Gallery of arithmetic pitch sequences: Difference between revisions

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This gallery can never realistically be complete as there are so many possible such scales, so feel free to add to it.
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]].
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==
==Terms==
Line 13: Line 13:
* [[Interval size measure]]
* [[Interval size measure]]
* [[Equal-step tuning]]
* [[Equal-step tuning]]
** [[Edo]], [[Edt]], [[Ed4]], [[Ed13]], [[Ed17]]
** [[Edo]], [[Edt]], [[Ed4]], [[Ed13]]
 
==APS of cents==
''See also: [[cent]].''


==APS of [[cent]]s==


; 1ed1c
; 1ed1c


Trivial case. This is exactly [[1200edo]].
Trivial case. This is exactly [[1200edo]].
; 1ed13.94—13.97c
Main page: [[8ed16/15|Delta scale]]




Line 40: Line 47:


Main page: [[1ed44c]]
Main page: [[1ed44c]]
; 1ed63.59—63.82c
Main page: [[Phoenix]]




Line 75: Line 87:


Main page: [[1ed125c]]
Main page: [[1ed125c]]
; 1ed148.5c
Main page: [[1ed148.5c]]




Line 81: Line 98:
Main page: [[1ed237.8c]]
Main page: [[1ed237.8c]]


==APS of [[farab]]s==
==APS of farabs==
''See also: [[farab]].''
 


; APS1farab
; APS1farab
Line 95: Line 114:
**Slightly worse 2/1
**Slightly worse 2/1


==APS of genes==
''See also: [[311edo|gene]].''


; 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 [[311edo|gene]]s==


;APS1gene
;APS1gene
Line 119: Line 129:
*Compared to 197edt:
*Compared to 197edt:
**197edt is not dual-2, so APS2.5gene completely restructures the tuning
**197edt is not dual-2, so APS2.5gene completely restructures the tuning
;APS6gene
*Step size: 6 genes (~23.15 cents)
*Close to: [[52edo]]
*Compared to 52edo:
**Much better 3/1, 17/1
**Slightly better 13/1
**Slightly worse 2/1, 5/1, 11/1
**Much worse 7/1
;APS8gene
*Step size: 8 genes (~30.87 cents)
*Close to: [[39edo]]
*Compared to 39edo:
**Much better 5/1, 7/1, 17/1, 19/1, 23/1, 29/1
**Slightly better 13/1
**Slightly worse 2/1
**Much worse 3/1, 11/1




Line 149: Line 139:
**Slightly worse 3/1, 17/1
**Slightly worse 3/1, 17/1
**Much worse 11/1, 23/1
**Much worse 11/1, 23/1
;APS11gene
*Step size: 11 genes (~42.44 cents)
*Close to: [[28edo]]
*Compared to 28edo:
**7/1, 13/1, 17/1 are about the same
**Much better 3/1, 23/1, 31/1
**Slightly worse 11/1, 19/1
**Much worse 2/1, 5/1, 29/1




Line 179: Line 159:
**Much worse 11/1, 13/1, 17/1
**Much worse 11/1, 13/1, 17/1


==APS of hekts==
''See also: [[hekt]].''


;APS16gene
*Step size: 16 genes (~61.74 cents)
*Close to: [[31edt]]
*Special properties: no-2s
*Compared to 31edt:
**Much better 5/1, 13/1
**Slightly better 11/1
**Much worse 3/1, 7/1
;APS19gene
*Step size: 19 genes (~73.31 cents)
*Close to: [[26edt]]
*Special properties: no-2s
*Compared to 26edt:
**Much better 23/1, 31/1
**Slightly better 5/1, 7/1
**Slightly worse 3/1, 11/1, 13/1, 17/1, 19/1, 29/1
==APS of [[hekt]]s==


; APS1hekt
; APS1hekt
*Step size: 1 hekt (~1.46304 cents)
*Step size: 1 hekt (~1.46304 cents)
*Trivial case. This is exactly [[1300edt]].
*Trivial case. This is exactly [[1300edt]].
; APS(pi)hekt
*Step size: ~3.14159 hekts (~4.59628 cents)
*Close to: [[261edo]]
*Compared to 261edo:
**2/1, 17/1 are about the same
**Slightly better 3/1, 7/1, 13/1, 19/1, 23/1
**Slightly worse 5/1, 11/1, 29/1, 31/1




Line 223: Line 176:
**Slightly worse 2/1, 11/1, 19/1
**Slightly worse 2/1, 11/1, 19/1
**Much worse 13/1
**Much worse 13/1
; APS11hekt
*Step size: 11 hekts (~16.09 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




Line 244: Line 186:
**Slightly better 7/1, 19/1
**Slightly better 7/1, 19/1
**Slightly worse 2/1, 3/1, 13/1
**Slightly worse 2/1, 3/1, 13/1
; APS17hekt
*Step size: 17 hekts (~24.87 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




Line 266: Line 198:
**Much worse 19/1
**Much worse 19/1


==APS of [[jot]]s==
==APS of jots==
''See also: [[jot]].''
 


; APS1jot
; APS1jot
Line 287: Line 221:
*Special properties: dual-2s
*Special properties: dual-2s
*Compared to 530edt:
*Compared to 530edt:
**2.3.5.7 is about the same
**Slightly better 5/1, 7/1, 11/1, 13/1, 17/1, 19/1, 23/1, 29/1, 31/1
**Much better 5/1, 7/1, 13/1, 19/1, 23/1, 29/1 and 31/1
**Slightly worse 3/1
**Slightly worse 3/1


Line 327: Line 260:




; APS720jot
; APS715jot
 
Main page: [[1ed28.5c|APS715jot]]


Main page: [[1ed28.7c|APS700jot]]


; APS720jot


;APS777jot
Main page: [[1ed28.7c|APS720jot]]
*Step size: 777 jot (~30.97 cents)
*Close to: [[39edo]]
*Compared to 39edo:
**Much better 5/1, 7/1
**Slightly better 11/1
**Slightly worse 2/1, 13/1
**Much worse 3/1




Line 346: Line 274:
*Close to: [[38edo]]
*Close to: [[38edo]]
*Compared to 38edo: Mostly slightly worse than 38edo, but a much better 11/1 than 38edo
*Compared to 38edo: Mostly slightly worse than 38edo, but a much better 11/1 than 38edo
;APS999jot
*Step size: 999 jot (~39.82 cents)
*Close to: [[30edo]]
*Compared to 30edo:
**Much better 3/1, 5/1, 17/1
**Slightly worse 2/1
**Much worse 7/1, 13/1




Line 373: Line 292:
**Much worse 7/1
**Much worse 7/1


 
==APS of mérides==
; APS2500jot
''See also: [[méride]].''
*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 [[méride]]s==
 
; 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




Line 407: Line 301:




; APS0.9méride
; APS11/12méride
*Step size: 0.9 mérides (~25.12 cents)
*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]]
*Close to: [[48edo]]
*Compared to 48edo:
*Compared to 48edo:
Line 416: Line 320:




; APS0.8méride
; APS7/8méride
*Step size: 0.8 mérides (~22.33 cents)
*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]]
*Close to: [[54edo]]
*Compared to 54edo:
*Compared to 54edo:
Line 426: Line 349:




; APS0.75méride
; APS8/11méride
*Step size: 0.75 mérides (~20.93 cents)
*Step size: 8/11 of a méride (~20.30 cents)
*Close to: [[57edo]]
*Close to: [[59edo]]
*Compared to 57edo:
*Compared to 59edo:
**7/1 is about the same
**Much better 3/1, 7/1, 19/1
**Much better 3/1, 5/1
**Slightly better 13/1
**Slightly worse 11/1, 13/1
**Slightly worse 2/1, 5/1, 17/1
**Much worse 2/1, 17/1, 19/1
**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-n|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-n|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


; APS0.3méride
==APS of tinas==
*Step size: 0.3 mérides (~8.37 cents)
''See also: [[tina]].''
*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


==APS of [[tina]]s==


;APS1tina
;APS1tina
Line 531: Line 504:
**Slightly better 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 29/1
**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
**Slightly worse 2/1, 19/1, 23/1, 31/1
;APS123tina
*Step size: 123 tina (~17.29 cents)
*Close to: [[110edt]]
*Special properties: dual-2s
*Compared to 110edt:
**Slightly better 7/1, 13/1, 17/1, 19/1, 23/1, 31/1
**Slightly worse 3/1, 5/1, 11/1, 29/1
;APS250tina
*Step size: 250 tina (~35.13 cents)
*Close to: [[34edo]]
*Compared to 34edo:
**3/1 is about the same
**Much better 7/1, 11/1, 19/1, 29/1, 31/1
**Slightly worse 2/1
**Much worse 5/1, 13/1, 17/1, 23/1
;APS300tina
*Step size: 300 tina (~42.16 cents)
*Close to: [[45edt]]
*Special properties: no-2s
*Compared to 45edt:
**5/1 is about the same
**Much better 7/1, 19/1, 23/1, 31/1
**Slightly worse 3/1
**Much worse 11/1, 13/1, 17/1
;APS400tina
*Step size: 400 tina (~56.21 cents)
*Close to: [[34edt]]
*Special properties: no-2s
*Compared to 34edt:
**Much better 7/1, 11/1, 13/1
**Slightly better 17/1, 31/1
**Slightly worse 29/1
**Much worse 3/1, 5/1, 19/1, 23/1




Line 596: Line 528:
;APS900tina
;APS900tina
*Step size: 900 tina (~126.48 cents)
*Step size: 900 tina (~126.48 cents)
*Close to: [[15-edt]]
*Close to: [[15edt]]
*Special properties: no-2s
*Special properties: no-2s
*Compared to 15edt:
*Compared to 15edt:
Line 606: Line 538:


==Non-integer equal divisions==
==Non-integer equal divisions==
''See also: [[EDO#Non-integer EDO]].''


;Pi-edo
;Pi-edo
Line 618: Line 552:
{{navbox scale gallery}}
{{navbox scale gallery}}


[[Category:Equal-step tunings]] [[Category:Lists of scales]]
[[Category:Equal-step tuning]] [[Category:Lists of scales]]

Latest revision as of 01:55, 6 August 2025

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:

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


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