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Dummy index

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{{interwiki
| en = User:Dummy index
| ja = User:Dummy index
}}
Hello. I'm a engineer and weekend mathematician.
== List of subpages ==


{{Special:Prefixindex|prefix=User:Dummy_index/|hideredirects=1|stripprefix=1}}
{{Special:Prefixindex|prefix=User:Dummy_index/|hideredirects=1|stripprefix=1}}


===memo===
== Memo ==
12ET-complementary comma pairs (e.g. syntonic-schismatic relation)
{| class="wikitable"
|+ Partialpyths
|-
! basis including !! examples !! remarks
|-
| 2.9 || [[Subgroup temperaments #2.9.5.7 subgroup]] ||
|-
| 4.3 || [[Subgroup temperaments #4.3.5 subgroup]],<br />not meanquad (p=4/1, g=4/3 or 3/1) -> named tetrameantone,<br />[https://x31eq.com/pyscript/rt.html?ets=q87_q111&limit=4_3_5_7 "mistyquad"], [https://x31eq.com/pyscript/rt.html?ets=q111_q144&limit=4_3_5_7 "mirquad"] ||
|-
| 4.6 = 4.3/2 = 6.3/2<br /> = 6.9 = 3/2.9 || [[meanquad]] (p=4/1, g=3/2), [https://x31eq.com/pyscript/rt.html?ets=q99_q144&limit=4_6_5_7 "ennealimmalquad"] || plain weave
|-
| 4.9 || [[Subgroup temperaments #Meansquared]] || 4.9.25 meansquared is [[Sane and insane temperaments|insane]]
|-
| 4/3.9 = 12.9 = 16.12 || [[39ed9]] ||
|-
| 4.9/2 = 4.18 || [https://x31eq.com/pyscript/rt.html?ets=q25_q12&limit=4_18_5_7 "israquad"] ||
|-
| 4/3.9/2 = 8.6 = 6.27 ||  || twill weave
|-
| 12.18 = 12.3/2 = 8.3/2<br /> = 18.3/2 = 27.3/2 = 8.12 || [https://x31eq.com/pyscript/rt.html?ets=q31_q26&limit=8_12_5_7 "septimal meanocto"] || twill weave
|-
| 12.9/2 = 8/3.9/2 = 8/3.12<br /> = 9/2.54 = 9/2.243<br /> = 8/3.32 = 12.32 ||  || satin weave
|-
| 4/3.18 = 4/3.27/2 = 18.27/2<br /> = 4/3.24 = 4/3.32 = 24.32 ||  || satin weave
|-
| 8.9 = 72.9 = 72.6<sup>6</sup><br /> = 9/8.9 = 9/8.(3/2)<sup>6</sup> || [[Subgroup temperaments #Sixscared]], [https://x31eq.com/pyscript/rt.html?ets=q19_q18&limit=8_9_10_14 "israocto"] ||
|}


{| class="wikitable"
{| class="wikitable"
! M3
|9/7||5/4|| 11/9||6/5||13/11||7/6||15/13||8/7
! A: 4*P5=M3+2*P8
! B: 8*P5+M3=5*P8
!
|-
|-
! 32/27
|13/10||14/11||16/13||17/14||19/16||20/17||22/19||23/20
| [[2187/2048]]={{monzo| -11 7 }}
| [[256/243]]={{monzo| 8 -5 }}
| A/B={{monzo| -19 12 }}, A: (7edo), B: (5edo)
|-
|-
! 6/5
|17/13||19/15||21/17||23/19||25/21||27/23
| [[135/128]]={{monzo| -7 3 1 }}
| [[20480/19683|(64/63)^2*(245/243)]]={{monzo| 12 -9 1 }}
| A/B={{monzo| -19 12 }}, A: [[Mavila]], B: [[Superpyth]]
|-
|-
! 11/9
| colspan="2" |<small>21/16 22/17 24/19</small>|| colspan="2" |26/21 27/22 || colspan="2" |32/27 33/28
| [[729/704]]={{monzo| -6 6 0 0 -1 }}
| [[8192/8019|(64/63)^2/(99/98)]]={{monzo| 13 -6 0 0 -1 }}
| A/B={{monzo| -19 12 }}, A: [[Meantone family #Meanenneadecal|Meanenneadecal]]?, B: [[Archytas clan #Supra|Supra]]
|-
|-
! 8192/6561
| colspan="2" |<small>32/25 33/26 34/27</small>|| colspan="2" |39/32 40/33
| [[531441/524288]]={{monzo| -19 12 }}
|}
| 1/1
 
| A: (12edo)
9-odd-limit diamond (not care of beginning at 1)
{|
|
|
|
|
|9/5
|
|
|
|
|-
|
|
|
|8/5
|
|<small>3/2</small>
|
|
|
|-
|-
! 5/4
|
| [[81/80]]={{monzo| -4 4 -1 }}
|
| [[32805/32768]]={{monzo| -15 8 1 }}
|7/5
| A*B={{monzo| -19 12 }}, A: [[Meantone]], B: [[Schismatic]]
|
|4/3
|
|9/7
|
|
|-
|-
! 81/64
|
| 1/1
|6/5
| [[531441/524288]]={{monzo| -19 12 }}
|
| B: (12edo)
|7/6
|
| 8/7
|
|9/8
|
|-
|-
! 9/7
|5/5
| [[64/63]]={{monzo| 6 -2 0 -1 }}
|
| [[59049/57344]]={{monzo| -13 10 0 -1 }}
|3/3
| B/A={{monzo| -19 12 }}, A: [[Archytas clan]], B: [[Septimal meantone]]
|
| 7/7
|
|1/1
|
|9/9
|-
|-
! 4/3
|
| [[256/243]]={{monzo| 8 -5 }}
|5/3
| [[2187/2048]]={{monzo| -11 7 }}
|
| B/A={{monzo| -19 12 }}, A: (5edo), B: (7edo)
|12/7
|
|7/4
|
|16/9
|
|-
|
|
|10/7
|
|3/2
|
|14/9
|
|
|-
|
|
|
|5/4
|
|<small>4/3</small>
|
|
|
|-
|
|
|
|
|10/9
|
|
|
|
|}
|}


Q: Mavila must have the fifth flatter than 7edo's, why be placed between 7edo and 5edo?
1.3.9.11.17 diamond, for 24edo (not cover 1\24, 5\24, 6\24, 8\24, ...)
{|
|
|
|
|
|18/11
|
|
|
|
|-
|
|
|
|17/11
|
|<small>3/2</small>
|
|
|
|-
|
|
|16/11
|
|17/12
|
|9/8
|
|
|-
|
|12/11
|
|4/3
|
|17/16
|
|18/17
|
|-
|11/11
|
|3/3
|
|1/1
|
|17/17
|
|9/9
|-
|
|11/6
|
|3/2
|
|32/17
|
|17/9
|
|-
|
|
|11/8
|
|24/17
|
|16/9
|
|
|-
|
|
|
|22/17
|
|<small>4/3</small>
|
|
|
|-
|
|
|
|
|11/9
|
|
|
|
|}
 
tritave 1.2.4.5.7 diamond
{|
|
|
|
|
|7/3
|
|
|
|
|-
|
|
|
|2/1
|
|7/4
|
|
|
|-
|
|
|5/3
|
|<small>3/2</small>
|
|7/5
|
|
|-
|
|4/3
|
|5/4
|
|6/5
|
|7/6
|
|-
|1/1
|
|4/4
|
|5/5
|
|2/2
|
|7/7
|-
|
|9/4
|
|12/5
|
|5/2
|
|18/7
|
|-
|
|
|9/5
|
|<small>2/1</small>
|
|15/7
|
|
|-
|
|
|
|3/2
|
|12/7
|
|
|
|-
|
|
|
|
|9/7
|
|
|
|
|}


A: I wrote the 32/27 in this table as a monzo-ish value. 32/27 constructed of P5 & P8 will much sharper when flatter P5 situation.
~~ hoge ~~
 
*[https://sintel.pythonanywhere.com/result?subgroup=3.7.31.127&reduce=on&weights=weil&target=&edos=3799%261324%268&submit_edo=submit&commas=] Mersenne prime basis


{| class="wikitable"
{| class="wikitable"
! (3/2)^(1/2)
|+Caption text
| [[2187/2048]]={{monzo| -11 7 }}
|-
| [[17-comma]]={{monzo| 27 -17 }}
!Subgroup!!Chord<br>(w/o implicit eqave)
| A/B={{monzo| -38 24 }}, A: (7edo), B: (17edo)
!Condition
!Comma!!Temperament
|-
|2.3||1:2:3:4
|3/2~4/3
|[[9/8|S3]]||2et
|-
|4.3.5||1:3:4:5
|4/3~5/4
|[[16/15|S4]]||tetrafather
|-
|2.3.5||
|(extension)
| ||father
|-
|4.6.5
|1:4:5:6
|5/4~6/5
|[[25/24|S5]]
|dicotquad
|-
|8.9.7
|1:7:8:9
|8/7~9/8
|[[64/63|S8]]
|sixscared
|-
|8.9.10
|1:8:9:10
|9/8~10/9
|[[81/80|S9]]
|israocto
|-
|
|
|
|
|
|-
|2.3
|1:2:3:4
|(2/1)/(3/2)~(3/2)/(4/3)
|[[32/27|S2/S3]]
|3et
|-
|2.3.5
|2:3:4:5
|(3/2)/(4/3)~(4/3)/(5/4)
|[[135/128|S3/S4]]
|mavila
|-
|2.3.5
|3:4:5:6
|(4/3)/(5/4)~(5/4)/(6/5)
|[[128/125|S4/S5]]
|augmented
|-
|3/2.5/4.7/4
|4:5:6:7
|(5/4)/(6/5)~(6/5)/(7/6)
|[[875/864|S5/S6]]
|
|-
|-
! (3/2)^(4/7)
|4.6.5.7
| [[531441/524288]]={{monzo| -19 12 }}
|
| [[531441/524288]]={{monzo| -19 12 }}
|(higher-rank expansion)
| A*B={{monzo| -38 24 }}, A: (12edo), B: (12edo)
|
|supermagicquad
|-
|-
! (3/2)^(2/3)
|10/7.20/11.20/17
| [[256/243]]={{monzo| 8 -5 }}
|[[11:14:17:20]]
| {{monzo| -41 26 }}
|(14/11)/(17/14)~(17/14)/(20/17)
| B/A={{monzo| -49 31 }}, A: (5edo), B: (26edo)
|54880/54043
|
|-
|4.14/5.11/5.17/5
|
| + 7''p''~4/1
| + ***
|
|-
|2.7/5.11/5.17/5
|
| (extension, 7''p''~2/1)
|
|non-over-1 greenwood
|}
|}


31ET-complementary comma pairs (e.g. 11-limit meantone-meanpop relation)
* 360edz 5ed12/11 10ed25/21 15ed13/10 36ed15/8 59ed14/5 69ed10/3 84ed13/3 95ed21/4 105ed25/4 278ed128
* I don't remember how I found it: 43-limit 1820105/1820104
* some cubismas: 61-limit 103823/103822 67-limit 50653/50652 300763/300762 79-limit 493039/493038
 
temperaments of [[7L 2s (3/1-equivalent)]]
* hyposoft:
** g = ~13/7, 2g = ~(8/7)*3, 6g = ~(3/2)*27
** 1029/1024, [[lemba]]
* hypohard:
** g = ~13/7, 2g = ~(15/13)*3, 4g = ~(4/3)*9
** b16&b23, [https://x31eq.com/pyscript/rt.html?ets=b16%26b23&limit=3.4.5.7.13 3.4.5.7.13], 3.5.7->enfactored canopus, [https://x31eq.com/pyscript/rt.html?ets=b16%26b23&limit=3.5.7.13 3.5.7.13], [https://x31eq.com/pyscript/rt.html?ets=b16%26b46cd&limit=2.3.5.7 2.3.5.7]
 
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