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Gallery of arithmetic pitch sequences: Difference between revisions

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Line 13: Line 13:
* [[Interval size measure]]
* [[Interval size measure]]
* [[Equal-step tuning]]
* [[Equal-step tuning]]
** [[Edo]], [[Edt]], [[Ed4]], [[Ed13]]
** [[Edo]], [[Edt]], [[Ed4]], [[Ed13]], [[Ed17]]


==APS of [[cent]]s==
==APS of [[cent]]s==
Line 105: Line 105:
**Slightly worse 2/1
**Slightly worse 2/1
**Much worse 11/1, 17/1
**Much worse 11/1, 17/1
==APS of [[311edo|gene]]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
;APS6gene
*Step size: 6 genes (~23.15 cents)
*Close to: [[52edo]]
*Compared to 52edo:
**??/?? are about the same
**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
;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
;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
;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
;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==
==APS of [[hekt]]s==
Line 111: Line 205:
*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




; 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:
Line 124: Line 227:


; APS11hekt
; APS11hekt
*Step size: 11 hekts (~16.09344 cents)
*Step size: 11 hekts (~16.09 cents)
*Close to: [[118edt]]
*Close to: [[118edt]]
*Special properties: dual-2s
*Special properties: dual-2s
Line 135: Line 238:


; 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:
Line 145: Line 248:


; APS17hekt
; APS17hekt
*Step size: 17 hekts (~24.87168 cents)
*Step size: 17 hekts (~24.87 cents)
*Close to: [[48edo]]
*Close to: [[48edo]]
*Special properties: dual-3
*Special properties: dual-3
Line 155: Line 258:


; 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
Line 163: Line 266:
**Slightly worse 3/1, 11/1, 13/1
**Slightly worse 3/1, 11/1, 13/1
**Much worse 19/1
**Much worse 19/1


==APS of [[jot]]s==
==APS of [[jot]]s==
Line 189: Line 291:
**Much better 5/1, 7/1, 13/1, 19/1, 23/1, 29/1 and 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




Line 218: Line 331:


Main page: [[1ed28.7c|APS700jot]]
Main page: [[1ed28.7c|APS700jot]]
;APS777jot
*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 224: Line 347:
*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 230: Line 362:
*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
Line 312: Line 445:
**Slightly better 3/1, 11/1
**Slightly better 3/1, 11/1
**Slightly worse 2/1, 5/1, 13/1
**Slightly worse 2/1, 5/1, 13/1
==APS of [[tina]]s==
;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]]
*Special properties: ??
*Compared to ??edo:
**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
;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
;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: [[15-edt]]
*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==


; Pi-edo
;Pi-edo


Main page: [[Pi-edo]]
Main page: [[Pi-edo]]
Retrieved from "https://en.xen.wiki/w/Gallery_of_arithmetic_pitch_sequences"