Yahoo Tuning Groups Ultimate Backup tuning Re:41-limit superparticular rational Werckmeister #3

Dear Andreas,

Thank you for these enlightning examples.
I am discovering these harmonic versions of Werckmeister temperaments
with deep resonance.
As our proverb says : "Qui n'entend qu'une cloche, n'entend qu'un
son..."
(He who hears (only) one bell, hears only one sound...)
I believe in any subject it is advantageous to hear more than one
version of a story.

I am pleased as well to learn that, without knowing it, when I
designed my own "meta" versions of WIII a few years ago :

! werckmeister3_eb.scl
!
Harmonic equal-beating meta-version of the famous Well Temperament
12
!
256/243
21995/19683
32/27
2740/2187
4/3
1024/729
9808/6561
128/81
10960/6561
16/9
1370/729
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

I was simply continuing an ancient tradition !

Thank you for your researches, and for sharing these precious documents.
- - - - - - -
Jacques

On Friday May 28, 2010, Andreas Sparschuh wrote :

> > > Andreas wrote :
> > > PC^(1/n)
> > Jacques replied :
> > ...So, if I follow you, WerckmeisterIII, IV, V would be of "modern
> > design" ??
>
> Salut Jaques,
> that depends on which (re)interpretation of Werckmeister #3
> one chooses. For some possible variants see my meassage:
> /tuning/topicId_73833.html#74576
> as for instance Daniel G. Tuerck's [1806] monochord stringlenths:
>
> C 8192
> # 7776
> D 7331
> # 6912
> E 6540
> F 6144
> # 5832
> G 5480
> # 5184
> A 4905
> # 4608
> C 4096
>
> !Tuerck_s_W3.scl
> Daniel Gottlob Tuerck's [1806] Werckmeister #3 compiled by A.Sparschuh
> ! by converting the monochord-stringlengths into ratio
> 12
> !
> ! 1/1 ! C 8192
> 256/243 ! C# 7776
> 8192/7331 ! D 7331
> 32/27 ! Eb 6912
> 2048/1635 ! E 6540
> 4/3 ! F 6144
> 1024/729 ! F# 5832
> 1024/685 ! G 5480
> 128/81 ! G# 5184
> 8192/4905 ! A 4905
> 16/9 ! Bb 4608
> 2/1 ! C' 4096
> !
> ![eof]
>
> Attend here the unequal distribution of the involved 5ths:
>
> C 2048/2055 G 21920/21993 D 14662/14715 A - E - B...
> ...B 2180/2187 F# - C# - G# - D# - A# - F - C
>
> that's in Cent-units:
>
> C ~-5.91cent ~G~ 5.76cent ~D~ 6.25cent ~A - E - B...
> ~ 5.55cent ~ F# - C# - G# - D# - A# - F - C
>
> which shown that all that four ones are different in seize,
> instead of the modern PC^(1/4), on that you do refer.
>
> >>> ...(Jacques) or rather possibly used by J.S.Bach as plausible
> >>> tuning techniques in comparison ?...
> > > ...(Andreas)..for instance Werckmeister's "septenarian" technique.
> > ...(Jacques)...interesting point of view.
>
> Because I considered even Tuerck's numbers as to high in range,
> see for my own lower nubers than Tuerck that I found in my:
> /tuning/topicId_68023.html#68047
> in order to distribute four superparticular ratios over
> Werckmeister's 'eight-pure-5ths- specification:
> C ~ G ~ D ~ A - E - B ~ F# - C# - G# - Eb - Bb - F - C
>
> 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 when expanded into absolute-ptich frequencies:
>
> 273.375 C_4 : ((17))2187:= 3^7
> 410 G_4 : (17*3=51,102,204<)205,410,820,1640,3280,6560(<6561:= 3^8)
> 306 D_4 : (19,38,76,152<)153:= 17*9
> 456 A_4 : 57:= 19*3
> 342 E_4 : 171:= 19*9
> 256.5 B_4 : (1,...,512<)513:= 19*27
> 384 F#4 : 3
> 288 C#4 : 9
> 432 G#4 : 27
> 324 Eb4 : 81
> 486 Bb4 : 243
> 364.5 F_4 : 729:= 3^6
> 273.375 C_4 : 2187:= 3^7
>
> that's in chromatique ascending pitch order
>
> pitch | name | ratio
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> 273.375 C_4 1/1
> 288 C#4 256/243
> 306 D_4 272/243
> 324 Eb4 32/27
> 342 E_4 304/243
> 364.5 F_4 4/3
> 384 F#4 1024/729
> 410 G_4 3280/2187 Bach's coeval Cammer-tone ~410cps
> 432 G#4 128/81
> 456 Hz A_4 1216/729 Bach's coeval Choir-tone ~456cps
> 486 Bb4 16/9
> 513 B_4 152/81
> 546.75 C_5 2/1
>
> The absolute-pitches in 41-limit "Monzo"s prime-number decomposition:
>
> 273.375 C_4 = |-3,7>
> 288 C#4 = |5,2>
> 306 D_4 = |1,2,0,0,0,0,1> = |1,2>*17
> 324 Eb4 = |2,4>
> 342 E_4 = |1,2,0,0,0,0,0,1> = |1,2>*19
> 364.5 F_4 = |-1,6>
> 384 F#4 = |7,1>
> 410 G_4 = |2,0,1,0,0,0,0,0,0,0,0,0,1> = |2,0,1>*41 !JSB!
> 432 G#4 = |4,3>
> 456 Hz A_4 = |3,1,0,0,0,0,0,1> = |3,1>*19
> 486 Bb4 = |1,5>
> 513 B_4 = |0,3,0,0,0,0,0,1> = 19*3^3
> 546.75 C_5 = |-2,7>
>
> or in scala normalized ratios
>
> !Sp41limW3.scl
> Sparschuh's 41-limit Werckmeister #3 interpretation for J.S.Bach
> 12
> !
> 256/243 ! C# |8,-5> the Pythagorean-Limma
> 272/243 ! D |4,-5,0,0,0,0,1> = |4,-5>*17
> 32/27 ! Eb |5,-3>
> 304/243 ! E |1,-5,0,0,0,0,0,1> = |1,-5>*19
> 4/3 ! F |2,-1>
> 1024/729 ! F# |10,-6>
> 3280/2187 ! G |4,-7,1,0,0,0,0,0,0,0,0,0,1> = |4,-7,1>*41 !JSB!
> 128/81 ! G# |-7,4>
> 1216/729 ! A |6,-6,0,0,0,0,0,1> = |6,-6>*19
> 16/9 ! Bb |4,-2>
> 152/81 ! B |3,-4,0,0,0,0,0,1> = |3,-4>*19
> 2/1
> !
> ![eof]
>
> Enjoy Bach's organ-music especially in that one.
> Hence now, it's up to the experts to consider how
> 'authentic' or even 'originally'
> they want do deem that as apt for Bach.
>
> > (Jacques:) Probably, and in rhythmic/melodic patterns, too ! :)
> Verly likely not only there,
> but also inbetween the beating of the 5ths an 3rds.
>
> I.m.h.o,
> A. Werckmeister, J.G.Walther and J.S.Bach used
> something close to the above tuning, or perhaps even
> exactly the same ratios, as an 41-limit refinement of the predecessor:
>
> http://groenewald-berlin.de/text/text_T126.html
>
> "Andreas Reinhard, [1604] - Abraham Bartulus, [1614]"
> 'Benutzt werden die fünf Quotienten':
> "By using the five quotients":
> 16/15, 17/16, 18/17, 19/18 und 20/19:
> almost at the same positions.
>
> Walther remarked in a letter that his teacher Werckmeister once
> donated him the works of Reinhard & Bartulus when he was W's student
> in Halberstadt, in order to study from that how to tune an well-
> temperament...
>
> au revoir, bye bye à bientôt
> Andy

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Thank you for these enlightning examples.
> I am discovering these harmonic versions of
> Werckmeister temperaments
> with deep resonance.

Salut Jacques,
unfortunately Werckmeister tells us only,
that he demands and insists in rational numbers for all
his tunings, but without giving the concrete proportions.

> As our proverb says : "Qui n'entend qu'une cloche, n'entend qu'un
> son..."
> (He who hears (only) one bell, hears only one sound...)
Fine saying, that hits the point.

> I believe in any subject it is advantageous to hear more than one
> version of a story....
...in order to benefit from all of the different views.
>
> I am pleased as well to learn that, without knowing it, when I
> designed my own "meta" versions of WIII a few years ago :

Here an mp3-example of yours variant in:
http://launch.dir.groups.yahoo.com/group/tuning/message/89308

Additional I enclose to yours ratios the prime-number decompositions:
View that under the option: [Use-Fixed-Width-Font]

! werckmeister3_eb.scl
!
Harmonic equal-beating meta-version of the famous Well Temperament
! prime-number 'Monzo' vectors compiled and added by A.Sparschuh
12
!
256/243 ! C# |8,-5>
21995/19683 ! D |0,-9,1> /(83*53)
32/27 ! Eb |5,-3>
2740/2187 ! E |2,-7,1> *137
4/3 ! F |2,-1>
1024/729 ! F# |4,-8>
9808/6561 ! G |4,-8> *613
128/81 ! G# |7,-4>
10960/6561 | A |4,-8,1> *137
16/9 | Bb |4,-2>
1370/729 | B |1,-6> *137
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
!
! in 5ths:
!...F-C 19616/19683 G 21995/22068 D 4384/4399 A-E-B 2048/2055 F#-C#...
!
![eof]

or the 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#...

shows: Your's distribution agrees almost with Tuerck's[1806] result:

> > C 2048/2055 G 21920/21993 D 14662/14715 A - E - B...
> > ...B 2180/2187 F# - C# - G# - D# - A# - F - C
> >
> > that's in Cent-units:
> >
> > C ~-5.91cent ~G~ 5.76cent ~D~ 6.25cent ~A - E - B...
> > ~ 5.55cent ~ F# - C# - G# - D# - A# - F - C

Attend:
Among that two solutions there appears an intresting common feature:

Türck [1806] and you had both found und used indepently
the same ratio 2048/2055, but at different places:

Türck[1086] inbetween the initial first 5th: C-G
but you within the last concluding tempered 5th : B-F#

That coincidence is an funny property of both ;-)

> I was simply continuing an ancient tradition !
You mastered that magnificent!

http://forum.wordreference.com/showthread.php?t=562255
"Was du ererbt von deinen Vätern hast, erwirb es, um es zu besitzen"
traducion:
"What you have inherited from your father[s], make it yours to possess it.". The meaning is that you should take the inheritance not to sell it, to spend it, to make fun of it, but to continue the work, to protect and develop it further. It is often used in context of cultural heritage and its conservation and its holding in esteem."

'Qu'est-ce que vous avez hérité de vos pères,
doit gagner afin de posséder.'

In deed. I.m.h.o:
Surely Werckmeister would love yours much more than
the common usual botching into 4 logarithmically parts.

> Thank you for your researches,
> and for sharing these precious documents.

Meanwhile,
I wonder and ask myself how many other interpretations
could there still exist of which we both know yet the slightest idea.

Hence my quest to all group-members:
Does any body else here in that forum
knows about others variants of Werckmeister's
most famous masterpiece with the 8 pure 5ths?

au revoir, bye bye à bientôt
Andy

🔗Jacques Dudon <fotosonix@...>

Thank you so much Andy,
for this fine analysis and all the complements !
I have to learn about Monzo' vectors symbolism and also this line :

> ! in 5ths:
> !...F-C 19616/19683 G 21995/22068 D 4384/4399 A-E-B 2048/2055 F#-C#...

... which I could not follow but seems interesting -
It seems to describe the commas, I have to decrypt this writing.

Funny that you found this same comma 2055/2048 in Tuerck's - you have
a trained eye !
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) is one among others,
since it is based on a recurrent sequence.
I choosed it because it is based on a perfect series of pure fifths
and manages luckily among them, but it has a quite high limit.
I never heard about rational versions of WIII before, and I wanted to
stick here to the 1/4 PCs in the most precise way...
But the same "Comptine" recurrent sequence x^3 = (3/5)x^2 + 2, that
approximates 1/4 PC tempered fifths, allows for as many versions as
we want.
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

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)

! comptine_h3.scl
!
1/4 pyth. comma meantone sequence between G and B, completed by 8
pure fifths
12
!
2225/2112
592/528
20025/16896
1323/1056
4/3
29667/21120
3/2
6675/4224
885/528
60075/33792
9889/5280
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

- - - - - - -
Jacques

Andreas Sparschuh wrote:

> > (Jacques) :
> > Thank you for these enlightning examples.
> > I am discovering these harmonic versions of
> > Werckmeister temperaments
> > with deep resonance.
>
> Salut Jacques,
> unfortunately Werckmeister tells us only,
> that he demands and insists in rational numbers for all
> his tunings, but without giving the concrete proportions.
>
> > As our proverb says : "Qui n'entend qu'une cloche, n'entend qu'un
> > son..."
> > (He who hears (only) one bell, hears only one sound...)
> Fine saying, that hits the point.
>
> > I believe in any subject it is advantageous to hear more than one
> > version of a story....
> ...in order to benefit from all of the different views.
> >
> > I am pleased as well to learn that, without knowing it, when I
> > designed my own "meta" versions of WIII a few years ago :
>
> Here an mp3-example of yours variant in:
> http://launch.dir.groups.yahoo.com/group/tuning/message/89308
>
> Additional I enclose to yours ratios the prime-number decompositions:
> View that under the option: [Use-Fixed-Width-Font]
>
> ! werckmeister3_eb.scl
> !
> Harmonic equal-beating meta-version of the famous Well Temperament
> ! prime-number 'Monzo' vectors compiled and added by A.Sparschuh
> 12
> !
> 256/243 ! C# |8,-5>
> 21995/19683 ! D |0,-9,1> /(83*53)
> 32/27 ! Eb |5,-3>
> 2740/2187 ! E |2,-7,1> *137
> 4/3 ! F |2,-1>
> 1024/729 ! F# |4,-8>
> 9808/6561 ! G |4,-8> *613
> 128/81 ! G# |7,-4>
> 10960/6561 | A |4,-8,1> *137
> 16/9 | Bb |4,-2>
> 1370/729 | B |1,-6> *137
> 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
> !
> ! in 5ths:
> !...F-C 19616/19683 G 21995/22068 D 4384/4399 A-E-B 2048/2055 F#-C#...
> !
> ![eof]
>
> or the 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#...
>
> shows: Your's distribution agrees almost with Tuerck's[1806] result:
>
> > > C 2048/2055 G 21920/21993 D 14662/14715 A - E - B...
> > > ...B 2180/2187 F# - C# - G# - D# - A# - F - C
> > >
> > > that's in Cent-units:
> > >
> > > C ~-5.91cent ~G~ 5.76cent ~D~ 6.25cent ~A - E - B...
> > > ~ 5.55cent ~ F# - C# - G# - D# - A# - F - C
>
> Attend:
> Among that two solutions there appears an intresting common feature:
>
> Türck [1806] and you had both found und used indepently
> the same ratio 2048/2055, but at different places:
>
> Türck[1086] inbetween the initial first 5th: C-G
> but you within the last concluding tempered 5th : B-F#
>
> That coincidence is an funny property of both ;-)
>
> > I was simply continuing an ancient tradition !
> You mastered that magnificent!
>
> http://forum.wordreference.com/showthread.php?t=562255
> "Was du ererbt von deinen Vätern hast, erwirb es, um es zu besitzen"
> traducion:
> "What you have inherited from your father[s], make it yours to
> possess it.". The meaning is that you should take the inheritance
> not to sell it, to spend it, to make fun of it, but to continue the
> work, to protect and develop it further. It is often used in
> context of cultural heritage and its conservation and its holding
> in esteem."
>
> 'Qu'est-ce que vous avez hérité de vos pères,
> doit gagner afin de posséder.'
>
> In deed. I.m.h.o:
> Surely Werckmeister would love yours much more than
> the common usual botching into 4 logarithmically parts.
>
> > Thank you for your researches,
> > and for sharing these precious documents.
>
> Meanwhile,
> I wonder and ask myself how many other interpretations
> could there still exist of which we both know yet the slightest idea.
>
> Hence my quest to all group-members:
> Does any body else here in that forum
> knows about others variants of Werckmeister's
> most famous masterpiece with the 8 pure 5ths?
>
> au revoir, bye bye à bientôt
> Andy

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

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

🔗monz <joemonz@...>

Hello Jacques,

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> I have to learn about Monzo' vectors symbolism

I have not been reading this list a lot for several years now,
but i am still here and if you ask me i will try to help.

My essential idea is that it is much simpler to represent
ratios by using a vector which lists the exponents of
the prime-factors.

Normally it is assumed that the primes simply start at 2
and include each one after that up to some limit. Thus
for example, the ratio 33:32 = [-5 1, 0 0 1>
because 32 = 2^5 and it is in the denominator, while
the numerator is 33 = 3 * 11. So the prime-factors
which are assumed in this vector are 2, 3, 5, 7, and 11.

As a convention, i like to put commas after the exponent
for prime-factor 3 and then every third exponent after that,
just to make it easier to read. For 3, 5, or 7-limit
tunings commas are not really necessary at all.

If there is a specific set of prime-factors involved
which does not follow the whole series, and one does not
wish to include all the factors which will always have
exponents of zero, then the factors must be explicitly
stated beforehand.

Another huge advantage of using this notation is that
when "adding" or "subtracting" intervals, which means
multiply or dividing the ratios, one can simply use
addition on each exponent to find the result. For example,
to find the interval which is a 8:7 above 33:32,

[-5 1, 0 0 1> = 33:32
+ [ 3 0, 0 -1 0> = 8:7
-------------------------
[-2 1, 0 -1 1> = 33:28

Much easier than reducing fractions and multiplying!

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software