Comma pump examples
Chord names are explained here: 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 triple gu)
See 12edo comma pump examples.
250/243 (porcupine or triple yo)
2048/2025 (diaschismic or small gugu)
Also see 12edo comma pump examples.
3125/3072 (magic or large quintyo)
15625/15552 (hanson or sixfold yo)
16875/16384 (negri or large quadyo)
20000/19683 (tetracot or small quadyo)
78732/78125 (sensi or sevenfold gu)
1600000/1594323 (amity or small quintyo)
2109375/2097152 (semicomma, 5-limit orwell or large sevenfold yo)
7-limit examples
Every s7 chord can also be thought of as a z7(zg5) chord, e.g. zVIIs7 could be Vz7(zg5).
126/125 (starling or zogugu)
pump126 chords: Ch7 -- zBbs7 -- gEbh7 -- gEbs7 -- [ggGb=ryF#]h7 -- Ch7 -- gEbh7 -- [ggGg=ryF#]h7 -- Ch7
Ih7 -- zVIIs7 -- gIIIh7 -- gIIIs7 -- [ggV=ryIV]h7 -- Ih7 -- gIIIh7 -- [ggV=ryIV]h7 -- Ih7
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | ~ | 125/126 | 625/504 | 125/84 | 125/72 |
| zVIIs7 | 21/20 | 21/16 | 3/2 | 7/4 | ~ | 25/24 | 125/96 | 125/84 | 125/72 |
| gIIIh7 | 21/20 | 6/5 | 3/2 | 9/5 | ~ | 25/24 | 25/21 | 125/84 | 25/14 |
| gIIIs7 | 36/35 | 6/5 | 36/25 | 9/5 | ~ | 50/49 | 25/21 | 10/7 | 25/14 |
| [ggV=ryIV]h7 | 27/25 | 63/50 | 36/25 | 9/5 | ~ | 15/14 | 5/4 | 10/7 | 25/14 |
| Ih7 | 126/125 | 63/50 | 189/125 | 441/250 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | ~ | 125/126 | 625/504 | 125/84 | 125/72 |
| gIIIh7 | 21/20 | 6/5 | 3/2 | 9/5 | ~ | 25/24 | 25/21 | 125/84 | 25/14 |
| [ggV=ryIV]h7 | 27/25 | 63/50 | 36/25 | 9/5 | ~ | 15/14 | 5/4 | 10/7 | 25/14 |
| Ih7 | 126/125 | 63/50 | 189/125 | 441/250 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
225/224 (marvel or ruruyo)
pump225 chords: Ch7 -- zEbs7 -- zgCbh7 -- [zAb=yyG#]s7 -- yEh7 -- yEs7 -- Ch7 -- zEbs7 -- [zAb=yyG#]s7 -- yEs7 -- Ch7
Ih7 -- zIIIs7 -- zgIh7 -- [zVI=yyV]s7 -- yIIIh7 -- yIIIs7 -- Ih7 -- zIIIs7 -- [zVI=yyV]s7 -- yIIIs7 -- Ih7
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | ~ | 225/224 | 1125/896 | 675/448 | 225/128 |
| zIIIs7 | 1/1 | 7/6 | 7/5 | 7/4 | ~ | 225/224 | 75/64 | 45/32 | 225/128 |
| zgIh7 | 14/15 | 7/6 | 7/5 | 49/30 | ~ | 15/16 | 75/64 | 45/32 | 105/64 |
| [zVI=yyV]s7 | 14/15 | 7/6 | 4/3 | 14/9 | ~ | 15/16 | 75/64 | 75/56 | 25/16 |
| yIIIh7 | 14/15 | 49/45 | 56/45 | 14/9 | ~ | 15/16 | 35/32 | 5/4 | 25/16 |
| yIIIs7 | 14/15 | 16/15 | 56/45 | 112/75 | ~ | 15/16 | 15/14 | 5/4 | 3/2 |
| Ih7 | 224/225 | 56/45 | 112/75 | 392/225 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | ~ | 225/224 | 1125/896 | 675/448 | 225/128 |
| zIIIs7 | 1/1 | 7/6 | 7/5 | 7/4 | ~ | 225/224 | 75/64 | 45/32 | 225/128 |
| [zVI=yyV]s7 | 14/15 | 7/6 | 4/3 | 14/9 | ~ | 15/16 | 75/64 | 75/56 | 25/16 |
| yIIIs7 | 14/15 | 16/15 | 56/45 | 112/75 | ~ | 15/16 | 15/14 | 5/4 | 3/2 |
| Ih7 | 224/225 | 56/45 | 112/75 | 392/225 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
245/243 (sensamagic or zozoyo)
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
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | ~ | 245/243 | 1225/972 | 245/162 | 1715/972 |
| rVIh7 | 15/14 | 9/7 | 3/2 | 12/7 | ~ | 175/162 | 35/27 | 245/162 | 140/81 |
| [rIII=zyIV]h7 | 27/28 | 9/8 | 9/7 | 45/28 | ~ | 35/36 | 245/216 | 35/27 | 175/108 |
| yIIh7 | 27/28 | 54/49 | 135/98 | 81/49 | ~ | 35/36 | 10/9 | 25/18 | 5/3 |
| yVIh7 | 405/392 | 243/196 | 81/56 | 81/49 | ~ | 25/24 | 5/4 | 35/24 | 5/3 |
| Ih7 | 243/245 | 243/196 | 729/490 | 243/140 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | ~ | 245/243 | 1225/972 | 245/162 | 1715/972 |
| [rIII=zyIV]h7 | 27/28 | 9/8 | 9/7 | 45/28 | ~ | 35/36 | 245/216 | 35/27 | 175/108 |
| yVIh7 | 405/392 | 243/196 | 81/56 | 81/49 | ~ | 25/24 | 5/4 | 35/24 | 5/3 |
| Ih7 | 243/245 | 243/196 | 729/490 | 243/140 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
686/675 (sengic or triple-zo gugu)
875/864 (keemic or zo triple-yo)
1029/1024 (gamelismic or large triple zo)
pump1029 chords: Ch7 -- yAs7 -- rDh7 -- ryBs7 -- [rrE=zF]h7 -- zFs7 -- Gh7 -- zBbs7 -- Ch7 -- rDh7 -- [rrE=zF]h7 -- Gh7 -- Ch7
Ih7 -- yVIs7 -- rIIh7 -- ryVIIs7 -- [rrIII=zIV]h7 -- zIVs7 -- Vh7 -- zVIIs7 -- Ih7 -- rIIh7 -- [rIII=zIV]h7 -- Vh7 -- Ih7
Ih7 = w1 - y3 - w5 - z7 = 1/1-5/4-3/2-7/4
yVIs7 = w1 - y3 - ry4 - y6 = 1/1-5/4-10/7-5/3
rIIh7 = w1 - r2 - ry4 - r6 = 1/1-8/7-10/7-12/7
ryVIIs7 = ry7 - r2 - ry4 - rry5 = 20/21-8/7-10/7-80/49
rrIIIh7 = rr7 - r2 - rr3 - rry5 = 48/49-8/7-64/49-80/49
zIVs7 = rrIIIs7 = rr7 - r31 - rr3 - rrg5 = 48/49-384/343-64/49-384/245
Vh7 = r3IVh7 = r31 - rr3 - r34 - r3y6 = 384/343-64/49-512/343-640/343
zVIIs7 = rrVIs7 = rrg1 - rr3 - r34 - rr6 = 256/245-64/49-512/343-256/147
Ih7 = sr3VIIh7 = sr37 - r3y2 - r34 - rr6 = 1024/1029-1280/1029-512/343-256/147
Ih7 = 1/1-5/4-3/2-7/4
rIIh7 = 1/1-8/7-10/7-12/7
rIIIh7 = 48/49-8/7-64/49-80/49
Vh7 = r3IVh7 = 384/343-64/49-512/343-640/343
Ih7 = sr3VIIh7 = 1024/1029-1280/1029-512/343-256/147
1728/1715 (orwellismic or triple-ru gu)
pump1728 Ch7 -- zEbs7 -- zEbh7 -- [zzGb=rgF]s7 - rgFh7 -- gAbs7 -- gAbh7 -- Cs7 -- Ch7 -- zEbh7 -- [zzGb=rgF]h7 -- gAbh7 -- Ch7
Ih7 -- zIIIs7 -- zIIIh7 -- [zzV=rgIV]s7 -- rgIVh7 -- gVIs7 -- gVIh7 -- Is7 -- Ih7 -- zIIIh7 -- [zzV=rgIV]h7 -- gVIh7 -- Ih7
Ih7 = w1 - y3 - w5 - z7 = 1/1-5/4-3/2-7/4
zIIIs7 = w1 - z3 - zg5 - z7 = 1/1-7/6-7/5-7/4
zIIIh7 = zz2 - z3 - zy5 - z7 = 49/48-7/6-35/24-7/4
zzVs7 = zz2 - z3 - zz5 - zzg7 = 49/48-7/6-49/36-49/30
rgIVh7 = zzVh7 = zz2 - z34 - zz5 - zzy7 = 49/48-343/288-49/36-245/144
gVIs7 = z3VIIs7 = z34 - zz5 - z37 - z3g9 = 343/288-49/36-343/216-343/180
gVIh7 = z3VIIh7 = z3y2 - z34 - z46 - zzz7 = 1715/1728-343/288-2401/1728-343/216
Is7 = z3yIIs7 = z3y2 - z34 - z3y6 - zzy7 = 1715/1728-343/288-1715/1152-245/144
Ih7 = z3yIIh7 = z3y2 - z3yy4 - z3y6 - z4y8 = 1715/1728-8575/6912-1715/1152-12005/6912
Ih7 = 1/1-5/4-3/2-7/4
zIIIh7 = 49/48-7/6-35/24-7/4
rgIVh7 = zzVh7 = 49/48-343/288-49/36-245/144
gVIh7 = z3VIIh7 = 1715/1728-343/288-2401/1728-343/216
Ih7 = z3yIIh7 = 1715/1728-8575/6912-1715/1152-12005/6912
2401/2400 (breed or quadzo gugu)
pump2401 chords: Ch7 -- zBbs7 -- zgGbh7 -- zzDbs7 -- [zzgFb=rryD#]h7 -- ryC#s7 -- rAh7 -- yEs7 -- Gh7 -- zBbs7 -- Ch7 -- zgGbh7 -- [zzgFb=rryD#]h7 -- rAh7 -- Gh7 -- Ch7
Ih7 -- zVIIs7 -- zgVh7 -- zzIIs7 -- [zzgIV=rryII]h7 -- ryIs7 -- rVIh7 -- yIIIs7 -- Vh7 -- zVIIs7 -- Ih7 -- zgVh7 -- [zzgIV=rryII]h7 -- rVIh7 -- Vh7 -- Ih7
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | |||||
| zVIIs7 | 21/20 | 21/16 | 3/2 | 7/4 | |||||
| zgVh7 | 21/20 | 49/40 | 7/5 | 7/4 | |||||
| zzIIs7 | 49/48 | 49/40 | 49/32 | 7/4 | |||||
| [zzgIV=rryII]h7 | 343/320 | 49/40 | 49/32 | 147/80 | ~ | 15/14 | 60/49 | 75/49 | 90/49 |
| ryIs7 | 343/320 | 1029/800 | 1029/640 | 147/80 | ~ | 15/14 | 9/7 | 45/28 | 90/49 |
| rVIh7 | 343/320 | 1029/800 | 2401/1600 | 343/200 | ~ | 15/14 | 9/7 | 3/2 | 12/7 |
| yIIIs7 | 343/320 | 2401/1920 | 2401/1600 | 2401/1280 | ~ | 15/14 | 5/4 | 3/2 | 15/8 |
| Vh7 | 7203/6400 | 16807/12800 | 2401/1600 | 2401/1280 | ~ | 9/8 | 21/16 | 3/2 | 15/8 |
| zVIIs7 | 16807/16000 | 16807/12800 | 2401/1600 | 16807/9600 | ~ | 21/20 | 21/16 | 3/2 | 7/4 |
| Ih7 | 2401/2400 | 2401/1920 | 2401/1600 | 16807/9600 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | |||||
| zVIIs7 | 21/20 | 21/16 | 3/2 | 7/4 | |||||
| zgVh7 | 21/20 | 49/40 | 7/5 | 7/4 | |||||
| zzIIs7 | 49/48 | 49/40 | 49/32 | 7/4 | |||||
| [zzgIV=rryII]h7 | 343/320 | 49/40 | 49/32 | 147/80 | ~ | 15/14 | 60/49 | 75/49 | 90/49 |
| ryIs7 | 343/320 | 1029/800 | 1029/640 | 147/80 | ~ | 15/14 | 9/7 | 45/28 | 90/49 |
| rVIh7 | 343/320 | 1029/800 | 2401/1600 | 343/200 | ~ | 15/14 | 9/7 | 3/2 | 12/7 |
| yIIIs7 | 343/320 | 2401/1920 | 2401/1600 | 2401/1280 | ~ | 15/14 | 5/4 | 3/2 | 15/8 |
| Vh7 | 7203/6400 | 16807/12800 | 2401/1600 | 2401/1280 | ~ | 9/8 | 21/16 | 3/2 | 15/8 |
| zVIIs7 | 16807/16000 | 16807/12800 | 2401/1600 | 16807/9600 | ~ | 21/20 | 21/16 | 3/2 | 7/4 |
| Ih7 | 2401/2400 | 2401/1920 | 2401/1600 | 16807/9600 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
Ih7 = w1 - y3 - w5 - z7 = 1/1-5/4-3/2-7/4
zVIIs7 = zg2 - z4 - w5 - z7 = 21/20-21/16-3/2-7/4
zgVh7 = zg2 - zzg4 - zg5 - z7 = 21/20-49/40-7/5-7/4
zzIIs7 = zz2 - zzg4 - zz6 - z7 = 49/48-49/40-49/32-7/4
rryII = zzgIVh7 = z3g3 - zzg4 - zz6 - zzg8 = 343/320-49/40-49/32-147/80
ryIs7 = z3gIIIs7 = z3g3 - z3gg5 - z3g7 - zzg8 = 343/320-1029/800-1029/640-147/80
rVIh7 = z3ggIh7 = z3g3 - z3gg5 - z4gg7 - z3gg8 = 343/320-1029/800-2401/1600-343/200
yIIIs7 = z4gVs7 = z3g3 - z4g5 - z4gg7 - z4g9 = 343/320-2401/1920-2401/1600-2401/1280
Vh7 = z4ggVIIh7 = z4gg4 - z5gg6 - z4gg7 - z4g9 = 7203/6400-16807/12800-2401/1600-2401/1280
zVIIs7 = z5ggIIs7 = z5g34 - z5gg6 - z4gg7 - z5gg9 = 16807/16000-16807/12800-2401/1600-16807/9600
Ih7 = z4ggIIIh7 = 2401/2400-2401/1920-2401/1600-16807/9600
Ih7 = 1/1-5/4-3/2-7/4
zgVh7 = 21/20-49/40-7/5-7/4
rryII = zzgIVh7 = 343/320-49/40-49/32-147/80
rVIh7 = 2401/1600-343/200-343/160-1029/400
Vh7 = z4ggVIIh7 = 7203/6400-16807/12800-2401/1600-2401/1280
Ih7 = z4ggIIIh7 = 2401/2400-2401/1920-2401/1600-16807/9600
3125/3087 (gariboh or triple-ru quintyo)
3136/3125 (hemimean or zozo quintgu)
pump3136 chords: Ch7 -- zEbs7 -- gAbh7 -- zgCbs7 -- zggEbbh7 -- zggEbbs7 -- [zg3Cbb=ryyA#]h7 -- yyG#s7 -- yEh7 -- yEs7 -- Ch7 -- -- Ch7 -- gAbh7 -- zggEbbh7 -- [zg3Cbb=ryyA#]h7 -- yEh7 -- Ch7
Ih7 -- zIIIs7 -- gVIh7 -- zgIs7 -- zggIIIh7 -- zggIIIs7 -- [zg3I=ryyVI]h7 -- yyVs7 -- yIIIh7 -- yIIIs7 -- Ih7 -- gVIh7 -- zggIIIh7 -- [zg3I=ryyVI]h7 -- -- yIIIh7 -- Ih7
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | |||||
| zIIIs7 | 1/1 | 7/6 | 7/5 | 7/4 | |||||
| gVIh7 | 1/1 | 6/5 | 7/5 | 8/5 | |||||
| zgIs7 | 14/15 | 28/25 | 7/5 | 8/5 | |||||
| zggIIIh7 | 49/50 | 28/25 | 7/5 | 42/25 | |||||
| zggIIIs7 | 24/25 | 28/25 | 168/125 | 42/25 | |||||
| [zg3I=ryyVI]h7 | 28/25 | 168/125 | 196/125 | 224/125 | ~ | 125/112 | 75/56 | 25/16 | 25/14 |
| yyVs7 | 147/125 | 168/125 | 196/125 | 1176/625 | ~ | 75/64 | 75/56 | 25/16 | 15/8 |
| yIIIh7 | 686/625 | 784/625 | 196/125 | 1176/625 | ~ | 35/32 | 5/4 | 25/16 | 15/8 |
| yIIIs7 | 672/625 | 784/625 | 4704/3125 | 1176/625 | ~ | 15/14 | 5/4 | 3/2 | 15/8 |
| Ih7 | 3136/3125 | 784/625 | 4704/3125 | 5488/3125 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
| Ih7 | 1/1 | 5/4 | 3/2 | 7/4 | |||||
| gVIh7 | 1/1 | 6/5 | 7/5 | 8/5 | |||||
| zggIIIh7 | 28/25 | 7/5 | 42/25 | 49/25 | |||||
| [zg3I=ryyVI]h7 | 28/25 | 168/125 | 196/125 | 224/125 | ~ | 125/112 | 75/56 | 25/16 | 25/14 |
| yIIIh7 | 686/625 | 784/625 | 196/125 | 1176/625 | ~ | 35/32 | 5/4 | 25/16 | 15/8 |
| Ih7 | 3136/3125 | 784/625 | 4704/3125 | 5488/3125 | ~ | 1/1 | 5/4 | 3/2 | 7/4 |
4000/3969 (octagar or ruru triple-yo)
1-5/4-3/2-7/4
1-5/4-10/7-5/3
15/14-5/4-10/7-25/14
1-5/4-10/7-5/3
20/21-25/21-10/7-5/3
50/49-25/21-10/7-25/14
50/49-25/21-200/147-250/147
1000/1029-500/441-200/147-250/147
500/441-200/147-100/63-800/441
125/126-500/441-250/189-100/63
500/441-250/189-2000/1323-2500/1323
200/189-250/189-2000/1323-1000/567
4000/3969-5000/3969-2000/1323-1000/567
1-5/4-3/2-7/4
15/14-5/4-10/7-25/14
20/21-25/21-10/7-5/3
50/49-25/21-200/147-250/147
500/441-200/147-100/63-800/441
500/441-250/189-2000/1323-2500/1323
4000/3969-5000/3969-2000/1323-1000/567
4375/4374 (ragismic or zo quadyo)
1-5/4-3/2-7/4
1-5/4-10/7-5/3
25/24-5/4-35/24-5/3
35/36-7/6-35/24-5/3
35/36-10/9-25/18-5/3
35/36-175/144-25/18-175/108
25/27-125/108-25/18-175/108
25/27-175/162-35/27-175/108
175/162-875/648-175/108-1225/648 ~ 27/25-27/20-81/50-189/100
175/162-875/648-125/81-875/486 ~ 27/25-27/20-54/35-9/5
4375/3888-875/648-6125/3888-875/486 ~ 9/8-27/20-63/40-9/5
4375/3888-625/486-4375/2916-875/486 ~ 9/8-9/7-3/2-9/5
6125/5832-875/729-4375/2916-875/486 ~ 21/20-6/5-3/2-9/5
4375/4374-875/729-4375/2916-1250/729 ~ 1-6/5-3/2-12/7
4375/4374-21875/17496-4375/2916-30625/17496 ~ 1-5/4-3/2-7/4
1-5/4-3/2-7/4
25/24-5/4-35/24-5/3
35/36-10/9-25/18-5/3
25/27-125/108-25/18-175/108
175/162-875/648-175/108-1225/648 ~ 27/25-27/20-81/50-189/100
4375/3888-875/648-6125/3888-875/486 ~ 9/8-27/20-63/40-9/5
6125/5832-875/729-4375/2916-875/486 ~ 21/20-6/5-3/2-9/5
4375/4374-21875/17496-4375/2916-30625/17496 ~ 1-5/4-3/2-7/4
5120/5103 (hemifamity or small ruyo)
1-5/4-3/2-7/4
35/32-5/4-35/24-7/4
25/24-5/4-35/24-5/3
25/24-25/21-25/18-5/3
35/36-10/9-25/18-5/3
20/21-10/9-4/3-5/3
20/21-10/9-80/63-100/63
400/441-200/189-80/63-100/63
200/189-80/63-40/27-320/189
80/81-32/27-40/27-320/189
80/81-640/567-800/567-320/189
80/81-640/567-320/243-128/81
1600/1701-640/567-320/243-2560/1701
256/243-320/243-2560/1701-1280/729
5120/5103-6400/5103-2560/1701-1280/729
1-5/4-3/2-7/4
25/24-5/4-35/24-5/3
35/36-10/9-25/18-5/3
20/21-10/9-80/63-100/63
200/189-80/63-40/27-320/189
80/81-640/567-800/567-320/189
1600/1701-640/567-320/243-2560/1701
256/243-320/243-2560/1701-1280/729
5120/5103-6400/5103-2560/1701-1280/729
6144/6125 (porwell or small ruru triple-gu)
1-5/4-3/2-7/4
35/32-5/4-35/24-7/4
35/32-5/4-25/16-15/8
35/32-21/16-105/64-15/8
35/32-175/128-105/64-245/128
525/512-175/128-75/64-105/64
525/512-1225/1024-175/128-875/512
6125/6144-1225/1024-6125/4096-875/512
8575/8192-1225/1024-6125/4096-3675/2048
18375/16384-2625/2048-6125/4096-3675/2048
18375/16384-42875/32768-6125/4096-30625/16384
4375/4096-30625/24576-6125/4096-30625/16384
6125/6144-30625/24576-6125/4096-42875/24576
1-5/4-3/2-7/4
35/32-5/4-25/16-15/8
35/32-175/128-105/64-245/128
525/512-1225/1024-175/128-875/512
8575/8192-1225/1024-6125/4096-3675/2048
18375/16384-42875/32768-6125/4096-30625/16384
6125/6144-30625/24576-6125/4096-42875/24576