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Hello. I'm a engineer and weekend mathematician, not for music.
==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"
! M3 or d4
! A: 4*P5=M3+2*P8
! B: 8*P5+d4=5*P8
! Remarks
|-
! 32/27
| [[2187/2048]]={{monzo| -11 7 }}
| [[256/243]]={{monzo| 8 -5 }}
| A/B={{monzo| -19 12 }}, A: (7edo), B: (5edo)
|-
! 6/5
| [[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
| [[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
| [[531441/524288]]={{monzo| -19 12 }}
| 1/1
| A: (12edo)
|-
! 5/4
| [[81/80]]={{monzo| -4 4 -1 }}
| [[32805/32768]]={{monzo| -15 8 1 }}
| A*B={{monzo| -19 12 }}, A: [[Meantone]], B: [[Schismatic]]
|-
! 81/64
| 1/1
| [[531441/524288]]={{monzo| -19 12 }}
| B: (12edo)
|-
! 9/7
| [[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
| [[256/243]]={{monzo| 8 -5 }}
| [[2187/2048]]={{monzo| -11 7 }}
| 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)
| [[531441/524288]]={{monzo| -19 12 }}
| [[531441/524288]]={{monzo| -19 12 }}
| A*B={{monzo| -38 24 }}, A: (12edo), B: (12edo)
|-
! (3/2)^(2/3)
| [[256/243]]={{monzo| 8 -5 }}
| {{monzo| -41 26 }}
| B/A={{monzo| -49 31 }}, A: (5edo), B: (26edo)
|}
 
===temperaments spectrum===
 
Respect to [[5L 2s/Temperaments]] and [[Epic Table 1]].
 
{| class="wikitable"
!
! colspan="4"|Ratios
! rowspan="2"|Remarks
! rowspan="2"|Mapping development
|-
! [[Fifthspan]]
! -8
! -6
! 4
! 6
|-
! [[Pelogic family#Pelogic|Pelogic]]
| 25/18
| 14/9
| 6/5<br />8/7
| 9/7
|
| [{{val| 1 0 ... }}, {{val| 0 1 -3 -4 -1 }}]
|-
! [[Pelogic family#Armodue|Armodue]]
| 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]]
| 7/5
| 25/16
| 6/5
| 32/25
|
| [{{val| 1 0 ... }}, {{val| 0 1 -3 -11 -1 }}] -16
|-
! [[Pelogic family#Hornbostel|Hornbostel]]
| 25/18<br />48/35
| 25/16
| 6/5
| 32/25
|
| [{{val| 1 0 ... }}, {{val| 0 1 -3 12 }}] +23
|-
! [[Meantone family#Plutus|Plutus]]
| 32/25<br />48/35
| 16/11
| 5/4<br />7/6
| 11/8
| 105/64 is at 10 fifthspan -> [[7edo]]
| [{{val| 1 0 ... }}, {{val| 0 1 4 5 6 }}] +7-7
|-
! [[Meantone family#Flattone|Flattone]]
| 21/16
| 16/11
| 5/4<br />11/9
| 11/8
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 -9 6 }}] -14
|-
! [[Meantone family#Meanenneadecal|Meanenneadecal]]
| 9/7
| 16/11
| 5/4<br />11/9
| 11/8
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 6 }}] +19
|-
! [[Septimal meantone]]
| 9/7
| 10/7
| 5/4
| 7/5
| Good 4:5:7 in 10 fifthspan<sub>p-p</sub>
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 }}]
|-
! [[Meantone family#Mohajira|Mohajira]]
| 14/11
|
| 5/4
|
| 7/5 is at -9.5 fifthspan
| [{{val| 1 0 ... }}, {{val| 0 2 8 -11 }}] *2-31
|-
! [[Meantone family#Unidecimal meantone aka Huygens|Undecimal meantone]]
| 14/11
| 10/7
| 5/4
| 7/5
| Good 4:5:7 in 10 fifthspan<sub>p-p</sub>
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 18 }}] +12
|-
! [[Meantone family#Dominant|Dominant]]
| 32/25
| 7/5
| 5/4
| 10/7
| inaccurate
| [{{val| 1 0 ... }}, {{val| 0 1 4 -2 }}] -12
|-
! [[Schismatic family#Schism|Schism]]
| 5/4
| 10/7
| 81/64
| 7/5
| inaccurate
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -2 }}] -12
|-
! [[Schismatic family#Grackle|Grackle]]
| 5/4
|
| 81/64
|
| 7/5 is at -18 fifthspan
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -26 }}] -24
|-
! [[Schismatic family#Garibaldi|Garibaldi]]
| 5/4
| 7/5
| 81/64<br />80/63
| 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>
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 }}] +12
|-
! [[Schismatic family#Andromeda|Andromeda]]
| 5/4
| 7/5
| 14/11
| 10/7
| 11/9 is at -20 fifthspan -> [[41edo]]
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 -18 -21 }}]
|-
! [[Hemififths]]
|
| 7/5
| 14/11
| 10/7
| 5/4 is at 12.5 fifthspan
| [{{val| 1 0 ... }}, {{val| 0 2 25 13 5 }}] *2+41
|-
! [[Chromatic pairs#Edson|Edson]]
|
| 7/5
| 14/11
| 10/7
| -> [[29edo]]
| [{{val| 1 0 ... }}, {{val| 0 1 no-five -14-(-8) -18-(-8) -21-(-8) }}]
|-
! [[Gentle region]]<br />[[No-fives subgroup temperaments#Leapfrog|Leapfrog]]
| 27/22
|
| 14/11
|
|
| [{{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
|
| [{{val| 1 0 ... }}, {{val| 0 1 no-five -2 -6 }}] -17
|-
! [[Superpyth]]
| 6/5
| 15/11
| 9/7
| 22/15
|
| [{{val| 1 0 ... }}, {{val| 0 1 9 -2 16 }}] +22
|-
! [[Archytas clan#Ultrapyth|Ultrapyth]]
| 33/28
| 112/81
| 9/7
| 81/56
|
| [{{val| 1 0 ... }}, {{val| 0 1 14 -2 -11 }}] +5-25
|}
 
zoom out
 
{| class="wikitable"
!
! rowspan="2"|Generator
! colspan="2"|Ratios
! rowspan="2"|Remarks
! rowspan="2"|Mapping
|-
! Fifth(?)span
! 1
! 2
|-
! [[Pluto]]
| 7/5
| 10/7
| 81/80
|
| [{{val| 1 0 ... }}, {{val| 0 -7 -26 -25 }}]
|-
! [[Tritonic]]
| 7/5
| 10/7,64/45
| 64/63
|
| [{{val| 1 0 ... }}, {{val| 0 -5 11 12 }}]
|-
! [[Liese]]
| 10/7
| 10/7,81/56
| 21/20,28/27
|
| [{{val| 1 0 ... }}, {{val| 0 3 12 11 }}]
|-
! [[Maquila]]
| 15/11
| 22/15,35/24
| 77/72
|
| [{{val| 1 0 ... }}, {{val| 0 17 -6 22 10 }}]
|-
! Pelogic
| 3/2
| 3/2,10/7,64/45
| 9/8,16/15,15/14,28/25,64/63
|
| [{{val| 1 0 ... }}, {{val| 0 1 -3 -4 }}]
|-
! Septimal mavila
| 3/2
| 3/2,16/11,22/15,64/45
| 12/11,11/10,9/8,16/15
|
| [{{val| 1 0 ... }}, {{val| 0 1 -3 -11 -1 }}]
|-
! [[Gravity family|Larry]]
| 40/27
| 40/27
| 11/10
|
| [{{val| 1 0 ... }}, {{val| 0 -6 -17 no-seven -15 }}]
|-
! Flattone
| 3/2
| 3/2,40/27
| 9/8,10/9
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 -9 }}]
|-
! Septimal meantone
| 3/2
| 3/2,40/27
| 9/8,10/9,28/25
|
| [{{val| 1 0 ... }}, {{val| 0 1 4 10 }}]
|-
! Garibaldi
| 3/2
| 3/2
| 9/8,28/25
|
| [{{val| 1 0 ... }}, {{val| 0 1 -8 -14 }}]
|-
! [[Leapday]]
| 3/2
| 3/2
| 9/8
|
| [{{val| 1 0 ... }}, {{val| 0 1 21 15 }}]
|-
! Superpyth
| 3/2
| 3/2,32/21
| 8/7,9/8
|
| [{{val| 1 0 ... }}, {{val| 0 1 9 -2 }}]
|-
! [[Hemiseven]]
| 320/243
| 243/160
| 8/7,55/48
|
| [{{val| 1 0 ... }}, {{val| 0 -6 -29 2 21 }}]
|-
! [[Father]]
| 3/2
| 3/2,8/5,14/9,45/32,81/56
| 6/5,7/6,9/8,32/25,56/45
|
| [{{val| 1 0 ... }}, {{val| 0 1 -1 3 }}]
|-
! [[Dicot family#Sidi|Sidi]]
| 9/7
| 14/9,45/28
| 5/4,6/5
|
| [{{val| 1 0 ... }}, {{val| 0 -4 -2 -9 }}]
|-
! [[Magic]]
| 5/4
| 8/5,45/28
| 9/7,32/25
|
| [{{val| 1 0 ... }}, {{val| 0 5 1 12 }}]
|}
 
 
===pan-5L2s tuning spectrum===
 
{| 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.)
|-
| 5/4 || -3(m3) || 671.229 || Mavila
|-
| 6/5 || +4(M3) || 678.910 || Mavila
|-
| 11/9 || +4(M3) || 686.852 || Flattone
|-
| 11/8 || +6(A4) || 691.886 || Flattone
|-
| 6/5 || -3(m3) || 694.786 || Meantone (1/3 comma)
|-
| 9/7 || -8(d4) || 695.614 || Septimal meantone
|-
| 7/6 || +9(A2) || 696.319 || Septimal meantone
|-
| 5/4 || +4(M3) || 696.578 || Meantone (1/4 comma)
|-
| 7/5 || +6(A4) || 697.085 || Septimal meantone
|-
| 11/8 || +18(AA3) || 697.295 || Undecimal meantone
|-
| 14/11 || -8(d4) || 697.812 || Undecimal meantone
|-
|  ||  ||  ||
|-
| 7/5 || -18(dd6) || 700.972 || Grackle
|-
| 5/4 || -8(d4) || 701.711 || Schismatic
|-
| 6/5 || +9(A2) || 701.738 || Schismatic
|-
| 3/2 || +1(P5) || 701.955 || Pythagorean
|-
| 11/8 || -18(dd6) || 702.705 || Andromeda
|-
| 7/5 || -6(d5) || 702.915 || Garibaldi
|-
| 13/11 || -3(m3) || 703.597 || Leapfrog
|-
| 14/11 || +4(M3) || 704.377 || Leapfrog
|-
| 27/22 || -8(d4) || 705.682 || Leapfrog
|-
| 11/9 || -8(d4) || 706.574 || Supra
|-
| 11/8 || -6(d5) || 708.114 || Supra
|-
| 9/7 || +4(M3) || 708.771 || Archy (1/4 comma)
|-
| 5/4 || +9(A2) || 709.590 || Superpyth
|-
| 6/5 || -8(d4) || 710.545 || Superpyth
|-
| 7/6 || -3(m3) || 711.043 || Archy (1/3 comma)
|}
 
Clarify [[Meantone#Tuning spectrum]]
 
{| class="wikitable"
| 19/17
| at +2(M2)
| 696.279
| 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.)
|-
| 18/17
| 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
| for regular temperament with nominal 18/17 at m2, see [[No-sevens subgroup temperaments#Photia]] for example.
|-
| 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.)
|-
| 51/38
| at -1(P4)
| 690.673
| for regular temperament with nominal 51/38 at P4, see some 19-limit variation in Meantone family (search {{map| 0 1 * * * * -5 -3 }} in mappings.)
|-
| 81/80
| at 0(P1)
| ---
| 81/80 is tempered out, return to 1/1, anyway.
|-
| 81/80
| at +12(A7)
| 701.792
| this interpretation is by the schismatic temperament.
|-
| 81/80
| at +19(AA7)
| 695.869
| this interpretation is by the [[Syntonic-enneadecal equivalence continuum|Lalayo]].
|-
| 256/243
| at -5(m2)
| rowspan="5"|701.955
| rowspan="5"|any 3-limit eigenmonzo results in the pythagorean tuning.
|-
| 1024/729
| at -6(d5)
|-
| 2187/2048
| at +7(A1)
|-
| 8192/6561
| at -8(d4)
|-
| {{monzo| 27 -17 }}
| at -17(dd3)
|-
| 256/243
| at +7(A1)
| 698.604
| flatten 2187/2048 of +7(A1) by {{monzo| -19 12 }}. (-1 pythagorean comma / +7 step)
|-
| 1024/729
| at +6(A4)
| 698.045
| flatten 729/512 of +6(A4) by {{monzo| -19 12 }}. (-1 pythagorean comma / +6 step)
|-
| 2187/2048
| at -5(m2)
| 697.263
| sharpen 256/243 of -5(m2) by {{monzo| -19 12 }}. (+1 pythagorean comma / -5 step)
|-
| 8192/6561
| at +4(M3)
| 696.090
| flatten 81/64 of +4(M3) by {{monzo| -19 12 }}. (-1 pythagorean comma / +4 step)
|-
| {{monzo| 27 -17 }}
| at +7(A1)
| 695.252
| flatten 2187/2048 of +7(A1) by two {{monzo| -19 12 }}. (-2 pythagorean comma / +7 step)
|}


"(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]]. Oh, 33/28 is the [[mediant]] of 13/11 and 20/17."
"(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]]. Oh, 33/28 is the [[mediant]] of 13/11 and 20/17. but... around 698 cents I can hardly any contribute."
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