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Gallery of arithmetic pitch sequences: Difference between revisions

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Created this to document a bunch of scales I found while tinkering, because they were not important enough to warrant individual pages and I didn’t want to overload recent changes with new pages, but they were interesting enough to warrant appearing in a gallery somewhere
 
Fredg999 (talk | contribs)
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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]]
** [[Edo]], [[Edt]], [[Ed4]], [[Ed13]]
** [[Edt]]
 
** [[Ed4]]
==APS of cents==
** [[Ed13]]
''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 43: Line 47:


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




Line 78: Line 87:


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




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Main page: [[1ed237.8c]]
Main page: [[1ed237.8c]]


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


; APS1farab
; APS1farab
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**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 [[hekt]]s==
;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
; APS1hekt
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; APS7hekt
; APS7hekt
*Step size: 7 hekts (~10.24128 cents)
*Step size: 7 hekts (~10.24 cents)
*Close to: [[117edo]]
*Close to: [[117edo]]
*Compared to 117edo:
*Compared to 117edo:
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**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 (~16.09344 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
; APS16hekt
*Step size: 16 hekts (~23.40864 cents)
*Step size: 16 hekts (~23.41 cents)
*Close to: [[51edo]]
*Close to: [[51edo]]
*Compared to 51edo:
*Compared to 51edo:
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**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 (~24.87168 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
; APS18hekt
*Step size: 18 hekts (~26.33472 cents)
*Step size: 18 hekts (~26.33 cents)
*Close to: [[72edt]]
*Close to: [[72edt]]
*Special properties: dual-2s
*Special properties: dual-2s
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**Much worse 19/1
**Much worse 19/1


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


==APS of [[jot]]s==


; APS1jot
; APS1jot
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*Step size: 90 jots (~3.58768 cents)
*Step size: 90 jots (~3.58768 cents)
*Close to: [[530edt]]
*Close to: [[530edt]]
*Special properties: no-2s or 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
;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




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*Close to: [[100edo]]
*Close to: [[100edo]]
*Compared to 100edo: Mostly about the same, but a much better 3/1 than 100edo
*Compared to 100edo: Mostly about the same, but a much better 3/1 than 100edo
; APS715jot
Main page: [[1ed28.5c|APS715jot]]




; APS720jot
; APS720jot


Main page: [[1ed28.7c|APS700jot]]
Main page: [[1ed28.7c|APS720jot]]




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*Close to: [[30edo]]
*Close to: [[30edo]]
*Compared to 30edo:
*Compared to 30edo:
**2/1 is about the same
**Much better 3/1, 5/1, 17/1
**Much better 3/1, 5/1, 17/1
**Much worse 7/1, 13/1
**Much worse 7/1, 13/1
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**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




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; 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:
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; 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:
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; 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
**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-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
 
==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
**Much better 3/1, 5/1
**Slightly worse 11/1, 13/1
**Slightly better 7/1, 13/1, 17/1, 19/1
**Much worse 2/1, 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


; APS0.3méride
 
*Step size: 0.3 mérides (~8.37 cents)
;APS750tina
*Close to: [[143edo]]
*Step size: 750 tina (~105.40 cents)
*Compared to 143edo:
*Close to: [[18edt]]
**7/1 is about the same
*Special properties: no-2s
**Much better 17/1, 19/1
*Compared to 18edt:
**Slightly better 3/1, 11/1
**Much better 7/1
**Slightly worse 2/1, 5/1, 13/1
**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==
==Non-integer equal divisions==
''See also: [[EDO#Non-integer EDO]].''
;Pi-edo
Main page: [[Pi-edo]]


; 13.888edo
; 13.888edo
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{{navbox scale gallery}}
{{navbox scale gallery}}


[[Category:Equal-step tunings]] [[Category:Lists of scales]]
[[Category:Equal-step tuning]] [[Category:Lists of scales]]
Retrieved from "https://en.xen.wiki/w/Gallery_of_arithmetic_pitch_sequences"