Gallery of arithmetic pitch sequences

This page is a gallery of interesting arithmetic pitch sequence (APS) scales.

This gallery can never realistically be complete as there are so many possible such scales, so feel free to add to it.

For a shorter, more strictly curated list of only the most widely used or most mathematically important APS scales, see APS#List of APSs.

Terms

These pages will help you understand terms used in this gallery:

• Subgroup
  • Dual-n
  • No-twos subgroup
• Prime harmonic
• Interval size measure
• Equal-step tuning
  • Edo, Edt, Ed4, Ed13

APS of cents

1ed1c

Trivial case. This is exactly 1200edo.

1ed28c

Main page: 1ed28c

1ed35.1c

Main page: Carlos Gamma

1ed39c

Main page: 1ed39c

1ed44c

Main page: 1ed44c

1ed63.8c

Main page: Carlos Beta

1ed65c

Main page: 1ed65c

1ed69c

Main page: 1ed69c

1ed78c

Main page: Carlos Alpha

1ed88c

Main page: 88cET

1ed97.5c

Main page: 1ed97.5c

1ed125c

Main page: 1ed125c

1ed237.8c

Main page: 1ed237.8c

APS of farabs

APS1farab

• Step size: 1 farab (~8.33 cents)
• Trivial case. This is exactly 144edo.

APS3.21farab

• Step size: 3.21 farabs (=26.75 cents)
• Close to: 45edo
• Compared to 45edo:
  • Much better 3/1, 5/1, 7/1, 11/1, 13/1, 17/1
  • Slightly worse 2/1

APS13farab

• Step size: 13 farabs (~108.33 cents)
• Close to: 11edo
• Special properties: no-3s
• Compared to 11edo:
  • 7/1 is about the same
  • Much better 5/1, 13/1, 19/1, 23/1, 29/1, 31/1
  • Slightly worse 2/1
  • Much worse 11/1, 17/1

APS of hekts

APS1hekt

• Step size: 1 hekt (~1.46304 cents)
• Trivial case. This is exactly 1300edt.

APS7hekt

• Step size: 7 hekts (~10.24128 cents)
• Close to: 117edo
• Compared to 117edo:
  • Much better 3/1, 5/1, 7/1, 23/1
  • Slightly better 17/1
  • Slightly worse 2/1, 11/1, 19/1
  • Much worse 13/1

APS11hekt

• Step size: 11 hekts (~16.09344 cents)
• Close to: 118edt
• Special properties: dual-2s
• Compared to 118edt:
  • 5/1, 19/1 are about the same
  • Much better 11/1, 13/1, 17/1
  • Slightly worse 3/1
  • Much worse 7/1

APS16hekt

• Step size: 16 hekts (~23.40864 cents)
• Close to: 51edo
• Compared to 51edo:
  • 17/1 is about the same
  • Much better 5/1, 11/1, 23/1, 29/1, 31/1
  • Slightly better 7/1, 19/1
  • Slightly worse 2/1, 3/1, 13/1

APS17hekt

• Step size: 17 hekts (~24.87168 cents)
• Close to: 48edo
• Special properties: dual-3
• Compared to 48edo:
  • 11/1, 13/1, 17/1, 19/1 are about the same
  • Much better 5/1
  • Much worse 3/1, 7/1

APS18hekt

• Step size: 18 hekts (~26.33472 cents)
• Close to: 72edt
• Special properties: dual-2s
• Compared to 72edt:
  • Much better 5/1, 7/1, 23/1
  • Slightly better 17/1
  • Slightly worse 3/1, 11/1, 13/1
  • Much worse 19/1

APS of jots

APS1jot

• Step size: 1 jot (~0.03986 cents)
• Trivial case. This is exactly 30103edo.

APS50jot

• Step size: 50 jots (~1.99316 cents)
• Close to: 602edo
• Compared to 602edo:
  • 2.3.5.7 is about the same
  • Much better 11/1, 13/1, 17/1, 29/1, 31/1
  • Much worse 19/1, 23/1

APS90jot

• Step size: 90 jots (~3.58768 cents)
• Close to: 530edt
• Special properties: dual-2s
• Compared to 530edt:
  • 2.3.5.7 is about the same
  • Much better 5/1, 7/1, 13/1, 19/1, 23/1, 29/1 and 31/1
  • Slightly worse 3/1

APS200jot

• Step size: 200 jots (~7.97262 cents)
• Close to: 303ed4, 557ed13
• Special properties: pretty much exactly 50% relative error on every harmonic in the 11-limit, making it a "dual-everything" tuning - dual-2, dual-3, dual-5, dual-7 and dual-11
• Comparison:
  • Step size (biggest to smallest): 303ed4, APS200jot, 557ed13
  • Properties:
    • 303ed4 is only dual-2
    • APS200jot is dual-3,5,7,11
    • 557ed13 is dual-3,5,7
  • Dual harmonics
    • APS200jot & 557ed13 have about equally-as-good dual 3/1, 5/1 and 7/1
  • Non-dual harmonics
    • 303ed4 is slightly better than the others at 11/1, 29/1
    • APS720jot is slightly better than the others at 17/1, 19/1
    • 557ed13 is slightly better than the others at 13/1, 23/1

APS300jot

• Step size: 300 jots (~11.96 cents)
• Close to: 100edo
• Compared to 100edo: Mostly about the same, but a much better 3/1 than 100edo

APS720jot

Main page: APS700jot

APS800jot

• Step size: 800 jots (~31.89 cents)
• Close to: 38edo
• Compared to 38edo: Mostly slightly worse than 38edo, but a much better 11/1 than 38edo

APS1000jot

• Step size: 1000 jots (~39.86 cents)
• Close to: 30edo
• Compared to 30edo:
  • Much better 3/1, 5/1, 17/1
  • Much worse 7/1, 13/1

APS2000jot

• Step size: 2000 jots (~79.73 cents)
• Close to: 15edo
• Compared to 15edo:
  • Much better 3/1, 5/1
  • Much worse 7/1

APS2500jot

• Step size: 2500 jots (~99.66 cents)
• Close to: 12edo
• Compared to 12edo:
  • Much better 5/1, 7/1, 11/1
  • Much worse 3/1, 17/1, 19/1

APS of mérides

APS4méride

• Step size: 4 méride (~111.63 cents)
• Close to: 17edt
• Special properties: no-2s
• Compared to 17edt:
  • 3/1 is about the same
  • Much better 5/1, 13/1, 17/1
  • Much worse 7/1, 11/1

APS2méride

• Step size: 2 méride (~55.81 cents)
• Close to: 34edt
• Compared to 34edt:
  • 3/1, 13/1 are about the same
  • Much better 5/1, 17/1
  • Much worse 7/1, 11/1

APS1méride

• Step size: 1 méride (~27.91 cents)
• Trivial case. This is exactly 43edo.

APS0.9méride

• Step size: 0.9 mérides (~25.12 cents)
• Close to: 48edo
• Compared to 48edo:
  • 17/1, 19/1, 23/1, 29/1, 31/1 are about the same
  • Much better 5/1, 7/1, 13/1
  • Slightly worse 2/1, 3/1, 11/1

APS0.8méride

• Step size: 0.8 mérides (~22.33 cents)
• Close to: 54edo
• Compared to 54edo:
  • 13/1, 17/1, 19/1 are about the same
  • Much better 3/1, 5/1, 7/1, 29/1, 31/1
  • Slightly better 11/1, 23/1
  • Slightly worse 2/1

APS0.75méride

• Step size: 0.75 mérides (~20.93 cents)
• Close to: 57edo
• Compared to 57edo:
  • 7/1 is about the same
  • Much better 3/1, 5/1
  • Slightly worse 11/1, 13/1
  • Much worse 2/1, 17/1, 19/1

APS0.3méride

• Step size: 0.3 mérides (~8.37 cents)
• Close to: 143edo
• Compared to 143edo:
  • 7/1 is about the same
  • Much better 17/1, 19/1
  • Slightly better 3/1, 11/1
  • Slightly worse 2/1, 5/1, 13/1

Non-integer equal divisions

13.888edo

Main page: 13.888edo