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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"
{| class="wikitable"
! M3 or d4
|+ Partialpyths
! A: 4*P5=M3+2*P8
|-
! B: 8*P5+d4=5*P8
! basis including !! examples !! remarks
! Remarks
|-
|-
! 32/27
| 2.9 || [[Subgroup temperaments #2.9.5.7 subgroup]] ||  
| [[2187/2048]]={{monzo| -11 7 }}
| [[256/243]]={{monzo| 8 -5 }}
| A/B={{monzo| -19 12 }}, A: (7edo), B: (5edo)
|-
|-
! 6/5
| 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"] ||
| [[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
| 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
| [[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 #Flattone|Flattone]], B: [[Archytas clan #Supra|Supra]]
|-
|-
! 8192/6561
| 4.9 || [[Subgroup temperaments #Meansquared]] || 4.9.25 meansquared is [[Sane and insane temperaments|insane]]
| [[531441/524288]]={{monzo| -19 12 }}
| 1/1
| A: (12edo)
|-
|-
! 5/4
| 4/3.9 = 12.9 = 16.12 || [[39ed9]] ||  
| [[81/80]]={{monzo| -4 4 -1 }}
| [[32805/32768]]={{monzo| -15 8 1 }}
| A*B={{monzo| -19 12 }}, A: [[Meantone]], B: [[Schismatic]]
|-
|-
! 81/64
| 4.9/2 = 4.18 || [https://x31eq.com/pyscript/rt.html?ets=q25_q12&limit=4_18_5_7 "israquad"] ||  
| 1/1
| [[531441/524288]]={{monzo| -19 12 }}
| B: (12edo)
|-
|-
! 9/7
| 4/3.9/2 = 8.6 = 6.27 || || twill weave
| [[64/63]]={{monzo| 6 -2 0 -1 }}
| [[59049/57344]]={{monzo| -13 10 0 -1 }}
| B/A={{monzo| -19 12 }}, A: [[Archytas clan]], B: [[Septimal meantone]]
|-
|-
! 4/3
| 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
| [[256/243]]={{monzo| 8 -5 }}
|-
| [[2187/2048]]={{monzo| -11 7 }}
| 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
| B/A={{monzo| -19 12 }}, A: (5edo), B: (7edo)
|}
 
Q: Mavila must have the fifth flatter than 7edo's, why be placed between 7edo and 5edo?
 
A: I wrote the 32/27 in this table as a monzo-ish nominal ratio. 32/27 constructed of P5 & P8 will much sharper when flatter P5 situation.
 
{| class="wikitable"
! (3/2)^(1/2)
| [[2187/2048]]={{monzo| -11 7 }}
| [[17-comma]]={{monzo| 27 -17 }}
| A/B={{monzo| -38 24 }}, A: (7edo), B: (17edo)
|-
|-
! (3/2)^(4/7)
| 4/3.18 = 4/3.27/2 = 18.27/2<br /> = 4/3.24 = 4/3.32 = 24.32 || || satin weave
| [[531441/524288]]={{monzo| -19 12 }}
| [[531441/524288]]={{monzo| -19 12 }}
| A*B={{monzo| -38 24 }}, A: (12edo), B: (12edo)
|-
|-
! (3/2)^(2/3)
| 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"] ||  
| [[256/243]]={{monzo| 8 -5 }}
| {{monzo| -41 26 }}
| B/A={{monzo| -49 31 }}, A: (5edo), B: (26edo)
|}
|}


===temperaments spectrum===
{| class="wikitable"
{| class="wikitable"
! [[Fifthspan]]
|9/7||5/4|| 11/9||6/5||13/11||7/6||15/13||8/7
! -8
! -6
! 4
! 6
! Remarks
! Mapping development
|-
|-
! [[Pelogic family#Pelogic|Pelogic]]
|13/10||14/11||16/13||17/14||19/16||20/17||22/19||23/20
| 25/18
|-
| 14/9
|17/13||19/15||21/17||23/19||25/21||27/23
| 6/5<br />8/7
| 9/7
|  
| [{{val| 1 0 ... }}, {{val| 0 1 -3 -4 -1 }}]
|-
|-
! [[Pelogic family#Armodue|Armodue]]
| colspan="2" |<small>21/16 22/17 24/19</small>|| colspan="2" |26/21 27/22 || colspan="2" |32/27 33/28
| 10/7
| 11/7
| 6/5<br />7/6
| 14/11
|  
| [{{val| 1 0 ... }}, {{val| 0 1 -3 5 -1 }}] +9
|-
|-
! [[Pelogic family#Septimal mavila|Septimal mavila]]
| colspan="2" |<small>32/25 33/26 34/27</small>|| colspan="2" |39/32 40/33
| 7/5
|}
| 25/16
 
| 6/5
9-odd-limit diamond (not care of beginning at 1)
| 32/25
{|
|  
|
| [{{val| 1 0 ... }}, {{val| 0 1 -3 -11 -1 }}] -16
|
|
|
|9/5
|
|
|
|
|-
|-
! [[Pelogic family#Hornbostel|Hornbostel]]
|
| 25/18<br />48/35
|
| 25/16
|
| 6/5
|8/5
| 32/25
|
|  
|<small>3/2</small>
| [{{val| 1 0 ... }}, {{val| 0 1 -3 12 }}] +23
|
|
|
|-
|-
! [[Meantone family#Plutus|Plutus]]
|
| 32/25<br />48/35
|
| 16/11
|7/5
| 5/4<br />7/6
|
| 11/8
|4/3
| 105/64 is at 10 fifthspan -> [[7edo]]
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 5 6 }}] +7-7
|9/7
|
|
|-
|-
! [[Meantone family#Flattone|Flattone]]
|
| 21/16
|6/5
| 16/11
|
| 5/4<br />11/9
|7/6
| 11/8
|
|  
| 8/7
| [{{val| 1 0 ... }}, {{val| 0 1 4 -9 6 }}] -14
|
|9/8
|
|-
|-
! [[Meantone family#Meanenneadecal|Meanenneadecal]]
|5/5
| 9/7
|
| 16/11
|3/3
| 5/4<br />11/9
|
| 11/8
| 7/7
|  
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 6 }}] +19
|1/1
|
|9/9
|-
|-
! [[Septimal meantone]]
|
| 9/7
|5/3
| 10/7
|
| 5/4
|12/7
| 7/5
|
| Good 4:5:7 in 10 fifthspan<sub>p-p</sub>
|7/4
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 }}]
|
|16/9
|
|-
|-
! [[Meantone family#Mohajira|Mohajira]]
|
| 14/11
|
|  
|10/7
| 5/4
|
|  
|3/2
| 7/5 is at -9.5 fifthspan
|
| [{{val| 1 0 ... }}, {{val| 0 2 8 -11 }}] *2-31
|14/9
|
|
|-
|-
! [[Meantone family#Unidecimal meantone aka Huygens|Undecimal meantone]]
|
| 14/11
|
| 10/7
|
| 5/4
|5/4
| 7/5
|
| Good 4:5:7 in 10 fifthspan<sub>p-p</sub>
|<small>4/3</small>
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 18 }}] +12
|
|
|
|-
|-
! [[Meantone family#Dominant|Dominant]]
|
| 32/25
|
| 7/5
|
| 5/4
|
| 10/7
|10/9
| inaccurate
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 -2 }}] -12
|
|
|
|}
 
1.3.9.11.17 diamond, for 24edo (not cover 1\24, 5\24, 6\24, 8\24, ...)
{|
|
|
|
|
|18/11
|
|
|
|
|-
|-
! [[Schismatic family#Schism|Schism]]
|
| 5/4
|
| 10/7
|
| 81/64
|17/11
| 7/5
|
| inaccurate
|<small>3/2</small>
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -2 }}] -12
|
|
|
|-
|-
! [[Schismatic family#Grackle|Grackle]]
|
| 5/4
|
|  
|16/11
| 81/64
|
|  
|17/12
| 7/5 is at -18 fifthspan
|
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -26 }}] -24
|9/8
|
|
|-
|-
! [[Schismatic family#Garibaldi|Garibaldi]]
|
| 5/4
|12/11
| 7/5
|
| 81/64<br />80/63
|4/3
| 10/7
|
| Good 4:5:6:7 in 15 fifthspan<sub>p-p</sub><br />Good 4:6 & 5:7 in 6 fifthspan<sub>p-p</sub>
|17/16
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 }}] +12
|
|18/17
|
|-
|-
! [[Schismatic family#Andromeda|Andromeda]]
|11/11
| 5/4
|
| 7/5
|3/3
| 14/11
|
| 10/7
|1/1
| 11/9 is at -20 fifthspan -> [[41edo]]
|
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 -18 -21 }}]
|17/17
|
|9/9
|-
|-
! [[Hemififths]]
|
|  
|11/6
| 7/5
|
| 14/11
|3/2
| 10/7
|
| 5/4 is at 12.5 fifthspan
|32/17
| [{{val| 1 0 ... }}, {{val| 0 2 25 13 5 }}] *2+41
|
|17/9
|
|-
|-
! [[Chromatic pairs#Edson|Edson]]
|
|  
|
| 7/5
|11/8
| 14/11
|
| 10/7
|24/17
| -> [[29edo]]
|
| [{{val| 1 0 ... }}, {{val| 0 1 no-five -14-(-8) -18-(-8) -21-(-8) }}]
|16/9
|
|
|-
|-
! [[Gentle region]]<br />[[No-fives subgroup temperaments#Leapfrog|Leapfrog]]
|
| 27/22
|
|  
|
| 14/11
|22/17
|  
|
|  
|<small>4/3</small>
| [{{val| 1 0 ... }}, {{val| 0 1 no-five 15 11 8 }}] +29
|
|
|
|-
|-
! [[Archytas clan#Supra|Supra]]
|
| 11/9
|
| 11/8
|
| 9/7<br />14/11
|
| 16/11
|11/9
|  
|
| [{{val| 1 0 ... }}, {{val| 0 1 no-five -2 -6 }}] -17
|
|
|
|}
|}


===pan-5L2s tuning spectrum===
tritave 1.2.4.5.7 diamond
 
{|
{| class="wikitable sortable"
|
! Eigenmonzo<br />(unchanged interval) !! data-sort-type="number"|at<br />(fifthspan) !! data-sort-type="number"|Generator<br />(cents) !! class="unsortable"|in this temperament<br />(e.g.)
|
|
|
|7/3
|
|
|
|
|-
|-
| 6/5 || +4(M3) || 678.910 || Mavila
|
|
|
|2/1
|
|7/4
|
|
|
|-
|-
| 11/9 || +4(M3) || 686.852 || Flattone
|
|
|5/3
|
|<small>3/2</small>
|
|7/5
|
|
|-
|-
| 11/8 || +6(A4) || 691.886 || Flattone
|
|4/3
|
|5/4
|
|6/5
|
|7/6
|
|-
|-
| 6/5 || -3(m3) || 694.786 || Meantone (1/3 comma)
|1/1
|
|4/4
|
|5/5
|
|2/2
|
|7/7
|-
|-
| 9/7 || -8(d4) || 695.614 || Septimal meantone
|
|9/4
|
|12/5
|
|5/2
|
|18/7
|
|-
|-
| 5/4 || +4(M3) || 696.578 || Meantone (1/4 comma)
|
|
|9/5
|
|<small>2/1</small>
|
|15/7
|
|
|-
|-
| 7/5 || +6(A4) || 697.085 || Septimal meantone
|
|
|
|3/2
|
|12/7
|
|
|
|-
|-
| 11/8 || +18(AA3) || 697.295 || Undecimal meantone
|
|
|
|
|9/7
|
|
|
|
|}
 
~~ 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"
|+Caption text
|-
|-
| 14/11 || -8(d4) || 697.812 || Undecimal meantone
!Subgroup!!Chord<br>(w/o implicit eqave)
!Condition
!Comma!!Temperament
|-
|-
| || || ||
|2.3||1:2:3:4
|3/2~4/3
|[[9/8|S3]]||2et
|-
|-
| 7/5 || -18(dd6) || 700.972 || Grackle
|4.3.5||1:3:4:5
|4/3~5/4
|[[16/15|S4]]||tetrafather
|-
|-
| 5/4 || -8(d4) || 701.711 || Schismatic
|2.3.5||
|(extension)
| ||father
|-
|-
| 6/5 || +9(A2) || 701.738 || Schismatic
|4.6.5
|1:4:5:6
|5/4~6/5
|[[25/24|S5]]
|dicotquad
|-
|-
| 3/2 || +1(P5) || 701.955 || Pythagorean
|8.9.7
|1:7:8:9
|8/7~9/8
|[[64/63|S8]]
|sixscared
|-
|-
| 11/8 || -18(dd6) || 702.705 || Andromeda
|8.9.10
|1:8:9:10
|9/8~10/9
|[[81/80|S9]]
|israocto
|-
|-
| 7/5 || -6(d5) || 702.915 || Garibaldi
|
|
|
|
|
|-
|-
| 13/11 || -3(m3) || 703.597 || Leapfrog
|2.3
|1:2:3:4
|(2/1)/(3/2)~(3/2)/(4/3)
|[[32/27|S2/S3]]
|3et
|-
|-
| 14/11 || +4(M3) || 704.377 || Leapfrog
|2.3.5
|2:3:4:5
|(3/2)/(4/3)~(4/3)/(5/4)
|[[135/128|S3/S4]]
|mavila
|-
|-
| 27/22 || -8(d4) || 705.682 || Leapfrog
|2.3.5
|3:4:5:6
|(4/3)/(5/4)~(5/4)/(6/5)
|[[128/125|S4/S5]]
|augmented
|-
|-
| 11/9 || -8(d4) || 706.574 || Supra
|3/2.5/4.7/4
|4:5:6:7
|(5/4)/(6/5)~(6/5)/(7/6)
|[[875/864|S5/S6]]
|
|-
|-
| 11/8 || -6(d5) || 708.114 || Supra
|4.6.5.7
|
|(higher-rank expansion)
|
|supermagicquad
|-
|-
| 9/7 || +4(M3) || 708.771 || Archy (1/4 comma)
|10/7.20/11.20/17
|[[11:14:17:20]]
|(14/11)/(17/14)~(17/14)/(20/17)
|54880/54043
|
|-
|-
| 5/4 || +9(A2) || 709.590 || Superpyth
|4.14/5.11/5.17/5
|
| + 7''p''~4/1
| + ***
|
|-
|-
| 6/5 || -8(d4) || 710.545 || Superpyth
|2.7/5.11/5.17/5
|-
|
| 7/6 || -3(m3) || 711.043 || Archy (1/3 comma)
| (extension, 7''p''~2/1)
|  
|non-over-1 greenwood
|}
|}


Clarify [[Meantone#Tuning spectrum]]
* 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


{| class="wikitable"
temperaments of [[7L 2s (3/1-equivalent)]]
| 19/17
* hyposoft:
| at +2(M2)
** g = ~13/7, 2g = ~(8/7)*3, 6g = ~(3/2)*27
| 696.279
** 1029/1024, [[lemba]]
| for regular temperament with nominal 19/17 at M2, see some 19-limit variation in Meantone family (search {{map| 0 1 * * * * -5 -3 }} in mappings.)
* hypohard:
|-
** g = ~13/7, 2g = ~(15/13)*3, 4g = ~(4/3)*9
| 18/17
** 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]
| at +7(A1)
| 699.850
| for regular temperament with nominal 18/17 at A1, see some 17-limit variation in Meantone family (search {{map| 0 1 * * * * -5 }} in mappings.)
|-
| 18/17
| at -5(m2)
| 700.209
| don't know regular temperament with nominal 18/17 at m2, but it will have {{map| 0 1 * * * * 7 }} in the mapping.
|-
| 17/16
| at -5(m2)
| 699.009
| for regular temperament with nominal 17/16 at m2, see some 17-limit variation in Meantone family (search {{map| 0 1 * * * * -5 }} in mappings.)
|-
| 19/16
| at -3(m3)
| 700.829
| for regular temperament with nominal 19/16 at m3, see some 19-limit variation in Meantone family (search {{map| 0 1 * * * * * -3 }} in mappings.)
|-
| 45/34
| at +11(A3)
| 698.661
| for regular temperament with nominal 45/34 at A3, see some 17-limit variation in Meantone family (search {{map| 0 1 4 * * * -5 }} in mappings.)
|-
| 17/15
| at -10(d3)
| 698.331
| for regular temperament with nominal 17/15 at d3, see some 17-limit variation in Meantone family (search {{map| 0 1 4 * * * -5 }} in mappings.)
|}


"(14/11)*(13/11) is for [[Gentle region]], then what is for meantone? 33/28, thrice, 99/84, flatly approx., 100/85, [[20/17]], (14/11)*(20/17)*(2/3)=[[561/560|560/561]]"
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