Comma pump examples

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Chord names are explained in Color notation #Chord names.

3-limit examples

256/243 (blackwood)

5-limit examples

81/80 (meantone)

These are very familiar from classical and popular music. See Meantone comma pump examples.

128/125 (augmented or trigu)

See 12edo comma pump examples.

250/243 (porcupine or triyo)

From Mizarian Ouverture by Herman Miller

2048/2025 (diaschismic or sagugu)

Pařízek's diaschisma pump play | xenpaper

Also see 12edo comma pump examples.

3125/3072 (magic or laquinyo)

15625/15552 (hanson or tribiyo)

16875/16384 (negri or laquadyo)

20000/19683 (tetracot or saquadyo)

78732/78125 (sensi or sepgu)

1600000/1594323 (amity or saquinyo)

2109375/2097152 (semicomma, 5-limit orwell or lasepyo)

7-limit examples

Every s6 chord can also be thought of as a s7 chord, e.g. zBbs6 could be Gs7.

126/125 (starling or zotrigu)

Monzo: [1 2 -3 1⟩

  • pump126 chords: Ch7–zBbs6–gEbh7–gEbs6–[ggGb=ryF#]h7–Ch7–gEbh7–[ggGg=ryF#]h7–Ch7

Ih7–zVIIs6–gIIIh7–gIIIs6–[ggV=ryIV]h7–Ih7–gIIIh7–[ggV=ryIV]h7–Ih7

Ch7 Ih7 1/1 5/4 3/2 7/4 ~ 125/126 625/504 125/84 125/72
zBbs6 zVIIs6 21/20 21/16 3/2 7/4 ~ 25/24 125/96 125/84 125/72
gEbh7 gIIIh7 21/20 6/5 3/2 9/5 ~ 25/24 25/21 125/84 25/14
gEbs6 gIIIs6 36/35 6/5 36/25 9/5 ~ 50/49 25/21 10/7 25/14
[ggGb=ryF#]h7 [ggV=ryIV]h7 27/25 63/50 36/25 9/5 ~ 15/14 5/4 10/7 25/14
Ch7 Ih7 126/125 63/50 189/125 441/250 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4 ~ 125/126 625/504 125/84 125/72
gEbs6 gIIIh7 21/20 6/5 3/2 9/5 ~ 25/24 25/21 125/84 25/14
[ggGb=ryF#]h7 [ggV=ryIV]h7 27/25 63/50 36/25 9/5 ~ 15/14 5/4 10/7 25/14
Ch7 Ih7 126/125 63/50 189/125 441/250 ~ 1/1 5/4 3/2 7/4

225/224 (marvel or ruyoyo)

Monzo: [5 2 2 -1⟩

  • pump225 chords: Ch7–zEbs6–zgCbh7–[zAb=yyG#]s6–yEh7–yEs6–Ch7–zEbs6–[zAb=yyG#]s6–yEs6–Ch7

Ih7–zIIIs6–zgIh7–[zVI=yyV]s6–yIIIh7–yIIIs6–Ih7–zIIIs6–[zVI=yyV]s6–yIIIs6–Ih7

Ch7 Ih7 1/1 5/4 3/2 7/4 ~ 225/224 1125/896 675/448 225/128
zEbs6 zIIIs6 1/1 7/6 7/5 7/4 ~ 225/224 75/64 45/32 225/128
zgCbh7 zgIh7 14/15 7/6 7/5 49/30 ~ 15/16 75/64 45/32 105/64
[zAb=yyG#]s6 [zVI=yyV]s6 14/15 7/6 4/3 14/9 ~ 15/16 75/64 75/56 25/16
yEh7 yIIIh7 14/15 49/45 56/45 14/9 ~ 15/16 35/32 5/4 25/16
yEs6 yIIIs6 14/15 16/15 56/45 112/75 ~ 15/16 15/14 5/4 3/2
Ch7 Ih7 224/225 56/45 112/75 392/225 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4 ~ 225/224 1125/896 675/448 225/128
zEbs6 zIIIs6 1/1 7/6 7/5 7/4 ~ 225/224 75/64 45/32 225/128
[zAb=yyG#]s6 [zVI=yyV]s6 14/15 7/6 4/3 14/9 ~ 15/16 75/64 75/56 25/16
yEs6 yIIIs6 14/15 16/15 56/45 112/75 ~ 15/16 15/14 5/4 3/2
Ch7 Ih7 224/225 56/45 112/75 392/225 ~ 1/1 5/4 3/2 7/4

245/243 (sensamagic or zozoyo)

Monzo: [0 -5 1 2⟩

  • pump245 chords: Ch7–rAh7–[rE=zyF]h7–yDh7–yAh7–Ch7–[rE=zyF]h7–yAh7–Ch7

Ih7–rVIh7–[rIII=zyIV]h7–yIIh7–yVIh7–Ih7–[rIII=zyIV]h7–yVIh7–Ih7

Ch7 Ih7 1/1 5/4 3/2 7/4 ~ 245/243 1225/972 245/162 1715/972
rAh7 rVIh7 15/14 9/7 3/2 12/7 ~ 175/162 35/27 245/162 140/81
[rE=zyF]h7 [rIII=zyIV]h7 27/28 9/8 9/7 45/28 ~ 35/36 245/216 35/27 175/108
yDh7 yIIh7 27/28 54/49 135/98 81/49 ~ 35/36 10/9 25/18 5/3
yAh7 yVIh7 405/392 243/196 81/56 81/49 ~ 25/24 5/4 35/24 5/3
Ch7 Ih7 243/245 243/196 729/490 243/140 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4 ~ 245/243 1225/972 245/162 1715/972
[rE=zyF]h7 [rIII=zyIV]h7 27/28 9/8 9/7 45/28 ~ 35/36 245/216 35/27 175/108
yAh7 yVIh7 405/392 243/196 81/56 81/49 ~ 25/24 5/4 35/24 5/3
Ch7 Ih7 243/245 243/196 729/490 243/140 ~ 1/1 5/4 3/2 7/4

686/675 (sengic or trizo-agugu)

Monzo: [1 -3 -2 3⟩

875/864 (keemic or zotriyo)

Monzo: [-5 -3 3 1⟩

1029/1024 (gamelismic or latrizo)

Monzo: [-10 1 0 3⟩

  • pump1029 chords: Ch7–yAs6–rDh7–ryBs6–[rrE=zF]h7–zFs6–Gh7–zBbs6–Ch7–rDh7–[rrE=zF]h7–Gh7–Ch7

Ih7–yVIs6–rIIh7–ryVIIs6–[rrIII=zIV]h7–zIVs6–Vh7–zVIIs6–Ih7–rIIh7–[rIII=zIV]h7–Vh7–Ih7

Ch7 Ih7 1/1 5/4 3/2 7/4
yAs6 yVIs6 1/1 5/4 10/7 5/3
rDh7 rIIh7 1/1 8/7 10/7 12/7
ryBs6 ryVIIs6 20/21 8/7 10/7 80/49
[rrE=zF]h7 [rrIII=zIV]h7 48/49 8/7 64/49 80/49 ~ 63/64 147/128 21/16 105/64
zFs6 zIVs6 48/49 384/343 64/49 384/245 ~ 63/64 9/8 21/16 63/40
Gh7 Vh7 384/343 64/49 512/343 640/343 ~ 9/8 21/16 3/2 15/8
zBbs6 zVIIs6 256/245 64/49 512/343 256/147 ~ 21/20 21/16 3/2 7/4
Ch7 Ih7 1024/1029 1280/1029 512/343 256/147 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
rDh7 rIIh7 1/1 8/7 10/7 12/7
[rrE=zF]h7 [rrIII=zIV]h7 48/49 8/7 64/49 80/49 ~ 63/64 147/128 21/16 105/64
Gh7 Vh7 384/343 64/49 512/343 640/343 ~ 9/8 21/16 3/2 15/8
Ch7 Ih7 1024/1029 1280/1029 512/343 256/147 ~ 1/1 5/4 3/2 7/4

1728/1715 (orwellismic or triru-agu)

Monzo: [6 3 -1 -3⟩

  • pump1728 Ch7–zEbs6–zEbh7–[zzGb=rgF]s6–rgFh7–gAbs6–gAbh7–Cs6–Ch7–zEbh7–[zzGb=rgF]h7–gAbh7–Ch7

Ih7–zIIIs6–zIIIh7–[zzV=rgIV]s6–rgIVh7–gVIs6–gVIh7–Is6–Ih7–zIIIh7–[zzV=rgIV]h7–gVIh7–Ih7

Ch7 Ih7 1/1 5/4 3/2 7/4
zEbs6 zIIIs6 1/1 7/6 7/5 7/4
zEbh7 zIIIh7 49/48 7/6 35/24 7/4
[zzGb=rgF]s6 [zzV=rgIV]s6 49/48 7/6 49/36 49/30 ~ 36/35 288/245 48/35 288/175
rgFh7 rgIVh7 49/48 343/288 49/36 245/144 ~ 36/35 6/5 48/35 12/7
gAbs6 gVIs6 343/288 49/36 343/216 343/180 ~ 6/5 48/35 8/5 48/25
gAbh7 gVIh7 1715/1728 343/288 2401/1728 343/216 ~ 1/1 6/5 7/5 8/5
Cs6 Is6 1715/1728 343/288 1715/1152 245/144 ~ 1/1 6/5 3/2 12/7
Ch7 Ih7 1715/1728 8575/6912 1715/1152 12005/6912 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
zEbh7 zIIIh7 49/48 7/6 35/24 7/4
[zzGb=rgF]h7 [zzV=rgIV]h7 49/48 343/288 49/36 245/144 ~ 36/35 6/5 48/35 12/7
gAbh7 gVIh7 1715/1728 343/288 2401/1728 343/216 ~ 1/1 6/5 7/5 8/5
Ch7 Ih7 1715/1728 8575/6912 1715/1152 12005/6912 ~ 1/1 5/4 3/2 7/4

2401/2400 (breed or bizozogu)

Monzo: [-5 -1 -2 4⟩

Ch7 Ih7 1/1 5/4 3/2 7/4
zBbs6 zVIIs6 21/20 21/16 3/2 7/4
zgGgh7 zgVh7 21/20 49/40 7/5 7/4
zzDbs6 zzIIs6 49/48 49/40 49/32 7/4
[zzgFb=rryD#]h7 [zzgIV=rryII]h7 343/320 49/40 49/32 147/80 ~ 15/14 60/49 75/49 90/49
ryC#s6 ryIs6 343/320 1029/800 1029/640 147/80 ~ 15/14 9/7 45/28 90/49
rAh7 rVIh7 343/320 1029/800 2401/1600 343/200 ~ 15/14 9/7 3/2 12/7
yEs6 yIIIs6 343/320 2401/1920 2401/1600 2401/1280 ~ 15/14 5/4 3/2 15/8
Gh7 Vh7 7203/6400 16807/12800 2401/1600 2401/1280 ~ 9/8 21/16 3/2 15/8
zBbs6 zVIIs6 16807/16000 16807/12800 2401/1600 16807/9600 ~ 21/20 21/16 3/2 7/4
Ch7 Ih7 2401/2400 2401/1920 2401/1600 16807/9600 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
zgGbh7 zgVh7 21/20 49/40 7/5 7/4
[zzgFb=rryD#]h7 [zzgIV=rryII]h7 343/320 49/40 49/32 147/80 ~ 15/14 60/49 75/49 90/49
rAh7 rVIh7 343/160 1029/400 2401/1600 343/200 ~ 15/7 18/7 3/2 12/7
Gh7 Vh7 7203/6400 16807/12800 2401/1600 2401/1280 ~ 9/8 21/16 3/2 15/8
Ch7 Ih7 2401/2400 2401/1920 2401/1600 16807/9600 ~ 1/1 5/4 3/2 7/4

3125/3087 (gariboh or triru-aquinyo)

Monzo: [0 2 5 -3⟩

3136/3125 (hemimean or zozoquingu)

Monzo: [6 0 -5 2⟩

Ch7 Ih7 1/1 5/4 3/2 7/4
zEbs6 zIIIs6 1/1 7/6 7/5 7/4
gAbh7 gVIh7 1/1 6/5 7/5 8/5
zgCbs6 zgIs6 14/15 28/25 7/5 8/5
zggEbbh7 zggIIIh7 49/50 28/25 7/5 42/25
zggEbbs6 zggIIIs6 24/25 28/25 168/125 42/25
[zg3Cbb=ryyA#]h7 [zg3I=ryyVI]h7 28/25 168/125 196/125 224/125 ~ 125/112 75/56 25/16 25/14
yyG#s6 yyVs6 147/125 168/125 196/125 1176/625 ~ 75/64 75/56 25/16 15/8
yEh7 yIIIh7 686/625 784/625 196/125 1176/625 ~ 35/32 5/4 25/16 15/8
yEs6 yIIIs6 672/625 784/625 4704/3125 1176/625 ~ 15/14 5/4 3/2 15/8
Ch7 Ih7 3136/3125 784/625 4704/3125 5488/3125 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
gAbh7 gVIh7 1/1 6/5 7/5 8/5
zggEbbh7 zggIIIh7 28/25 7/5 42/25 49/25
[zg3Cbb=ryyA#]h7 [zg3I=ryyVI]h7 28/25 168/125 196/125 224/125 ~ 125/112 75/56 25/16 25/14
yEh7 yIIIh7 686/625 784/625 196/125 1176/625 ~ 35/32 5/4 25/16 15/8
Ch7 Ih7 3136/3125 784/625 4704/3125 5488/3125 ~ 1/1 5/4 3/2 7/4

4000/3969 (octagar or rurutriyo)

Monzo: [5 6 3 -2⟩

Ch7 Ih7 1/1 5/4 3/2 7/4
yEs6 yIIIs6 1/1 5/4 10/7 5/3
ryF#h7 ryIVh7 15/14 5/4 10/7 25/14
yEs6 yIIIs6 1/1 5/4 10/7 5/3
ryBh7 ryVIIh7 20/21 25/21 10/7 5/3
[ryyD#=zgEb]s6 [ryyII=zgIII]s6 50/49 25/21 10/7 25/14 ~ 81/80 189/160 567/400 567/320
gFh7 gIVh7 50/49 25/21 200/147 250/147 ~ 81/80 189/160 27/20 27/16
wDs6 wIIs6 1000/1029 500/441 200/147 250/147 ~ 27/14 9/8 27/20 27/16
gBbh7 gVIIh7 500/441 200/147 100/63 800/441 ~ 9/8 27/20 63/40 9/5
zFs6 zIVs6 125/126 500/441 250/189 100/63 ~ 63/64 9/8 21/16 63/40
Gh7 Vh7 500/441 250/189 2000/1323 2500/1323 ~ 9/8 21/16 3/2 15/8
zBbs6 zVIIs6 200/189 250/189 2000/1323 1000/567 ~ 21/20 21/16 3/2 7/4
Ch7 Ih7 4000/3969 5000/3969 2000/1323 1000/567 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
ryF#h7 ryIVh7 15/14 5/4 10/7 25/14
ryBh7 ryVIIh7 20/21 25/21 10/7 5/3
[rryyE#=gF]h7 [rryyIII=gIV]h7 50/49 25/21 200/147 250/147 ~ 81/80 189/160 27/20 27/16
gBbh7 gVIIh7 500/441 200/147 100/63 800/441 ~ 9/8 27/20 63/40 9/5
Gh7 Vh7 500/441 250/189 2000/1323 2500/1323 ~ 9/8 21/16 3/2 15/8
Ch7 Ih7 4000/3969 5000/3969 2000/1323 1000/567 ~ 1/1 5/4 3/2 7/4

4375/4374 (ragismic or zoquadyo)

Monzo: [-1 -7 4 1⟩

Ch7 Ih7 1/1 5/4 3/2 7/4
yAs6 yVIs6 1/1 5/4 10/7 5/3
yAh7 yVIh7 25/24 5/4 35/24 5/3
zyCs6 zyIs6 35/36 7/6 35/24 5/3
yDh7 yIIh7 35/36 10/9 25/18 5/3
zyyAs6 zyyVIs6 35/36 175/144 25/18 175/108
yyBh7 yyVIIh7 25/27 125/108 25/18 175/108
[zyyD=ggDb]s6 [zyyII=ggII]s6 25/27 175/162 35/27 175/108 ~ 324/350 27/25 162/125 81/50
ggDbh7 ggIIh7 175/162 875/648 175/108 1225/648 ~ 27/25 27/20 81/50 189/100
gBbs6 gVIIs6 175/162 875/648 125/81 875/486 ~ 27/25 27/20 54/35 9/5
gBbh7 gVIIh7 4375/3888 875/648 6125/3888 875/486 ~ 9/8 27/20 63/40 9/5
Gs6 Vs6 4375/3888 625/486 4375/2916 875/486 ~ 9/8 9/7 3/2 9/5
gEbh7 gIIIh7 6125/5832 875/729 4375/2916 875/486 ~ 21/20 6/5 3/2 9/5
Cs6 Is6 4375/4374 875/729 4375/2916 1250/729 ~ 1/1 6/5 3/2 12/7
Ch7 Ih7 4375/4374 21875/17496 4375/2916 30625/17496 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
yAh7 yVIh7 25/24 5/4 35/24 5/3
yDh7 yIIh7 35/36 10/9 25/18 5/3
yyBh7 yyVIIh7 25/27 125/108 25/18 175/108
[zyyD=ggDb]h7 [zyyII=ggII]h7 175/162 875/648 175/108 1225/648 ~ 27/25 27/20 81/50 189/100
gBbh7 gVIIh7 4375/3888 875/648 6125/3888 875/486 ~ 9/8 27/20 63/40 9/5
gEbh7 gIIIh7 6125/5832 875/729 4375/2916 875/486 ~ 21/20 6/5 3/2 9/5
Ch7 Ih7 4375/4374 21875/17496 4375/2916 30625/17496 ~ 1/1 5/4 3/2 7/4

5120/5103 (hemifamity or saruyo)

Monzo: [9 -6 1 -1⟩

Ch7 Ih7 1/1 5/4 3/2 7/4
yEs6 yIIIs6 35/32 5/4 35/24 7/4
yAh7 yVIh7 25/24 5/4 35/24 5/3
yyIVs6 yyF#s6 25/24 25/21 25/18 5/3
yDh7 yIIh7 35/36 10/9 25/18 5/3
yDs6 yIIs6 20/21 10/9 4/3 5/3
[ry=w]Eh7 [ry=Lw]IIIh7 20/21 10/9 80/63 100/63 ~ 243/128 567/512 81/64 405/256
yC#s6 LyIs6 400/441 200/189 80/63 100/63 ~ 405/224 135/128 81/64 405/256
Ah7 wVIh7 200/189 80/63 40/27 320/189 ~ 135/128 81/64 189/128 27/16
zCs6 zIs6 80/81 32/27 40/27 320/189 ~ 63/64 189/160 189/128 27/16
Dh7 wIIh7 80/81 640/567 800/567 320/189 ~ 63/64 9/8 45/32 27/16
zDs6 zIVs6 80/81 640/567 320/243 128/81 ~ 63/64 9/8 21/16 63/40
Gh7 Vh7 1600/1701 640/567 320/243 2560/1701 ~ 15/16 9/8 21/16 3/2
zBbs6 zVIIs6 256/243 320/243 2560/1701 1280/729 ~ 21/20 21/16 3/2 7/4
Ch7 Ih7 5120/5103 6400/5103 2560/1701 1280/729 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
yAh7 yVIh7 25/24 5/4 35/24 5/3
yDh7 yIIh7 35/36 10/9 25/18 5/3
[ry=w]Eh7 [ry=Lw]IIIh7 20/21 10/9 80/63 100/63 ~ 243/128 567/512 81/64 405/256
Ah7 wVIh7 200/189 80/63 40/27 320/189 ~ 135/128 81/64 189/128 27/16
Dh7 wIIIh7 80/81 640/567 800/567 320/189 ~ 63/64 9/8 45/32 27/16
Gh7 Vh7 1600/1701 640/567 320/243 2560/1701 ~ 15/16 9/8 21/16 3/2
zBbs6 zVIIs6 256/243 320/243 2560/1701 1280/729 ~ 21/20 21/16 3/2 7/4
Ch7 Ih7 5120/5103 6400/5103 2560/1701 1280/729 ~ 1/1 5/4 3/2 7/4

6144/6125 (porwell or saruru-trigu)

Monzo: [11 1 -3 -2⟩

Ch7 Ih7 1/1 5/4 3/2 7/4
zyGs6 zyVs6 35/32 5/4 35/24 7/4
yEh7 yIIIh7 35/32 5/4 25/16 15/8
zyDs6 zyIIs6 35/32 21/16 105/64 15/8
zyDh7 zyIIh7 35/32 175/128 105/64 245/128
[zyyF#=rgF]s6 [zyy=rg]IVs6 525/512 175/128 75/64 105/64 ~ 36/35 48/35 288/245 288/175
rgFh7 rgIVh7 525/512 1225/1024 175/128 875/512 ~ 36/35 6/5 48/35 12/7
Cs6 Is6 6125/6144 1225/1024 6125/4096 875/512 ~ 1/1 6/5 3/2 12/7
gEbh7 gIIIh7 8575/8192 1225/1024 6125/4096 3675/2048 ~ 21/20 6/5 3/2 9/5
Gs6 Vs6 18375/16384 2625/2048 6125/4096 3675/2048 ~ 9/8 9/7 3/2 9/5
Gh7 Vh7 18375/16384 42875/32768 6125/4096 30625/16384 ~ 9/8 21/16 3/2 15/8
yEs6 yIIIs6 4375/4096 30625/24576 6125/4096 30625/16384 ~ 15/14 5/4 3/2 15/8
Ch7 Ih7 6125/6144 30625/24576 6125/4096 42875/24576 ~ 1/1 5/4 3/2 7/4
Ch7 Ih7 1/1 5/4 3/2 7/4
yEh7 yIIIh7 35/32 5/4 25/16 15/8
zyDh7 zyIIh7 35/32 175/128 105/64 245/128
[zyyF#=rgF]h7 [zyy=rg]IVh7 525/512 1225/1024 175/128 875/512 ~ 36/35 6/5 48/35 12/7
gEbh7 gIIIh7 8575/8192 1225/1024 6125/4096 3675/2048 ~ 21/20 6/5 3/2 9/5
Gh7 Vh7 18375/16384 42875/32768 6125/4096 30625/16384 ~ 9/8 21/16 3/2 15/8
Ch7 Ih7 6125/6144 30625/24576 6125/4096 42875/24576 ~ 1/1 5/4 3/2 7/4

10976/10935 (hemimage or satrizo-agu)

Monzo: [5 -7 -1 3⟩

64827/64000 (squalentine or laquadzo-atrigu)

Monzo: [-9 3 -3 4⟩

13-limit examples

676/675 (parizeksmic or bithogu)

Monzo: [2 -3 -2 0 0 2⟩

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