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 Equal-step tuning#List of notable APS.

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

See also: cent.

1ed1c

Trivial case. This is exactly 1200edo.

1ed13.94—13.97c

Main page: Delta scale

1ed28c

Main page: 1ed28c

1ed35.1c

Main page: Carlos Gamma

1ed39c

Main page: 1ed39c

1ed44c

Main page: 1ed44c

1ed63.59—63.82c

Main page: Phoenix

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

1ed148.5c

Main page: 1ed148.5c

1ed237.8c

Main page: 1ed237.8c

APS of farabs

See also: farab.

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

APS of genes

See also: gene.

APS1gene

• Step size: 1 gene (~ 3.85852 cents)
• Trivial case. This is exactly 311edo.

APS2.5gene

• Step size: 2.5 genes (~9.64630 cents)
• Close to: 197edt
• Special properties: dual-2s
• Compared to 197edt:
  • 197edt is not dual-2, so APS2.5gene completely restructures the tuning

APS9gene

• Step size: 9 genes (~34.73 cents)
• Close to: 55edt
• Special properties: no-2s
• Compared to 55edt:
  • Much better 5/1, 7/1, 13/1, 19/1, 29/1
  • Slightly worse 3/1, 17/1
  • Much worse 11/1, 23/1

APS12gene

• Step size: 12 genes (~46.30 cents)
• Close to: 26edo
• Compared to 26edo:
  • 2/1 is about the same
  • Much better 3/1, 5/1, 13/1, 17/1, 19/1, 23/1, 29/1
  • Much worse 7/1, 11/1, 31/1

APS15gene

• Step size: 15 genes (~57.88 cents)
• Close to: 33edt
• Special properties: no-2s
• Compared to 33edt:
  • Much better 5/1, 7/1, 19/1
  • Slightly worse 3/1
  • Much worse 11/1, 13/1, 17/1

APS of hekts

See also: hekt.

APS1hekt

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

APS7hekt

• Step size: 7 hekts (~10.24 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

APS16hekt

• Step size: 16 hekts (~23.41 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

APS18hekt

• Step size: 18 hekts (~26.33 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

See also: jot.

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:
  • Slightly better 5/1, 7/1, 11/1, 13/1, 17/1, 19/1, 23/1, 29/1, 31/1
  • Slightly worse 3/1

APS99jot

• Step size: 99 jot (~3.94614 cents)
• Close to: 304edo
• Compared to 304edo:
  • 2/1 are about the same
  • Much better 7/1, 17/1
  • Slightly better 3/1, 5/1, 11/1, 19/1
  • Slightly worse 13/1, 29/1
  • Much worse 23/1, 31/1

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

APS715jot

Main page: APS715jot

APS720jot

Main page: APS720jot

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:
  • 2/1 is about the same
  • 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

APS of mérides

See also: méride.

APS1méride

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

APS11/12méride

• Step size: 11/12 of a méride (~25.58 cents)
• Close to: 47edo
• Compared to 47edo:
  • Much better: 3/1, 11/1, 23/1
  • Slightly better 5/1, 19/1
  • Slightly worse 2/1, 7/1, 13/1, 17/1
  • Much worse 13/1

APS9/10méride

• Step size: 9/10 of a méride (~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

APS7/8méride

• Step size: 7/8 of a méride (~24.42 cents)
• Close to: 49edo
• Compared to 49edo:
  • Much better 3/1, 7/1, 11/1, 13/1, 11/1
  • Slightly better 5/1, 23/1
  • Slightly worse 2/1, 19/1

APS5/6méride

• Step size: 5/6 of a méride (~23.26 cents)
• Close to: 52edo
• Compared to 52edo:
  • Better 3/1, 13/1, 17/1
  • Slightly better 5/1
  • Slightly worse 7/1, 19/1, 23/1
  • Worse 2/1, 11/1

APS4/5méride

• Step size: 4/5 of a méride (~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

APS8/11méride

• Step size: 8/11 of a méride (~20.30 cents)
• Close to: 59edo
• Compared to 59edo:
  • Much better 3/1, 7/1, 19/1
  • Slightly better 13/1
  • Slightly worse 2/1, 5/1, 17/1
  • Much worse 11/1, 23/1

APS2/3méride

• Step size: 2/3 of a méride (~18.60 cents)
• Close to: 103edt, 130ed4, 181ed7
• Special properties: pretty much exactly 50% relative error (but less than 10c absolute) on 2/1, and low relative error on everything else, making it a great dual-2 tuning.
• Comparison:
  • Step size (biggest to smallest): 181ed7, APS2/3méride, 130ed4, 103edt
  • Dual harmonics
    • APS2/3méride & 130ed4 have slightly better dual-2 than the others
  • Non-dual harmonics
    • 181ed7 is slightly better than the others at 7/1, 11/1
    • APS2/3méride is slightly better than the others at 7/1, 13/1, 17/1, 19/1
    • 130ed4 is slightly better than the others at 3/1, 5/1, 17/1, 23/1
    • 103edt is slightly better than the others at 3/1, 5/1, 23/1

APS7/11méride

• Step size: 7/11 of a méride (~17.76 cents)
• Close to: 107edt, 135ed4, 157ed5
• Special properties: almost 50% relative error (but less than 10c absolute) on 2/1, and low relative error on everything else, making it a great dual-2 tuning.
• Comparison:
  • Step size (biggest to smallest): 135ed4, 107edt, APS7/11méride, 175ed5
  • Dual-2
    • Excellent: 135ed4, 107edt
    • Good: APS7/11méride
    • Weak: 175ed5
  • Non-dual harmonics
    • 135ed4 is slightly better than the others at 3/1, 17/1
    • 107edt is slightly better than the others at 3/1, 17/1
    • APS7/11méride is slightly better than the others at 3/1, 5/1, 7/1, 11/1, 13/1, 19/1
    • 175ed5 is slightly better than the others at 5/1, 7/1, 11/1, 23/1

APS5/8méride

• Step size: 5/8 of a méride (~17.44 cents)
• Close to: 69edo
• Compared to 69edo:
  • 5/1 is about the same
  • Slightly better 3/1, 7/1, 11/1
  • Slightly worse 2/1, 13/1, 17/1, 19/1, 23/1

APS7/12méride

• Step size: 7/12 of a méride (~16.28 cents)
• Close to: 74edo
• Compared to 74edo:
  • 11/1 is about the same
  • Better 3/1, 5/1, 7/1, 17/1, 19/1
  • Worse 2/1, 13/1, 23/1

APS6/11méride

• Step size: 6/11 of a méride (~15.22 cents)
• Close to: 79edo
• Compared to 79edo:
  • Better 3/1, 5/1, 11/1, 13/1, 19/1
  • Worse 2/1, 7/1, 17/1, 23/1

APS of tinas

See also: tina.

APS1tina

• Step size: 1 tina (~ 0.14053 cents)
• Trivial case. This is exactly 8539edo.

APS40tina

• Step size: 40 tina (~5.6212 cents)
• Close to: 427ed4, 790ed13
• 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): APS40tina, 790ed13, 427ed4
  • Properties:
    • all three tunings are dual-everything: a bit shaky on 5/1 and 7/2, but still more than dual enough
  • Dual harmonics
    • APS40tina has the most dual 5/1, 11/1
    • 790ed13 is a balanced compromise between both other tunings
    • 427ed4 has the most dual 2/1, 3/1, 7/1
  • Non-dual harmonics
    • APS40tina has the best 29/1
    • 790ed13 has the best 13/1
    • 427ed4 has the best 17/1, 19/1, 23/1, 31/1

The Argentina scale (APS47tina)

• Step size: 47 tina (~6.60491 cents)
• Close to: 182edo
• Compared to 182edo:
  • 19/1, 23/1, 29/1, 31/1 are about the same
  • Much better 3/1, 5/1, 13/1
  • Slightly better 7/1
  • Slightly worse 2/1, 11/1, 17/1
• Name reasoning: Argent means silver. 47 is the atomic number of silver.

APS60tina

• Step size: 60 tina (~8.43180 cents)
• Close to: 285ed4, 582ed17
• Special properties: close to 50% relative error on every harmonic in the 11-limit, making it a "dual-everything" tuning - dual-2, dual-3, dual-5, dual-7, dual-11 and dual-13
• Comparison:
  • Step size (biggest to smallest): APS60tina, 582ed17, 285ed4
  • Properties:
    • APS60tina is dual-2, -3, -5, -7, -11 and -13
    • 582ed17 is dual-2, -3, -5, -7 and -11
    • 285ed4 is only dual-2
  • Dual harmonics (ignoring 285ed4)
    • APS60tina has the most dual 3/1, 5/1 and 7/1
    • 582ed17 has the most dual 2/1 and 11/1
  • Non-dual harmonics (ignoring 285ed4)
    • APS60tina has the best 31/1
    • 582ed17 has the best 17/1, 19/1, 23/1, 29/1

APS70tina

• Step size: 70 tina (~9.84 cents)
• Close to: 122edo
• Compared to 122edo:
  • Slightly better 3/1, 5/1, 7/1, 11/1, 13/1, 19/1
  • Slightly worse 2/1, 17/1

APS80tina

• Step size: 80 tina (~11.24 cents)
• Close to: 107edo
• Compared to 107edo:
  • Much better 3/1, 5/1
  • Slightly better 7/1, 13/1, 17/1, 19/1
  • Slightly worse 2/1, 11/1, 23/1

APS90tina

• Step size: 90 tina (~12.65 cents)
• Close to: 95edo
• Compared to 95edo:
  • Much better 13/1, 19/1, 29/1, 31/1
  • Slightly better 3/1, 5/1, 11/1, 17/1, 23/1
  • Slightly worse 2/1, 7/1

APS99tina

• Step size: 99 tina (~13.91 cents)
• Close to: 86edo
• Compared to 86edo:
  • Slightly better 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 29/1
  • Slightly worse 2/1, 19/1, 23/1, 31/1

APS500tina

• Step size: 500 tina (~70.27 cents)
• Close to: 17edo
• Compared to 17edo:
  • 3/1 is about the same
  • Much better 7/1, 11/1, 17/1, 29/1
  • Slightly better 5/1
  • Slightly worse 2/1
  • Much worse 13/1, 19/1, 23/1, 31/1

APS750tina

• Step size: 750 tina (~105.40 cents)
• Close to: 18edt
• Special properties: no-2s
• Compared to 18edt:
  • Much better 7/1
  • Slightly worse 3/1, 5/1, 11/1, 13/1

APS900tina

• Step size: 900 tina (~126.48 cents)
• Close to: 15edt
• Special properties: no-2s
• Compared to 15edt:
  • 5/1 is about the same
  • Much better 11/1, 23/1, 31/1
  • Slightly better 7/1, 17/1
  • Slightly worse 3/1, 19/1
  • Much worse 13/1, 29/1

Non-integer equal divisions

See also: EDO#Non-integer EDO.

Pi-edo

Main page: Pi-edo

13.888edo

Main page: 13.888edo