Re:Learning wedgie-Monzo vector symbolism Re:Jacques 613-limit rationa

Andreas Sparschuh wrote :

> Quest:
> Why does appear here inbetween B-F#
> an 35600/35721 wide 5th,
> which turns out to be about ~5.748...Cents to much sharpend?

Hi Andy (and Monz and all),

Thanks for these complete tutorial links and explanations.

About B:F# in my "89-limit W3 rational variant" :
B = 11907, not 11097.
and B*3 is as you wrote exactly, 35721
So B:F# is 17800/11907 = 1.494918955 = 696.0807273 c.
(normal 1/4 PC tempered fifth-like)
Corrected here :

C: (7*7 = A/27 = 49 98 <) 99 198 _396_
G: 37 74 (A/9 = 147 <)148 (D/3=295 <) 296 (<297 := C*3 ) _592_
D: (A/3 = 441 882 <) _885_ := 295*3
A: _1323_ := 441*3
E: _3969_ := A*3 := 49*81
B: _11907_ := E*3 := 49*243
F#: _2225_ 4450 8900 17800 35600 (<35721 := B*3)
C#: 6675 := F#*3
G#: 20025 := C#*3
Eb: 60075 := G#*3
Bb: 11 22 44 88 176 .... 180224 (< 180225 := Eb*3)
F : 33 66 132 264 ...

Now what I miss to understand is what is a "3n-1 'Werckmeister-Collatz'-
"septenarian" sequence, with epimoric temperings of the 5ths" ...?

I have a vague idea about "epimoric tempering", but I would appreciate to know a definition, if there is.
Was not my precedent 613-limit variant epimoric-tempered as well ?
Subsidiary question : would this one be considered a correct WIII also ?
and according to you which one would be more valuable, between this 89-limit and my previous 613-limit variant ??

Here is the Scala file :

! werckmeister3_eb89-l.scl
!
Harmonic equal-beating version of the famous Well Temperament
12
!
2225/2112 ! C#
295/264 ! D
6675/5632 ! Eb
441/352 ! E
4/3 ! F
2225/1584 ! F#
148/99 ! G
2225/1408 ! G#
147/88 ! A
16/9 ! Bb
1323/704 ! B
2/1
! achieved with Comptine recurrent sequence x^3 = (3/5)x^2 + 2
! (x = 1.494929263 or 696.09266484 c.), Dudon 2006
! triple equal-beating property : 3A - 5C = 4E - 5C = 3D - 2A = 9

Thank you so much !
- - - - - - -
Jacques

> Salut Jacques,
> > I have to learn about Monzo' vectors symbolism:
> Here some tutorial links, that might be useful as introduction:
> http://www.io.com/~hmiller/music/regular-temperaments.html
> http://lumma.org/tuning/gws/wedgie.html
> http://lumma.org/tuning/gws/wedge.html
> http://tonalsoft.com/enc/w/wedgie.aspx
> /tuning-math/message/14024
> http://x31eq.com/primerr.pdf
>
> > and also this line :
> >> ! in 5ths:
> >>..F-C 19616/19683 G 21995/22068 D 4384/4399 A-E-B 2048/2055 F#-C#...
>
> with: the 5ths flattend down by the rational amounts of
> 19616/19683 := 613*2^5 / 3^9 intbetween 5th: C-G
> 21995/22068 := 83*53*5 / (613 * 3^2 * 2^2) within 5th: G-D
> 4384/4399 := 137*2 / (83*53 ) among 5th: D-A
> 2048/2055 := 2^11 / (137 * 5 *3 ) in the 5th: B-F#
>
> or in Cent-units [c] approximations of all that four ratios
>
> ....F-C ~-5.903...c G ~-5.736...c D ~-5.913...c A ~-5.907...c F#-C#...
> >
> >
> > It seems to describe the commas,
> Yes, in deed, that ratios determine yours four-fold splitting of the
> PC = 3^12/2^19 into four subparts.
>
> 2^19/3^12
> = |19,-12>
> =(19616/19683)* (21995/22068) * (4384/4399) * (2048/2055)
> =(|5,-9>/613)*(|-2,-2,1>*83*53/613)*(|1>*137/83/53)*(|11,-1,-1>/137)
> = |19,-12>
>
> > I have to decrypt this writing.
> In order to do that, just analyze of each ratio
> the correspoding prime-decompostion, that yields
> the the representation as so called "Monzo" wedgie.
>
> >
> > Funny that you found this same comma 2055/2048 in Tuerck's -
> > you have a trained eye !
> Once you have the ratios of both variants, it is easy to see that.
>
> > One thing I want to add, since you ask if we know about other
> > variants of WIII,
> > is that this precise model (613-limit as it is)
> '613-limit', because '613' is the highest prime that occurs in yours
> W3 interpretation....
>
> > This is a simpler other one, inspired from another Comptine > sequence :
> >
> > C 396 (11*9)
> > G 592 (37)
> > D 885 (59 *5 *3)
> > A 1323 (7^2 *3^3)
> > E 3969 (7^2 *3^4)
> > B 11907 (7^2 *3^5)
> > F# 2225 (89 *5^2)
> > C# 6675 (89 *5^2 *3)
> > G# 20025
> > Eb 60075
> > Bb 11 (octaves reduced...)
> > F 33
>
> or more en detail: as an 3n-1 'Werckmeister-Collatz'-
> "septenarian" sequence, with epimoric temperings of the 5ths:
>
> C: (7*7 = A/27 = 49 98 <) 99 198 _396_
> G: 37 74 (A/9 = 147 <)148 (D/3=295 <) 296 (<297 := C*3 ) _592_
> D: (A/3 = 441 882 <) _885_ := 295*3
> A: _1323_ := 441*3
> E: _3969_ := A*3 := 49*81
> B: _11097_ := E*3
> F#: _2225_ 4450 8900 17800 35600 (<35721 := B*3)
> C#: 6675 := F#*3
> G#: 20025 := C#*3
> Eb: 60075 := G#*3
> Bb: 11 22 44 88 176 .... 180224 (< 180225 := Eb*3)
>
> Quest:
> Why does appear here inbetween B-F#
> an 35600/35721 wide 5th,
> which turns out to be about ~5.748...Cents to much sharpend?
>
> > 3A - 5C = 4E - 5C = 3D - 2A = 9
> >
> > It has an "unperfect" pure fifth between Eb and Bb but should have a
> > "lower limit" of 89...
> >
> > Actually, when we use 4 "Comptine fifths" and 8 pure or quasi-pure
> > fifths, we have a very simple temperament that I enjoy very much :
> > (variant of the rational WIII above but with C = 33 instead of 99)
> >
>
> Attend the simplifications of the ratios and the additional wedgies:
>
> ! comptine_h3.scl
> !
> 1/4 pyth. comma meantone sequence between G and B, completed by 8
> pure fifths
> 12
> !
> 2225/2112 ! C# = 2^(-6)x 3^(-1)x 5^2x 11^(-1)x 89 = |-6,-1,2,0-1>*89
> 592/528 ! D = 37/33 = 3^(-1)x11^(-1)x37 = |0,-1,0,0,-1>*37
> 20025/16896! Eb = 2^(-9)x3x5^2x11^(-1)x89 = |-9,1,2,0,-1>*89
> 1323/1056 ! E = 441/352 = 2^(-5)x3^2x7^2x11^(-1) = |-5,2,0,2,-1>
> 4/3 ! F = = |-2,1>
> 29667/21120! F# = 899/640 = 2^(-7)x5^(-1)x29x31 = |-7,0,-1>*29*31
> 3/2 ! G = = |-1,1>
> 6675/4224 ! G# = 2225/1408 = 2^(-7)x5^2x11^(-1)x89= |-7,0,2,0,-1>*89
> 885/528 ! A = 295/176 = 2^(-4)x5x11^(-1)x59 = |-4,0,1,0,-1>*59
> 60075/33792! Bb = 20025/11264=
> ! ! Bb = 2^(-10)x3^2x5^2x11^(-1)x89 =|-10,2,2,0,-1>*89
> 9889/5280 ! B = 899/480=2^(-5)x3^(-1)x5^(-1)x29x31 =|-5,-1,-3>*29*31
> 2/1
> ! Quasi well-temperament (except for G : B a schisma below 5/4)
> ! C : G pure facilitates the tuning of open-tuning instruments.
> ! Comptine recurrent sequence x^3 = (3/5)x^2 + 2,
> ! x = 1.494929263, Dudon 2006
> ! Equal-beating properties :
> ! 6E - 5G = 3A - 4E, 6B - 10D = 8F# - 10D = 6E - 4B
>
> au revoir, bye bye, a bientot
> Andy

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> Corrected here :
>
> C: (7*7 = A/27 = 49 98 <) 99 198 _396_
> G: 37 74 (A/9 = 147 <)148 (D/3=295 <) 296 (<297 := C*3 ) _592_
> D: (A/3 = 441 882 <) _885_ := 295*3
> A: _1323_ := 441*3
> E: _3969_ := A*3 := 49*81
> B: _11907_ := E*3 := 49*243
> F#: _2225_ 4450 8900 17800 35600 (<35721 := B*3)
> C#: 6675 := F#*3
> G#: 20025 := C#*3
> Eb: 60075 := G#*3
> Bb: 11 22 44 88 176 .... 180224 (< 180225 := Eb*3)
> F : 33 66 132 264 ...
>
> Now what I miss to understand is what is a "3n-1" sequence:

Salut Jacques,
for instance, yours above scheme can be converted into an
3xN+-1 sequence by some little changes:

Overtake the 5ths fron C:..to..A: as they are unaltered,
then continue only with epimoric ratios:

E: 31 62 124 248 496 892 1984 3968 (<3969 := 49*81)
B: 93
F#: 139 278 (<279 := B*3)
C#: 13 26 52 104 206 416 (< 417 := F#*3)
G#: 39
Eb: 117
Bb: 11 (> 351 := D#*3)
F: 33 66 132 264 528 from that you had once started

Already A.Werckmeister, did so in his:
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings
/tuning/topicId_89066.html#89216
"
C: 49 := 7*7 98 C=196
F: F=147 := C*3
B:(German) 55 B=110 220 440 (<441:=F*3) English: "Bb"
Eb: Eb=165 := B*3
G#: 31 62 G#=124 := Eb*3
C#: 93 C#=186 := G#*3
F#: F#=139 278 (<279 := C#*3)
H: 13 26 52 H=104 208 416 (< 417 := F#*3) Engl.:"B"
E: 39 78 E=156 := H*3
A: A=117 := E*3
D: 11 22 44 88 D=176 352(> 351 := A*3) or ???"175"???
G: G=131 (132 := D44 * 3)
C: C=196 392 (< 393 := G*3)
"

or in todays modern terms:
http://en.wikipedia.org/wiki/Collatz_conjecture
For deeper understanding, here you can play with that:
http://l.pellegrino.free.fr/syracuse/
http://www.nitrxgen.net/collatz.php
http://www.gfredericks.com/sandbox/arith/collatz
http://fr.wikipedia.org/wiki/Conjecture_de_Syracuse

> I have a vague idea about "epimoric tempering", but I would
> appreciate to know a definition, if there is.

It means that in all tempered 5ths,
there occur only barely temperings that are
restricted to epimoric-ratios of the form: n/(n+-1)
with any arbitrary-chosen integral number n
out of the whole number-set: N.

> Was not my precedent 613-limit variant epimoric-tempered as well ?
Not quite fully yet, because yours choice:

> F#: _2225_ 4450 8900 17800 35600 (<35721 := B*3)
differs about 121 := 35600-35721
with 121, which is quite different from the demanded case of +-1.

> Subsidiary question :
> would this one be considered a correct WIII also ?
Only almost,
because there appears an tiny tempering within Eb-Bb

> Eb: 60075 := G#*3
> Bb: 11 22 44 88 176 .... 180224 (< 180225 := Eb*3)

but that is absent in Werckmeister's original specification:
that allows temperings only @ the 5ths: C~G~D~A ... B~F#
Therefore: When taken in W's strict sense: Not so at all.

> and according to you which one would be more valuable,
> between this 89-limit and my previous 613-limit variant ??
Difficult to judge about here on a question of personal taste.

But in doubt I would then decide but for the simpler version
of the lower 89-limit than instead of the higher 613-limit.

Contered question:
What do you think about my own 41-limit variant version?
/tuning/topicId_89550.html#89588
"
Concise solution:
C 6560/6561 G 204/205 D 152/153 A - E - B...
...B 512/513 F# - C# - G# - Eb - Bb - F - C
"
or that epimotic-ratios do amount in Cent-units:

C ~-0.264...cents G ~-8.47...cents D ~-11.35...cents A - E - B...
...B ~-3.38...Cents F# - C# - G# - Eb - Bb - F - C

which are all four differ considerable from the modern
ahistorically PC^(1/4) reinterpretation that was
created by somebody else after W's death:
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_III_.28V.29:_an_additional_temperament_divided_up_through_1.2F4_comma

>
>
But when retuning back to yours actual tuning:
Consider the above changements, when transferred into...
>
> ...the Scala file :
>
> ! werckmeister3_eb89-l.scl
> !
> Harmonic equal-beating version of the famous Well Temperament
> 12
> !
104/99 ! C# instead of > 2225/2112 ! C#
> 295/264 ! D
13/11 ! Eb , which is much simpler than: > 6675/5632 ! Eb
> 441/352 ! E
> 4/3 ! F
139/99 ! > 2225/1584 ! F#
> 148/99 ! G
52/33 ! G# replaces > 2225/1408 ! G#
> 147/88 ! A
> 16/9 ! Bb
> 1323/704 ! B
> 2/1
all others unaffected do remain unchanged.

Quest:
What do you think about that proposed alterations?

au revoir, bye bye, a bientot
Andy

--- In tuning@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:

> Salut Jacques,
> for instance, yours above scheme can be converted into an
> 3xN+-1 sequence by some little changes:

Where's the Collatz sequence? I'm not seeing it.

Andreas Sparschuh wrote :

> > (Jacques) :
> > Corrected here :
> >
> > C: (7*7 = A/27 = 49 98 <) 99 198 _396_
> > G: 37 74 (A/9 = 147 <)148 (D/3=295 <) 296 (<297 := C*3 ) _592_
> > D: (A/3 = 441 882 <) _885_ := 295*3
> > A: _1323_ := 441*3
> > E: _3969_ := A*3 := 49*81
> > B: _11907_ := E*3 := 49*243
> > F#: _2225_ 4450 8900 17800 35600 (<35721 := B*3)
> > C#: 6675 := F#*3
> > G#: 20025 := C#*3
> > Eb: 60075 := G#*3
> > Bb: 11 22 44 88 176 .... 180224 (< 180225 := Eb*3)
> > F : 33 66 132 264 ...
> >
> > Now what I miss to understand is what is a "3n-1" sequence:
>
> Salut Jacques,
> for instance, yours above scheme can be converted into an
> 3xN+-1 sequence by some little changes:
>
> Overtake the 5ths fron C:..to..A: as they are unaltered,
> then continue only with epimoric ratios:
>
> E: 31 62 124 248 496 892 1984 3968 (<3969 := 49*81)
> B: 93
> F#: 139 278 (<279 := B*3)
> C#: 13 26 52 104 206 416 (< 417 := F#*3)
> G#: 39
> Eb: 117
> Bb: 11 (> 351 := D#*3)
> F: 33 66 132 264 528 from that you had once started
>
> Already A.Werckmeister, did so in his:
> http://en.wikipedia.org/wiki/> Werckmeister_temperament#Werckmeister_IV_.28VI.> 29:_the_Septenarius_tunings
> /tuning/topicId_89066.html#89216
> "
> C: 49 := 7*7 98 C=196
> F: F=147 := C*3
> B:(German) 55 B=110 220 440 (<441:=F*3) English: "Bb"
> Eb: Eb=165 := B*3
> G#: 31 62 G#=124 := Eb*3
> C#: 93 C#=186 := G#*3
> F#: F#=139 278 (<279 := C#*3)
> H: 13 26 52 H=104 208 416 (< 417 := F#*3) Engl.:"B"
> E: 39 78 E=156 := H*3
> A: A=117 := E*3
> D: 11 22 44 88 D=176 352(> 351 := A*3) or ???"175"???
> G: G=131 (132 := D44 * 3)
> C: C=196 392 (< 393 := G*3)

Splendid temperament !

> or in todays modern terms:
> http://en.wikipedia.org/wiki/Collatz_conjecture
> For deeper understanding, here you can play with that:
> http://l.pellegrino.free.fr/syracuse/
> http://www.nitrxgen.net/collatz.php
> http://www.gfredericks.com/sandbox/arith/collatz
> http://fr.wikipedia.org/wiki/Conjecture_de_Syracuse
>
> > I have a vague idea about "epimoric tempering", but I would
> > appreciate to know a definition, if there is.
>
> It means that in all tempered 5ths,
> there occur only barely temperings that are
> restricted to epimoric-ratios of the form: n/(n+-1)
> with any arbitrary-chosen integral number n
> out of the whole number-set: N.
>
> > Was not my precedent 613-limit variant epimoric-tempered as well ?
> Not quite fully yet, because yours choice:
>
> > F#: _2225_ 4450 8900 17800 35600 (<35721 := B*3)
> differs about 121 := 35600-35721
> with 121, which is quite different from the demanded case of +-1.

Of course.

> > Subsidiary question :
> > would this one be considered a correct WIII also ?
> Only almost,
> because there appears an tiny tempering within Eb-Bb
>
> > Eb: 60075 := G#*3
> > Bb: 11 22 44 88 176 .... 180224 (< 180225 := Eb*3)
>
> but that is absent in Werckmeister's original specification:
> that allows temperings only @ the 5ths: C~G~D~A ... B~F#
> Therefore: When taken in W's strict sense: Not so at all.
>
> > and according to you which one would be more valuable,
> > between this 89-limit and my previous 613-limit variant ??
> Difficult to judge about here on a question of personal taste.
>
> But in doubt I would then decide but for the simpler version
> of the lower 89-limit than instead of the higher 613-limit.

Thank you for this point of view. I thought the 613 was better but you made me try the 89 and it is also very good. Also knowing what you revealed to me about the history of rational temperaments, I am about to like both as much now.
A funny fractal thing : 613/89 ~62/9 (you just mentionned 62)

Contered question:
What do you think about my own 41-limit variant version?
/tuning/topicId_89550.html#89588

Very good, simple and effective ! its very much in the style of many I did also. But perhaps I would not say it's a "W3" in the common sense.

> Concise solution:
> C 6560/6561 G 204/205 D 152/153 A - E - B...
> ...B 512/513 F# - C# - G# - Eb - Bb - F - C
> "
> or that epimotic-ratios do amount in Cent-units:
>
> C ~-0.264...cents G ~-8.47...cents D ~-11.35...cents A - E - B...
> ...B ~-3.38...Cents F# - C# - G# - Eb - Bb - F - C
>
> which are all four differ considerable from the modern
> ahistorically PC^(1/4) reinterpretation

- that's what I said ...

> that was created by somebody else after W's death:
> http://en.wikipedia.org/wiki/> Werckmeister_temperament#Werckmeister_III_.28V.> 29:_an_additional_temperament_divided_up_through_1.2F4_comma
>
> >
> >
> But when retuning back to yours actual tuning:
> Consider the above changements, when transferred into...
> >
> > ...the Scala file :
> >
> > ! werckmeister3_eb89-l.scl
> > !
> > Harmonic equal-beating version of the famous Well Temperament
> > 12
> > !
> 104/99 ! C# instead of > 2225/2112 ! C#
> > 295/264 ! D
> 13/11 ! Eb , which is much simpler than: > 6675/5632 ! Eb
> > 441/352 ! E
> > 4/3 ! F
> 139/99 ! > 2225/1584 ! F#
> > 148/99 ! G
> 52/33 ! G# replaces > 2225/1408 ! G#
> > 147/88 ! A
> > 16/9 ! Bb
> > 1323/704 ! B
> > 2/1
> all others unaffected do remain unchanged.
>
> Quest:
> What do you think about that proposed alterations?

Why not, it looks simpler, and could be a good temperament also ! My purpose was only to make a rational version of W3 as much as realistic as possible according to the canons of 1/4 PC tempering, in order to not change its design but only optimising it acoustically.
I believe, by experience and for many reasons that rational tunings have better accoustic qualities.
But the central idea here is the equal-beating property 3A - 5C = 4E - 5C = 3D - 2A, that serves completely this design. The Comptine recurrent sequence does it in a way that generates quasi perfect 1/4 PC tempered fifths, and the W3 versions that support that eq-b are so perfectly close to the 1/4 PC irrational temperament that it indicates in fact a natural property of W3.
When tuning strict W3 by ear and wether you know it or not, the eq-b of these three important dyads is unescapable and can only make tuning easier and more harmonious.
As a historical temperament it is not unique in that aspect, but it is a good example.
So high or low prime limits in this regard was not important for me here, and both have their qualities.
I liked the 613-limit version because it's the only one (I think) that supports this precise Eq-b AND has 8 perfectly pure 3/2 fifths.
I like the 89-limit version for other reasons : because C = 99 would allow simple transpositions by n/11 and therefore, suggests very interesting 24-quartertones tunings...
Complementing for example 44 66 99 148 of this W3 by
16 24 36 54 81 121 181 generates
16 (20) 24 (30) 36 44 54 66 81 99 121 148 181 = superb Mohajira -c sequence !!
(where the 99 - 81 difference tone is in tune with the 3A - 5C = 4E - 5C = 3D - 2A eq-b frequency = 9, etc...)
And I like the idea of having a 24 quartertones tuning based on two well-temperaments ; if one is a W3, it should not hurt !

> au revoir, bye bye, a bientot
> Andy

a bientot !
- - - - - - -
Jacques