.636363636

1.6875

1.777777778

1.851851852

1.875

2

2

2

2

2

2

2

2

2

2

2

2

FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION FRACTION

1/2 1/2 1/2 1/2 1/2 1/2 1/2

2

2

2

2

2

2

2

CHORD FRACTION FINDER

108

108 cm; 64 ET u

72

Chord LENGTH

CENTS

Chrys Log Chrys Metric

Chord FRACTION

64

108 106.904711 105.8205299 104.7473442 103.6850422 102.6335136 101.5926492 100.5623407 99.54248122 98.53296469 97.53368624 96.54454203 95.5654293 94.5962463 93.63689233 92.68726771 91.74727378 90.81681285 89.89578825

0

0

1 1.010889286 1.021897149 1.033024879 1.044273782 1.055645178 1.067140401 1.078760798 1.090507733 1.102382583 1.114386743 1.126521619 1.138788635 1.15118923 1.163724859 1.176396992 1.189207115 1.202156731 1.21524736

17.64705882 1.058824 35.29411765 2.117647 52.94117647 3.176471 70.58823529 4.235294 88.23529412 5.294118 105.8823529 6.352941 123.5294118 7.411765 141.1764706 8.470588 158.8235294 9.529412 176.4705882 10.58824 194.1176471 11.64706 211.7647059 12.70588 229.4117647 13.76471 247.0588235 14.82353 264.7058824 15.88235 282.3529412 16.94118 300 18

317.6470588 19.05882

88.98410428 88.08166622 87.18838028 86.30415367 85.4288945 84.56251182 83.70491562 82.85601679 82.01572712 81.18395931 80.36062692 79.54564441 78.7389271 77.94039116 77.14995363 76.36753237 75.59304609 74.8264143 74.06755737 73.31639643 72.57285344 71.83685114 71.10831306 70.38716349 69.6733275 68.96673093 68.26730036 67.5749631

335.2941176 20.11765 352.9411765 21.17647 370.5882353 22.23529 388.2352941 23.29412 405.8823529 24.35294 423.5294118 25.41176 441.1764706 26.47059 458.8235294 27.52941 476.4705882 28.58824 494.1176471 29.64706 511.7647059 30.70588 529.4117647 31.76471 547.0588235 32.82353 564.7058824 33.88235 582.3529412 34.94118 600 36

1.228480536 1.241857812 1.255380757 1.269050957 1.282870016 1.296839555 1.310961212 1.325236643 1.339667524 1.354255547 1.369002423 1.383909882 1.398979673 1.414213562 1.429613338 1.445180807 1.460917794 1.476826146 1.492907728 1.509164428 1.525598151 1.542210825 1.5590044 1.575980845 1.593142151 1.610490332 1.628027422 1.645755478

617.6470588 37.05882 635.2941176 38.11765 652.9411765 39.17647 670.5882353 40.23529 688.2352941 41.29412 705.8823529 42.35294 723.5294118 43.41176 741.1764706 44.47059 758.8235294 45.52941 776.4705882 46.58824 794.1176471 47.64706 811.7647059 48.70588

66.88964723 66.21128154 65.53979553 64.87511944 64.21718421 63.56592147 62.92126356 62.28314348 61.65149494 61.02625231 60.40735061 59.79472554 59.18831346 58.58805134 57.99387681 57.40572815 56.82354423 56.24726458 55.67682929 55.11217912 54.55325538 54 53.4523555 52.91026497 52.37367208 51.84252109 51.3167568 50.79632458

829.4117647 49.76471 847.0588235 50.82353 864.7058824 51.88235 882.3529412 52.94118 900 54

1.66367658 1.681792831 1.700106354 1.718619298 1.737333835 1.75625216 1.775376493 1.794709075 1.814252176 1.834008086 1.853979125 1.874167634 1.894575982 1.915206561 1.936061793 1.957144124 1.978456026 2 2.021778572 2.043794297 2.066049758 2.088547565 2.111290357 2.134280801 2.157521596 2.181015465 2.204765167 2.228773485

917.6470588 55.05882 935.2941176 56.11765 952.9411765 57.17647 970.5882353 58.23529 988.2352941 59.29412 1005.882353 60.35294 1023.529412 61.41176 1041.176471 62.47059 1058.823529 63.52941 1076.470588 64.58824 1094.117647 65.64706 1111.764706 66.70588 1129.411765 67.76471 1147.058824 68.82353 1164.705882 69.88235 1182.352941 70.94118 1200 72

1217.647059 73.05882 1235.294118 74.11765 1252.941176 75.17647 1270.588235 76.23529 1288.235294 77.29412 1305.882353 78.35294

50.28117036 49.77124061 49.26648235 48.76684312 48.27227102 47.78271465 47.29812315 46.81844617 46.34363386 45.87363689 45.40840642 44.94789412 44.49205214 44.04083311 43.59419014 43.15207684 42.71444725 42.28125591 41.85245781 41.4280084 41.00786356 40.59197965 40.18031346 39.7728222 39.36946355 38.97019558 38.57497681 38.18376618

1323.529412 79.41176 1341.176471 80.47059 1358.823529 81.52941 1376.470588 82.58824 1394.117647 83.64706 1411.764706 84.70588 1429.411765 85.76471 1447.058824 86.82353 1464.705882 87.88235 1482.352941 88.94118 1500 90

2.253043237 2.27757727 2.30237846 2.327449718 2.352793983 2.37841423 2.404313463 2.43049472 2.456961072 2.483715624 2.510761514 2.538101914 2.565740032 2.593679109 2.621922423 2.650473286 2.679335048 2.708511094 2.738004846 2.767819764 2.797959345 2.828427125 2.859226677 2.890361614 2.921835588 2.953652292 2.985815457 3.018328855

1517.647059 91.05882 1535.294118 92.11765 1552.941176 93.17647 1570.588235 94.23529 1588.235294 95.29412 1605.882353 96.35294 1623.529412 97.41176 1641.176471 98.47059 1658.823529 99.52941 1676.470588 100.5882 1694.117647 101.6471 1711.764706 102.7059 1729.411765 103.7647 1747.058824 104.8235 1764.705882 105.8824 1782.352941 106.9412 1800 108

37.79652304 37.41320715 37.03377869 36.65819822 36.28642672 35.91842557

1817.647059 109.0588 1835.294118 110.1176 1852.941176 111.1765 1870.588235 112.2353 1888.235294 113.2941 1905.882353 114.3529

3.051196301 3.084421651 3.1180088 3.15196169 3.186284303 3.220980664

108

0.00 203.91 203.91

0 12.2346 12.2346 12 8 9 7 12 10 8

1

96

1.125

96

100

133.24 7.994254 150.64 9.038224 116.01 6.960673 203.91 12.2346

99

101

96

98

168.21 10.09279 133.24 7.994254 #DIV/0! #DIV/0! 0

100

#DIV/0!

96

203.91

1.125

88

354.55 21.27282 364.81 21.88845

1.227272727

87.48

1.234567901

86.4

386.31 23.17882 498.04 701.96 29.8827 42.1173

1.25

81

1.333333333

72

1.5

66

852.59 51.15552 905.87 996.09 54.3519 59.7654

1.636363636

64

1.6875

60.75

1.777777778

58.32

1066.76 64.00575 1088.27 65.29612 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 72 72 72 72 72 72 72 72 72 72 72 72

1.851851852

57.6

1.875

54

2

54

2

54

2

54

2

54

2

54

2

54

2

54

2

54

2

54

2

54

2

54

2

54

1200.00 1200.00 1200.00 1200.00 1200.00 1200.00 1200.00

72 72 72 72 72 72 72

2

54

2

54

2

54

2

54

2

54

2

54

2

108 cm; 64 ET units

108

108 cm; 68 ET units

72

Chord LENGTH

CENTS

Chord FRACTION

Chord LENGTH

68

108 106.8366254 105.6857827 104.5473368 103.4211543 102.307103 101.2050522 100.1148728 99.03643667 97.96961748 96.91429005 95.8703306 94.83761666 93.8160271 92.8054421 91.8057431 90.81681285 89.83853534 88.87079582

0 18.75 37.5 56.25 75 93.75 112.5 131.25 150 168.75 187.5 206.25 225 243.75 262.5 281.25 300 318.75 337.5

0 1.125 2.25 3.375 4.5 5.625 6.75 7.875 9 10.125 11.25 12.375 13.5 14.625 15.75 16.875 18 19.125 20.25

1 1.01024547 1.02059591 1.031052394 1.041616011 1.052287856 1.06306904 1.073960682 1.084963914 1.096079879 1.107309732 1.118654641 1.130115783 1.141694351 1.153391546 1.165208584 1.177146694 1.189207115 1.201391101

108 106.904711 105.8205299 104.7473442 103.6850422 102.6335136 101.5926492 100.5623407 99.54248122 98.53296469 97.53368624 96.54454203 95.5654293 94.5962463 93.63689233 92.68726771 91.74727378 90.81681285 89.89578825

87.91348078 86.96647793 86.02967617 85.10296564 84.18623761 83.27938457 82.38230014 81.49487909 80.61701733 79.74861188 78.88956089 78.03976358 77.19912027 76.36753237 75.54490232 74.73113363 73.92613084 73.12979953 72.3420463 71.56277873 70.79190542 70.02933595 69.27498087 68.52875169 67.79056088 67.06032185 66.33794895 65.62335744

356.25 375 393.75 412.5 431.25 450 468.75 487.5 506.25 525 543.75 562.5 581.25 600 618.75 637.5 656.25 675 693.75 712.5 731.25 750 768.75 787.5 806.25 825 843.75 862.5

21.375 22.5 23.625 24.75 25.875 27 28.125 29.25 30.375 31.5 32.625 33.75 34.875 36 37.125 38.25 39.375 40.5 41.625 42.75 43.875 45 46.125 47.25 48.375 49.5 50.625 51.75

1.213699917 1.226134843 1.238697171 1.251388206 1.264209266 1.277161684 1.290246806 1.303465991 1.316820612 1.330312058 1.34394173 1.357711045 1.371621433 1.385674339 1.399871224 1.414213562 1.428702845 1.443340577 1.45812828 1.473067489 1.488159758 1.503406654 1.518809762 1.534370681 1.55009103 1.565972441 1.582016565 1.598225068

88.98410428 88.08166622 87.18838028 86.30415367 85.4288945 84.56251182 83.70491562 82.85601679 82.01572712 81.18395931 80.36062692 79.54564441 78.7389271 77.94039116 77.14995363 76.36753237 75.59304609 74.8264143 74.06755737 73.31639643 72.57285344 71.83685114 71.10831306 70.38716349 69.6733275 68.96673093 68.26730036 67.5749631

64.9164635 64.21718421 63.52543755 62.84114237 62.16421842 61.49458628 60.8321674 60.1768841 59.5286595 58.88741756 58.25308308 57.62558164 57.00483963 56.39078425 55.78334347 55.18244603 54.58802145 54 53.41831271 52.84289135 52.27366842 51.71057716 51.1535515 50.60252612 50.05743638 49.51821833 48.98480874 48.45714503

881.25 900 918.75 937.5 956.25 975 993.75 1012.5 1031.25 1050 1068.75 1087.5 1106.25 1125 1143.75 1162.5 1181.25 1200 1218.75 1237.5 1256.25 1275 1293.75 1312.5 1331.25 1350 1368.75 1387.5

52.875 54 55.125 56.25 57.375 58.5 59.625 60.75 61.875 63 64.125 65.25 66.375 67.5 68.625 69.75 70.875 72 73.125 74.25 75.375 76.5 77.625 78.75 79.875 81 82.125 83.25

1.614599635 1.631141967 1.647853783 1.664736819 1.681792831 1.699023588 1.716430884 1.734016525 1.751782339 1.769730172 1.78786189 1.806179375 1.824684531 1.843379282 1.862265569 1.881345355 1.900620622 1.920093374 1.939765633 1.959639443 1.97971687 2 2.02049094 2.041191819 2.062104789 2.083232021 2.104575712 2.12613808

66.88964723 66.21128154 65.53979553 64.87511944 64.21718421 63.56592147 62.92126356 62.28314348 61.65149494 61.02625231 60.40735061 59.79472554 59.18831346 58.58805134 57.99387681 57.40572815 56.82354423 56.24726458 55.67682929 55.11217912 54.55325538 54 53.4523555 52.91026497 52.37367208 51.84252109 51.3167568 50.79632458

47.9351653 47.41880833 46.90801355 46.40272105 45.90287155 45.40840642 44.91926767 44.43539791 43.95674039 43.48323896 43.01483809 42.55148282 42.09311881 41.63969229 41.19115007 40.74743955 40.30850866 39.87430594 39.44478044 39.01988179 38.59956014 38.18376618 37.77245116 37.36556681 36.96306542 36.56489977 36.17102315 35.78138937

1406.25 1425 1443.75 1462.5 1481.25 1500 1518.75 1537.5 1556.25 1575 1593.75 1612.5 1631.25 1650 1668.75 1687.5 1706.25 1725 1743.75 1762.5 1781.25 1800 1818.75 1837.5 1856.25 1875 1893.75 1912.5

84.375 85.5 86.625 87.75 88.875 90 91.125 92.25 93.375 94.5 95.625 96.75 97.875 99 100.125 101.25 102.375 103.5 104.625 105.75 106.875 108 109.125 110.25 111.375 112.5 113.625 114.75

2.147921363 2.169927827 2.192159758 2.214619465 2.237309282 2.260231567 2.283388701 2.306783092 2.330417169 2.354293388 2.37841423 2.402782202 2.427399834 2.452269687 2.477394342 2.502776411 2.528418532 2.554323368 2.580493611 2.606931981 2.633641224 2.660624116 2.687883461 2.71542209 2.743242866 2.771348678 2.799742448 2.828427125

50.28117036 49.77124061 49.26648235 48.76684312 48.27227102 47.78271465 47.29812315 46.81844617 46.34363386 45.87363689 45.40840642 44.94789412 44.49205214 44.04083311 43.59419014 43.15207684 42.71444725 42.28125591 41.85245781 41.4280084 41.00786356 40.59197965 40.18031346 39.7728222 39.36946355 38.97019558 38.57497681 38.18376618

35.39595271 35.01466798 34.63749043 34.26437584 33.89528044 33.53016093

1931.25 1950 1968.75 1987.5 2006.25 2025

115.875 117 118.125 119.25 120.375 121.5

2.85740569 2.886681154 2.916256559 2.946134978 2.976319516 3.006813308

37.79652304 37.41320715 37.03377869 36.65819822 36.28642672 35.91842557

108

0 203.91

0 10.8752

1

108

96

1.125

108

96

203.91 354.5471 364.8074

10.8752 18.90918 19.4564

1.125

108

88

1.227272727

108

87.48

1.234567901

108

86.4

386.3137 498.045 701.955 852.5921 905.865 996.09 1066.762 1088.269 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200

20.6034 26.5624 37.4376 45.47158 48.3128 53.1248 56.894 58.041 64 64 64 64 64 64 64 64 64 64 64 64

1.25

108

81

1.333333333

108

72

1.5

108

66

1.636363636

108

64

1.6875

108

60.75

1.777777778

108

58.32

1.851851852

108

57.6

1.875

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

1200 1200 1200 1200 1200 1200 1200

64 64 64 64 64 64 64

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

54

2

108

08 cm; 68 ET units

108

108 cm; 72 ET units

72

CENTS

Chord FRACTION

Chord LENGTH

CENTS

72

0 17.647 35.294 52.941 70.588 88.235 105.88 123.53 141.18 158.82 176.47 194.12 211.76 229.41 247.06 264.71 282.35 300 317.65

0 1.059 2.118 3.176 4.235 5.294 6.353 7.412 8.471 9.529 10.59 11.65 12.71 13.76 14.82 15.88 16.94 18 19.06

1 1.009673533 1.019440644 1.029302237 1.039259226 1.049312535 1.059463094 1.069711846 1.080059739 1.090507733 1.101056795 1.111707905 1.122462048 1.133320222 1.144283433 1.155352697 1.16652904 1.177813497 1.189207115

108 106.9652679 105.9404495 104.9254496 103.9201744 102.9245305 101.9384258 100.9617687 99.99446893 99.03643667 98.08758317 97.1478205 96.21706156 95.29522008 94.38221063 93.47794859 92.58235015 91.6953323 90.81681285

0 16.66667 33.33333 50 66.66667 83.33333 100 116.6667 133.3333 150 166.6667 183.3333 200 216.6667 233.3333 250 266.6667 283.3333 300

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

335.29 352.94 370.59 388.24 405.88 423.53 441.18 458.82 476.47 494.12 511.76 529.41 547.06 564.71 582.35 600 617.65 635.29 652.94 670.59 688.24 705.88 723.53 741.18 758.82 776.47 794.12 811.76

20.12 21.18 22.24 23.29 24.35 25.41 26.47 27.53 28.59 29.65 30.71 31.76 32.82 33.88 34.94 36 37.06 38.12 39.18 40.24 41.29 42.35 43.41 44.47 45.53 46.59 47.65 48.71

1.20071095 1.212326067 1.224053543 1.235894466 1.247849932 1.25992105 1.272108938 1.284414726 1.296839555 1.309384575 1.32205095 1.334839854 1.347752472 1.36079 1.373953647 1.387244634 1.400664191 1.414213562 1.427894004 1.441706784 1.455653183 1.469734492 1.483952018 1.498307077 1.512801 1.527435131 1.542210825 1.557129453

89.94671036 89.08494419 88.23143448 87.38610211 86.54886875 85.71965681 84.89838942 84.08499047 83.27938457 82.48149707 81.69125401 80.90858215 80.13340896 79.36566258 78.60527187 77.85216636 77.10627623 76.36753237 75.6358663 74.91121022 74.19349696 73.48266001 72.77863348 72.08135213 71.39075132 70.70676706 70.02933595 69.35839521

316.6667 333.3333 350 366.6667 383.3333 400 416.6667 433.3333 450 466.6667 483.3333 500 516.6667 533.3333 550 566.6667 583.3333 600 616.6667 633.3333 650 666.6667 683.3333 700 716.6667 733.3333 750 766.6667

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

829.41 847.06 864.71 882.35 900 917.65 935.29 952.94 970.59 988.24 1005.9 1023.5 1041.2 1058.8 1076.5 1094.1 1111.8 1129.4 1147.1 1164.7 1182.4 1200 1217.6 1235.3 1252.9 1270.6 1288.2 1305.9

49.76 50.82 51.88 52.94 54 55.06 56.12 57.18 58.24 59.29 60.35 61.41 62.47 63.53 64.59 65.65 66.71 67.76 68.82 69.88 70.94 72 73.06 74.12 75.18 76.24 77.29 78.35

1.572192397 1.587401052 1.602756829 1.61826115 1.633915453 1.649721189 1.665679821 1.681792831 1.698061709 1.714487966 1.731073122 1.747818715 1.764726298 1.781797436 1.799033713 1.816436725 1.834008086 1.851749425 1.869662384 1.887748625 1.906009824 1.924447674 1.943063882 1.961860175 1.980838295 2 2.019347066 2.038881287

68.69388265 68.03573669 67.38389633 66.73830116 66.09889134 65.46560761 64.83839128 64.21718421 63.60192884 62.99256814 62.38904563 61.79130539 61.199292 60.61295061 60.03222687 59.45706695 58.88741756 58.3232259 57.76443967 57.2110071 56.66287687 56.11999821 55.58232078 55.04979476 54.52237079 54 53.48263396 52.97022473

783.3333 800 816.6667 833.3333 850 866.6667 883.3333 900 916.6667 933.3333 950 966.6667 983.3333 1000 1016.667 1033.333 1050 1066.667 1083.333 1100 1116.667 1133.333 1150 1166.667 1183.333 1200 1216.667 1233.333

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 73 74

1323.5 1341.2 1358.8 1376.5 1394.1 1411.8 1429.4 1447.1 1464.7 1482.4 1500 1517.6 1535.3 1552.9 1570.6 1588.2 1605.9 1623.5 1641.2 1658.8 1676.5 1694.1 1711.8 1729.4 1747.1 1764.7 1782.4 1800

79.41 80.47 81.53 82.59 83.65 84.71 85.76 86.82 87.88 88.94 90 91.06 92.12 93.18 94.24 95.29 96.35 97.41 98.47 99.53 100.6 101.6 102.7 103.8 104.8 105.9 106.9 108

2.058604473 2.078518452 2.098625069 2.118926189 2.139423692 2.160119478 2.181015465 2.202113591 2.22341581 2.244924097 2.266640444 2.288566866 2.310705394 2.333058079 2.355626994 2.37841423 2.401421899 2.424652134 2.448107087 2.471788932 2.495699864 2.5198421 2.544217876 2.568829452 2.593679109 2.61876915 2.644101901 2.669679708

52.46272482 51.96008719 51.46226526 50.96921288 50.48088437 49.99723446 49.51821833 49.04379159 48.57391025 48.10853078 47.64761004 47.19110532 46.73897429 46.29117507 45.84766615 45.40840642 44.97335518 44.54247209 44.11571724 43.69305106 43.27443438 42.8598284 42.44919471 42.04249523 41.63969229 41.24074853 40.845627 40.45429108

1250 1266.667 1283.333 1300 1316.667 1333.333 1350 1366.667 1383.333 1400 1416.667 1433.333 1450 1466.667 1483.333 1500 1516.667 1533.333 1550 1566.667 1583.333 1600 1616.667 1633.333 1650 1666.667 1683.333 1700

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

1817.6 1835.3 1852.9 1870.6 1888.2 1905.9

109.1 110.1 111.2 112.2 113.3 114.4

2.695504944 2.72158 2.747907295 2.774489267 2.801328382 2.828427125

40.06670448 39.68283129 39.30263594 38.92608318 38.55313812 38.18376618

1716.667 1733.333 1750 1766.667 1783.333 1800

103 104 105 106 107 108

0 203.91

0 11.55

1

108

0 203.91

0 12.2346

1.125

108

203.91 354.55 364.81

11.55 20.09 20.67

1.125

108

203.91

12.2346

1.227272727

108

354.5471 21.27282 364.8074 21.88845

1.234567901

108

386.31 498.04 701.96 852.59 905.87 996.09 1066.8 1088.3 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200

21.89 28.22 39.78 48.31 51.33 56.45 60.45 61.67 68 68 68 68 68 68 68 68 68 68 68 68

1.25

108

386.3137 23.17882 498.045 701.955 29.8827 42.1173

1.333333333

108

1.5

108

1.636363636

108

852.5921 51.15552 905.865 996.09 54.3519 59.7654

1.6875

108

1.777777778

108

1.851851852

108

1066.762 64.00575 1088.269 65.29612 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 72 72 72 72 72 72 72 72 72 72 72 72

1.875

108

2

108

2

108

2

108

2

108

2

108

2

108

2

108

2

108

2

108

2

108

2

108

2

108

1200 1200 1200 1200 1200 1200 1200

68 68 68 68 68 68 68

2

108

1200 1200 1200 1200 1200 1200 1200

72 72 72 72 72 72 72

2

108

2

108

2

108

2

108

2

108

2

108

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) ((1+(1/100)l og10/(2)):1) (1224440064 :1220703125 ) (225:224) (15625:1555 56:2635 2) (393216:390 625) (126:125) (121:120) (100:99) (99:98) (2048:2025) 211:3452 (81:80)

3.600 3.990 5.292 7.712 8.107 11.445 13.795 14.367 17.399 17.576 19.553 21.506

0.22 0.24 0.32 0.46 0.49 0.69 0.83 0.86 1.04 1.05 1.17 1.29

0.20 0.23 0.30 0.44 0.46 0.65 0.78 0.81 0.99 1.00 1.11 1.22

parakleisma

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

minor comma, diaschisma 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

34:245

(21/53:1) (531441:524 312:219 288)

22.640 23.460

1.36 1.41

1.28 1.33

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

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

2 :5

7

3

(525:512) (40:39)

43.408 43.831 44.970 45.561 48.770 49.166

2.60 2.63 2.70 2.73 2.93 2.95 3.00

2.46 2.48 2.55 2.58 2.76 2.79 2.83

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

2 3 :57

2

2

50.000

(35:34) (34:33) (50331648:4 8828125)

50.184 51.682 52.504

3.01 3.10 3.15

2.84 2.93 2.98

equal temperament (ET) 1/4tone approximation

(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)

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

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

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

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)

5 :2 3

2

3

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 37:2 52

4.42 4.54 4.62 4.76 4.83 5.07 5.26 5.33 5.41

4.18 4.28 4.36 4.49 4.56 4.79 4.97 5.03 5.11

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

(21:20) (81:77) (20:19) (256:243)

84.467 87.676 88.801 90.225

28:35

(58:55) (135:128)

3 5:2

3

7

91.946 92.179

5.52 5.53

5.21 5.22

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

(96:91) (6442450944 :6103515625 ) (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

115.458 116.234 119.443 123.529

6.93 6.97 7.17 7.41

6.54 6.59 6.77 7.00

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

Cowell just semitone (halfstep), septimal diatonic semitone

(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)

133.810

8.03

7.58

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

138.573 140.828 142.373 143.159 143.498

8.31 8.45 8.54 8.59 8.61

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)

2 3:11

2

150.637

9.04

8.54

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

(59:54) (35:32) (23:21)

153.307 155.140 157.493 158.824

9.20 9.31 9.45 9.53

8.69 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 : 59049

216:310

168.213 170.423 173.268 176.210 176.646 178.636 179.697 180.450

10.09 10.23 10.40 10.57 10.60 10.72 10.78 10.83

9.53 9.66 9.82 9.99 10.01 10.12 10.18 10.23

71st harmonic Pythagorean diminished third, Pythagorean minor tone

(10:9)

25:3

2

182.404

10.94

10.34

minor whole-tone, just minor tone, smaller step

(49:40) (39:35) (29:26) (125:112) (48:43) (19:17) (160:143) (28:25) (121:108) (55:49) (21/6:1)

186.338 187.343 189.050 190.115 190.437 192.558 194.468 196.198 196.771 199.980

200.000

11.18 11.24 11.34 11.41 11.43 11.55 11.67 11.77 11.81 12.00 12.00

10.56 10.62 10.71 10.77 10.79 10.91 11.02 11.12 11.15 11.33 11.33

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

(64:57) (9:8)

3 :2

2

3

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

13.52 13.61 13.67

12.77 12.85 12.91

73rd harmonic

229.412

13.76

13.00

(8:7)

2 :7

3

231.174 233.333

13.87 14.00

13.10 13.22

septimal whole-tone, septimal major second

(63:55) (55:48) (39:34) (225:196) (31:27) (147:128) (169:147) (23:20) (2187:1900) (38:33) (144:125) (121:105)

235.104 235.677 237.527 238.886 239.171 239.607 241.449 241.961 243.545 244.240 244.969 245.541 247.059

14.11 14.14 14.25 14.33 14.35 14.38 14.49 14.52 14.61 14.65 14.70 14.73 14.82

13.32 13.36 13.46 13.54 13.55 13.58 13.68 13.71 13.80 13.84 13.88 13.91 14.00diminished 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) (32:27)

25:33

284.447 285.792 289.210 291.925 292.711 294.135

17.07 17.15 17.35 17.52 17.56 17.65

16.12 16.19 16.39 16.54 16.59 16.67

Pythagorean minor third, Pythagorean semiditone

(19:16)

297.513

17.85

16.86

overtone minor third

(21/4:1)

300.000

18.00

17.00

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) (40:33)

320.144 325.562 327.622 329.547 329.832 330.008 330.761 332.208 333.041 333.333

19.21 19.53 19.66 19.77 19.79 19.80 19.85 19.93 19.98 20.00

18.14 18.45 18.57 18.67 18.69 18.70 18.74 18.83 18.87 18.89

77th harmonic

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)

11:3

2

347.408 350.000

20.84 21.00

19.69 19.83

undecimal "median" third, undecimal neutral third

(60:49) (49:40) (38:31) (27:22)

350.617 351.338 352.477 354.547

21.04 21.08 21.15 21.27

19.87 19.91 19.97 20.09

(16:13)

359.472 356.250

21.57 21.38

20.37 20.19

24:13(79:64)

359.470 364.537

21.57 21.87

20.37 20.66

Tridecimal neutral third 79th harmonic

(100:81)

364.807

21.89

20.67

(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)

368.914 369.747 370.588

22.13 22.18 22.24

20.91 20.95 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) (21/3:1)

393.090 395.169 396.178 397.447 399.090 400.000

23.59 23.71 23.77 23.85 23.95 24.00

22.28 22.39 22.45 22.52 22.62 22.67

equal-tempered major third, exact

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

3 :2

4

6

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

31.29 31.40 31.54 31.72

29.55 29.65 29.79 29.96

529.412

31.76

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)/(100 :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)

547.211 550.000

32.83 33.00

31.01 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)

600.088 603.000 606.282 607.623 608.352 609.354 609.776

36.01 36.18 36.38 36.46 36.50 36.56 36.59

34.00 34.17 34.36 34.43 34.47 34.53 34.55

91st harmonic low 5-limit tritone

(729:512)

611.730

36.70

34.66

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) (23:16) (36:25)

621.418 624.999 626.343 628.274 631.283

37.29 37.50 37.58 37.70 37.88

35.21 35.42 35.49 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)

636.618 638.994 640.119

38.20 38.34 38.41

36.08 36.21 36.27

(42:29)

641.204 641.543

38.47 38.49

36.33 36.35

(29:20) (45:31) (93:64) (16:11)

24 : 11

643.263 645.188 646.991 648.682

38.60 38.71 38.82 38.92

36.45 36.56 36.66 36.76

93rd harmonic 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)

2 5:3

3

3

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) (25:16)

724.886 726.865 729.219 732.064 733.149 733.722 735.572 736.931 737.652 740.006 743.014 745.786 747.516 750.725 751.121 753.637 755.228 756.919 758.722 760.647 764.916 768.125 769.855 772.627

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 46.36

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 43.78

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 augmented fifth

(36:23) (11:7) (63:40)

11:7

775.636 782.492 786.422

46.54 46.95 47.19

43.95 44.34 44.56

undecimal minor sixth

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

2 :3

7

4

787.255 789.854 790.756 792.180 792.616 794.134 795.558 798.697 799.892 800.000

47.24 47.39 47.45 47.53 47.56 47.65 47.73 47.92 47.99 48.00

44.61 44.76 44.81 44.89 44.91 45.00 45.08 45.26 45.33 45.33

101st harmonic Pythagorean minor sixth

equal-tempered minor sixth, exact

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

2 :5

3

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

232:11

818.189 821.398 823.801 825.667 827.592 828.053 830.253 832.676 834.175 835.193 838.797 840.528 844.328 845.453 847.523 848.662 849.383 852.592 854.745

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 51.28

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 48.44

103rd harmonic

overtone sixth

undecimal "median" sixth, undecimal neutral sixth

(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) (77:45) (12:7)

908.075 910.790 914.208 915.553 918.642 921.821 922.409 923.264 924.622 925.418 929.920 933.129 937.632 941.126 941.562 943.050 946.195 946.924 947.496 949.696 952.259 953.299 955.031 955.760 958.039 960.829 964.323 964.896

54.48 54.65 54.85 54.93 55.12 55.31 55.34 55.40 55.48 55.53 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

51.46 51.61 51.81 51.88 52.06 52.24 52.27 52.32 52.40 52.44 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

109th harmonic

diminished seventh (16/9 x 24/25) septimal major sixth (inversion of a minimal third) 55th harmonic

223:7

(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)

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

(7:4)

7:2

2

968.826

58.13

54.90

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

2 :3

4

2

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

septimal minor seventh, harmonic seventh, (inversion of major tone). [some assert that 7:4 is one of the blue notes used in jazz] 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 1,017.596

59.97 60.00 60.00 60.23 60.45 60.66 60.82 60.88 61.06

56.64 56.67 56.67 56.88 57.09 57.29 57.44 57.49 57.66

(98:55) (25:14) (34:19) (52:29) (88:49) (115:64) (9:5)

115th harmonic 5-limit large minor seventh, greater just minor seventh, tonic seventh

32:5

(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)

1,052.572 1,054.432 1,055.647 1,056.502 1,057.627 1,059.172 1,061.427

63.15 63.27 63.34 63.39 63.46 63.55 63.69

59.65 59.75 59.82 59.87 59.93 60.02 60.15

59th harmonic inversion of minor tone

(50:27) (63:34) (13:7) (119:64) (54:29) (28:15) (58:31) (15

GKM 2009 Diastemata Intervals Psaltiki ALL 001 g

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