Yahoo Tuning Groups Ultimate Backup tuning Dieterich Buxtehude and the Mean-Tone Organ

There are lotsa LucyTuned (meantone-type) tracks from lullabies to rave out there on the net.

You can find some links from:

http://www.lucytune.com
and
http://www.lullabies.co.uk
http://www.harmonics.com

One day soon I'll sort them out and update the sites, after they finish digging up my street, which has been disturbing me for the past week.

On 9 Jul 2008, at 11:22, Mike Battaglia wrote:

> Has anyone ever listened to this? This album is beautiful. I'm just
> listening to it for the first time now. I understand why quarter comma
> meantone is so popular now - the mood of this album is very relaxed
> and beautiful. I do hear what George Secor meant by the just leading
> tone being less effective than that of 12-et, but it's not less
> effective - it's just a different kind of leading tone. It creates
> much less tension before it leads, which partially contributes to the
> extremely relaxed feel of the music.
>
> Do any of the baroque and early classical music specialists on this
> group have any other quarter-comma or other meantone recordings they
> could recommend? I've never paid much attention to it before,
> preferring to focus on 31-tet and such instead, but it just occured to
> me this is a pretty easy way to get a feel for part of 31-tet anyways.
>
> Or, if anyone has any recommendations for recordings of classical
> works done in 53-tet or other temperaments, or even perhaps JI... That
> would be amazing. Acoustic or electric recordings is what I'm after
> specifically, not too much MIDI.
>
> I'm slowly becoming one of those people that hates 12-tet. This album
> got me pretty good. Even the feel of the harmony is just MUCH more
> complex than 12-tet. Quarter comma meantone/31-tet would be very
> useful in all styles of music, especially shoegaze and pop and such.
> Quarter comma emo maybe. Maybe not jazz, which has built up a huge
> vocabulary of stacked-fifth chords, but we have 12-tet for that anyway
> - for now.
>
> -Mike
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Mike Battaglia <battaglia01@...>

Ah yeah, I've heard these... Really good. There's one, I don't
remember which, in which the sample clip is in F# phrygian dominant...
Really liked that one.

On Wed, Jul 9, 2008 at 6:45 AM, Charles Lucy <lucy@...> wrote:
> There are lotsa LucyTuned (meantone-type) tracks from lullabies to rave out
> there on the net.
>
> You can find some links from:
> http://www.lucytune.com
> and
> http://www.lullabies.co.uk
> http://www.harmonics.com
> One day soon I'll sort them out and update the sites, after they finish
> digging up my street, which has been disturbing me for the past week.
>
>
> On 9 Jul 2008, at 11:22, Mike Battaglia wrote:
>
> Has anyone ever listened to this? This album is beautiful. I'm just
> listening to it for the first time now. I understand why quarter comma
> meantone is so popular now - the mood of this album is very relaxed
> and beautiful. I do hear what George Secor meant by the just leading
> tone being less effective than that of 12-et, but it's not less
> effective - it's just a different kind of leading tone. It creates
> much less tension before it leads, which partially contributes to the
> extremely relaxed feel of the music.
>
> Do any of the baroque and early classical music specialists on this
> group have any other quarter-comma or other meantone recordings they
> could recommend? I've never paid much attention to it before,
> preferring to focus on 31-tet and such instead, but it just occured to
> me this is a pretty easy way to get a feel for part of 31-tet anyways.
>
> Or, if anyone has any recommendations for recordings of classical
> works done in 53-tet or other temperaments, or even perhaps JI... That
> would be amazing. Acoustic or electric recordings is what I'm after
> specifically, not too much MIDI.
>
> I'm slowly becoming one of those people that hates 12-tet. This album
> got me pretty good. Even the feel of the harmony is just MUCH more
> complex than 12-tet. Quarter comma meantone/31-tet would be very
> useful in all styles of music, especially shoegaze and pop and such.
> Quarter comma emo maybe. Maybe not jazz, which has built up a huge
> vocabulary of stacked-fifth chords, but we have 12-tet for that anyway
> - for now.
>
> -Mike
>
> Charles Lucy
> lucy@...
> - Promoting global harmony through LucyTuning -
> for information on LucyTuning go to:
> http://www.lucytune.com
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>
>
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Ah yeah, I've heard these... Really good. There's one, I don't
> remember which, in which the sample clip is in F# phrygian dominant...
> Really liked that one.
>
> On Wed, Jul 9, 2008 at 6:45 AM, Charles Lucy <lucy@...> wrote:
> > There are lotsa LucyTuned (meantone-type) tracks from lullabies to
rave out
> > there on the net.
> >
> > You can find some links from:
> > http://www.lucytune.com
> > and
> > http://www.lullabies.co.uk
> > http://www.harmonics.com
> > One day soon I'll sort them out and update the sites, after they
finish
> > digging up my street, which has been disturbing me for the past week.
> >
> >
> > On 9 Jul 2008, at 11:22, Mike Battaglia wrote:
> >
> > Has anyone ever listened to this? This album is beautiful. I'm just
> > listening to it for the first time now. I understand why quarter comma
> > meantone is so popular now - the mood of this album is very relaxed
> > and beautiful. I do hear what George Secor meant by the just leading
> > tone being less effective than that of 12-et, but it's not less
> > effective - it's just a different kind of leading tone. It creates
> > much less tension before it leads, which partially contributes to the
> > extremely relaxed feel of the music.
> >
> > Do any of the baroque and early classical music specialists on this
> > group have any other quarter-comma or other meantone recordings they
> > could recommend? I've never paid much attention to it before,
> > preferring to focus on 31-tet and such instead, but it just occured to
> > me this is a pretty easy way to get a feel for part of 31-tet anyways.
> >
> > Or, if anyone has any recommendations for recordings of classical
> > works done in 53-tet or other temperaments, or even perhaps JI... That
> > would be amazing. Acoustic or electric recordings is what I'm after
> > specifically, not too much MIDI.
> >
> > I'm slowly becoming one of those people that hates 12-tet. This album
> > got me pretty good. Even the feel of the harmony is just MUCH more
> > complex than 12-tet. Quarter comma meantone/31-tet would be very
> > useful in all styles of music, especially shoegaze and pop and such.
> > Quarter comma emo maybe. Maybe not jazz, which has built up a huge
> > vocabulary of stacked-fifth chords, but we have 12-tet for that anyway
> > - for now.
> >
> > -Mike
> >
> > Charles Lucy
> > lucy@...
> > - Promoting global harmony through LucyTuning -
> > for information on LucyTuning go to:
> > http://www.lucytune.com
> > For LucyTuned Lullabies go to:
> > http://www.lullabies.co.uk
> >
> >Digging up the street hah. Very annoying. Probably the local
council creating another unnecessary speed hump? Yes, those lullabies
are very interesting and beautiful indeed.

Just to play devils advocate, wouldn't you say that the advantage of
Bach over Buxtehude is all of those wonderful modulations and
counterpoint so characteristic of Bach and the 12-tet tuning system?
I'm just wondering, is this possible in the meantone system? For eg,
given maj 3 as 5/4 and min as 19/16, then the fifth is the product of
the two giving 95/64. Since multiplication is commutative (A x B = B x
A) then the maj3 from the min3 = the min3 from the maj3 and both = the
fifth. But then applying this 12 times does not reach the octave
(giving around 114).

A better approx. is min3 = 609/512 and maj3 = 645/512. Their product
(the fifth) gives 392805/262144 which is very close to the tempered
fifth and applying this 12 times gives 128.128...which is very close
to 128, the 7'th octave. Yet it is still harmonic. Of course, we are
here dealing with just a few notes and it becomes much more
complicated when we extend this problem from all points of view. It
still seems to me that for all its faults the 'democratic' 12-tet
system resolves this issue so nicely. But as Mike said, we still have
it anyway.
> >
>PS. Charles, I don't need new glasses. Ends up I'm allergic to
wattle, which being one of our natural faunas is so un-Australian of me.

Rick

🔗Mike Battaglia <battaglia01@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > Just to play devils advocate, wouldn't you say that the advantage of
> > Bach over Buxtehude is all of those wonderful modulations and
> > counterpoint so characteristic of Bach and the 12-tet tuning system?
> > I'm just wondering, is this possible in the meantone system? For eg,
> > given maj 3 as 5/4 and min as 19/16, then the fifth is the product of
> > the two giving 95/64. Since multiplication is commutative (A x B = B x
> > A) then the maj3 from the min3 = the min3 from the maj3 and both = the
> > fifth. But then applying this 12 times does not reach the octave
> > (giving around 114).
>
> Bach didn't compose in 12-tet. From what I know, he started off in
> meantone like everyone else, and then switched to one or more forms of
> well temperament later on. But he never messed around with 12-equal,
> as far as I know.
>
> > A better approx. is min3 = 609/512 and maj3 = 645/512. Their product
> > (the fifth) gives 392805/262144 which is very close to the tempered
> > fifth and applying this 12 times gives 128.128...which is very close
> > to 128, the 7'th octave. Yet it is still harmonic. Of course, we are
> > here dealing with just a few notes and it becomes much more
> > complicated when we extend this problem from all points of view. It
> > still seems to me that for all its faults the 'democratic' 12-tet
> > system resolves this issue so nicely. But as Mike said, we still have
> > it anyway.
>
> That's a decently rational intonation version of 12-tet.
>
> -Mike

>I suppose that he was just the first to compose on it with the well
tempered clavier. But I am no expert.

In fact, all of the intervals from 640 to 648 over 512 seem to
correspond to different versions of our major 3rd. Can someone please
remind me of the just noticeable difference. (I've re-ordered
Helmholtz' sensation of tone but it takes months to get here to
Australia). I'm just thinking that entire 'chunks' of the harmonic
series can be assigned to each interval within each respective octave
and that there is a limit to the number of octaves we can perceive.
The same might apply to any other tuning system.

Rick

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Or, if anyone has any recommendations for recordings of classical
> works done in 53-tet or other temperaments, or even perhaps JI...

Dear Mike & all others,

for Newton's "horogramm" in 53:
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

in the
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
style, i do reccomend the epimoric
stepwise cycle of 5ths modulo octaves in Bosanquet's notation
in terms of the corresponding absolute pitches:

0; c-_-4 = 1 ... c-_4=256Hz unison as general reference to the unit
1; g-_-2 = 3 ! 5th
2; d-_-1 = 9 ! major-tone
3; a-_0 = 27 ! Pythagorean 6th
4; e-_2 = 81 ! ditone
5; b-_3 = 243 ! octave:limma
6; gB_6 = 729 ! tritone
7; dB_7 = 2,187 ! apotome 4,374 [< 4,375 = 5*a\_5] the 'ragisma'
8; aB_3 = 205 410 820 1,640 3,280 6,560 (<6,561 = 3^8)
9; eB_5 = 615
10; bB_5 = 1,845
11; f\_7 = 2,767 5,534 (<5,535)
12; c\_7 = 2,075 4,150 8,300 (<8,301)
13; g\_4 = 389 778 1,556 3,112 6.224 (<6,225)
14; d\_6 = 1,167
15; a\_5 = 875=5*f._3 ; 1,750 3,500 (<3,501) JI 3rds: F. -> A\
16; e\_7 = (41 82 164 328 656 1,312 2,624<) 2625 = 5*c._3
17; b\_8 = (123 ... 7.872<) 7875 = 5*g._6
18; gb_10 = (369 ... 23,616<) 23,625 = 5*d._8 last JI 3rd among 4
19; db_6 = 1107
20; ab_4 = 415 830 1,660 3,320 (<3321) neoBaroque tuning-forks
21; eb_6 = 1,245
22; bb_6 = 1,867 3,734 (<3,735)
23; f._3 = 175 350 700 1,400 2,800 5,600(<5601) instead W's "176"
24; c._5 = 525 tenor_C5 ; middle_C4 = 262.5 Hz
25; g._6 = 1,575
26; d._8 = 4,725
27; a._9 = 14,175
28; e._11 = 42,525
29; b._12 = 127,575
30; f#_12 = (1,495 ... 95,680<) 95,681 ... 382,724 (<382,725)
31; c#_8 = 4,485
32; g#_9 = 13,455
33; d#_11 = 40,365
34; a#_12 = 121,095
35; f/_12 = 90,821 181,642 363,284 (<363,285)
36; c/_13 = 136,231 272,462 (<272,463)
37; g/_12 = 102,173 204,346 408,692 (<408,693)
38; d/_13 = 153,259 306,518 (<306,519)
39; a/_4 = 449 ... 459,776 (<459,777)
40; e/_6 = 1,347
41; b/_7 = 4,041
42; f&_9 = 12,123 := f#/ with '&'='#'*'/' about 6 commata sharper
43; c&_11 = 36,369
44; g&_12 = 109,107
45; d&_11 = 40,915 ... 327,320 (<327,321)
46; a&_9 = 15,343 ... 122,744 (<122,745)
47; f+_11 = 46,029 := f//_11 with '+':= '//' double comma elevation
48; c+_12 = 69,043 138,086 (<138,087)
49; g+_10 = 25,891 ... 207,128 (<207,129)
50; d+_12 = 77,673
51; a+_12 = 116,509 233,018 (<233,019)
52; ( e+ = f- )_13 = 177,763 349,526 (<349,527) enharmoic change
53=0'; b+_3 = c-_4 = 256Hz=2^8 ... 2^19=524,288 (<524,299) back home

that cycle matches almost
http://en.wikipedia.org/wiki/53_equal_temperament
it also subdivides the
"Mercator's Comma. Mercator's Comma is of such small value to begin
with (~3.615 cents)"
into the above 23 epimoric subfactors instead of 53 equal ones.
Attend within that the schisma 32805:32768 inbetween:

...Gb 2624:2625 Db 3320:3321 Ab Eb 3734:3735 Bb 5600:5601 F...

gaining JI heptatonics for C-major in

1. major and minor 3rds:
[F.] 5:4 [A\] 6:5 [C.] 5:4 [E\] 6:5 [G.] 5:4 [B\] 6:5 [D.]

2. as scale of whole&semi-tones:
[C.] 9:8 [D.] 10:9 [E\]16:15[F.] 9:8 [G.] 10:9 [A\] 9:8 [B\]16:15[c.]

or in commatic ascending order, as in Newton's 1664 drawing:

!septenarian53well.scl
Sparschuh's 53 generalization of Werckmeister's septenarius
53
2075/2048 ! 1; c\_7 : 2^11
525/512 ! 2; c._5 : 2^9 ~tenor_C in ET in reference to a4=440Hz
136231/131072! 3; c\_13 : 2^17
69043/65536 ! 4; c+_12 : 2^16
2187/2048 ! 5; dB_7 : 2^11 apotome
1107/1024 ! 6; db_6 : 2^10
4485/4096 ! 7; c#_8 : 2^12
36369/32768 ! 8; c&_11 : 2^15
9/8 ! 9; d-_3 : 2^3 Pythaogorean major-tone
1167/1024 !10; d\_6 : 2^10
4725/4096 !11; d._8 : 2^12
153259/131072!12; d\_6 : 2^10
77673/65536 !13; d+_12 : 2^16
615/512 !14; eB_5 : 2^9 (5:4)*(123:128)
1245/1024 !15; eb_6 : 2^10 (5:4)*(249:256)
40365/32768 !16; d#_11 : 2^15 (5:4)*(8073:8192)
40915/32768 !17; d&_11 : 2^15 (5:4)*(8183:8192)
81/64 !18; e-_2 : 2^6 Pythagorean ditone
2625/2048 !19; e\_7 : 2^11 = C.*(5:4) JI 3rds in [C.]->[E\]
42525/32768 !20; e._11 : 2^15
1347/1024 !21; e/_6 : 2^10
174763/131072!22;(e+=f-)_13:2^17 = (4:3)*(524,289:524,288) enharm.ch.
2767/2048 !23; f\_7 : 2^11
175/128 !24; f._3 : 2^7 instead of Werckmeister's "176" choice
90821/65536 !25; f/_12 : 2^16
46029/32768 !26; f+_11 : 2^15
729/512 !27; gB_6 : 2^9 tritone
23625/16384 !28; gb_10 : 2^14 = D.*(5:4) JI 3rds in [D]->[F#\]
95681/65536 !29; f#_12 : 2^16
12123/8192 !30; f&_12 : 2^16
3/2 !31; g-_-2 : 2 the initial 5th step at begin
389/256 !32; g\_4 : 2^8
1575/1024 !33; g-_5 : 2^10
102173/65536 !34; g/_12 : 2^16
25891/16384 !35; g+_10 : 2^14
205/128 !36; aB_3 : 2^7
415/256 !37; ab_4 : 2^8 neoBaroque tuning-fork in 415Hz
13455/8192 !38; g#_9 : 2^13
109107/65536 !39; g&_12 : 2^16
27/16 !40; a-_0 : 2^4 Pythagorean 6th
875/512 !41; a\_5 : 2^9 = F.*(5:4) JI 3rds [F.]->[A\]
14175/8192 !42; a._9 : 2^13
449/256 !43; a/_4 : 2^8
116509/65536 !44; a+-12 : 2^16
1845/1024 !45; bB_6 : 2^10
1867/1024 !46; bb-6 : 2^10
121095/65536 !47; a#_12 : 2^16
15343/8192 !48; c&_9 : 2^13
243/128 !49; b-_3 : 2^7 Pythagorean 7th or octave:limma
7875/4096 !50; b\-8 : 2^12
127575/65536 !51; b._12 : 2^16
4041/2048 !52; b/_7 : 2^11
2/1 ! 53==0 ; ( b+ = 2*c+ )_3 Helmholtz's enharmonics: B// = C\\
!
!
that yiedls -compared against 53EDO- an intersting
53-comma key-charcteristics
http://www.wmich.edu/mus-theo/courses/keys.html
because all the 3rds range in fine graduation inbetween

8192:6561 ~384Cents (schimatic 3rd) <<<???<<< and 5:4 ~386Cents

Attend for instance the 3rd [A\] -> [DB]
that becomes about an
http://en.wikipedia.org/wiki/Ragisma
flattend.
That small interval was also historically also used for coin-making:
http://de.wikipedia.org/wiki/Karlspfund
"Bei historischen LÃ¤ngenmaÃen liegt der Variationskoeffizient im
allgemeinen unter 1/500, was eine Genauigkeit von Â± 0,2 % bedeutet. So
gelten bei den LÃ¤ngenmaÃen z.B. 1/2400 oder 1/4374, also die 7-glatten
Ratios 2401 : 2400 und 4375 : 4374, sowie ihre Reziprokwerte nicht als
eigentliche Ratios, sondern nur als Kommata."

Yours Sincerely
A.S.

🔗threesixesinarow <CACCOLA@...>

🔗Carl Lumma <carl@...>

🔗threesixesinarow <CACCOLA@...>

🔗Carl Lumma <carl@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@...> wrote:
> > Newton's "horogramm" in 53:
> >
> >http://mto.societymusictheory.org/issues/mto.93.0.3/
> mto.93.0.3.lindley7.gif
>
> Can someone help these lazy people?
>
> http://en.wikipedia.org/wiki/
> Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F
>
Dear CACCOLA & all others that want to understand N's draft sketch,

here comes some more work for "lazy people" beyond
my earlier interpretations of Newton's layout:
/tuning/topicId_71935.html#72161
/tuning/topicId_73506.html#73536
Provenace:
"
Newton's autograph drawing is dated November 1665:
Literature reference:
The original manuscript is located at:
Cambridge Univ.Lib. Signature: Ms.Add.4000,fol.105v
"
Has anybody in that group access to that original or an copy of that,
and could please offer an reprint to us group-members here in that forum?

All i know about N's diagramm bases barely
on the views of my old colleauge and friend and expert in that field:
http://en.wikipedia.org/wiki/Mark_Lindley
He refers to N's delineation
in his encyclopecic ripely in depth article
'tuning & temperature' standard reference article:
"Stimmung und Temperatur"
within the book:
(F. Zaminer, Editor, Geschichte der
Musiktheorie, Vol. 6: Hören, Messen und Rechnen in der Frühen Neuzeit
(Wissenschaftliche Buchgesellschaft, 1987) pp.205-210
Sadly -sorry i'm afraid-
in print that's available solely in German :-(
How unfortunately!

Lindley's scholar article reproduces also an similar
even earlier "horogramm"
-appearently an forerunning predecessor-
made by
http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes
that appeared in his famous:
"1618. Compendium Musicae.
A treatise on music theory and the
aesthetics of music written for Descartes's
early collaborator Isaac Beeckman."
http://en.wikipedia.org/wiki/Isaac_Beeckman

R. Descartes reports about that encounter when meeting Beekman personally:
http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/descartes.html
" Around 1618 I believe, I began to study mathematics once again under
the Dutch scientist Isaac Beekman, who I had met one day walking
through the streets when he translated a Dutch placard for me that
turned out to be a math problem."

Probably D. refers in his advanced blueprint
-written already at age of 22 years-
to the questionable treatment of the subject by an precursor:
http://en.wikipedia.org/wiki/Robert_Fludd
's outlines:
http://www.celestialmonochord.org/log/images/celestial_monochord.jpg
or if you prefer the same pic in somewhat higher resolution:
http://www.imaginatorium.org/books/monochd.gif
Attend God's divine hand out of the clouds that tunes the string.
http://www.bach1722.com/
'Il temperamento di Dio' == ???"God's temperament"???
including an Italian foro di propagare:
http://www.edumus.com/forum/read.php?21,54338,page=1
Parlare a vuoto di un agrumento putativo inprofessionale:
Che palle!
.......
......
.....
....
...
..
.

But better let's retrun back at that point of no return
-away from that nonsensial layman's flubdub-
toggeling reverse to our's historically real serious D&N:

Beyond Fludd's heptatonics
Descartes explains there in his compedium en detail
-alike in Newton's later refinement too-
how to obtain the 53-commata scale from shifting
hexachords by 4ths(4:3) from hard(durum) to soft(molle) modes
by changeing the keys: G-C-F in refernce to
the incomlete C-major scale,
because in coeval hexachords the pitch-class of 'B' is left out there.

http://en.wikipedia.org/wiki/Hexachord

In order to understand N's conceptual design
read the 5 concentric circles from inside to outside as scales over:
outest = 44Bb(ut,) > 21F(ut) > 53=0C(Ut) > 31G(UT) > 9(UT') = innerst

so that Descartes's C-major 'scala-naturalis' is located in the
middle at the center of N's 5 concentric circles:
That old gamut (GAMMA-UT) is labeld in today's modern concept as:

1. extended tonic durum-hexachord C-major-Mixolydian 7-tone scale:
consisting of 2
http://en.wikipedia.org/wiki/Tetrachord
s and an major-tone 9:8 intbetween both of them:
Yilding an C-major
http://en.wikipedia.org/wiki/Tonic_%28music%29
scale with legenda:

C=0Ut=1:1 +8 8Re=10:9 +9 17Mi=5:4 +5 22Fa=4:3 first-tetrachord22Fa4:3
22Fa=4:3 +9 31Sol=3:2 major-tone9commata inbetween both tetrachords
31Sol=3:2 +8 39La +5 44--16:9 +9 (53=0)Ut'=2:1 second-tetrachord

On the one hand:
Step from that one cirle nearer to the main focal point,
that corresponds to an transition an 5th upwards into the:
http://en.wikipedia.org/wiki/Dominant_%28music%29
2. dominant hard(sharp)-hexachord G-major-Mixolydian 7-tone scale
with legenda:
G=31UT=3:2 +8 39RE=5:3 +9 48MI=15:8 +5 53=0FA=1:1 first-tetrachord
53=0FA=1:1 +9 9SOL=9:8 major-tone 9:8 consists of 9 commata steps
9SOL=9:8 +8 17LA=5:4 +5 22--=4:3 +9 31UT second-tetrachord

On the other hand:
By stepping in reverse direction by
one 4th (4:3) from the middle-C-major cicle
into outwards direction we yield an change of key into the:
http://en.wikipedia.org/wiki/Subdominant
3. subdominant moll(soft)-hexachord F-major-Mixolydian 7-tone scale:
Legenda:
F=22ut=4:3 +8 30re40:27 +9 39mi=5:3 +5 44fa=16:9 first-tetrachord
44fa=16:9 +9 53=0sol=1:1 major-tone 9:8 of 9 commatas
0sol1:1 +8 8la=10:9 +5 13--=50:27 +9 22ut4:3

So far N's scribble agrees in the 3 innerst circles fully with
Descartes's elaborated original scheme.
But beyond D's hexachords in barely the kernel keys
F > C > G
N. delivers additional
Bb > (F > C > G) > D
in extending D's range from the
double-subdomiant=Bb>(subdom.=F > tonic=C > dom.=G)>double-dom.=D
by 2 additional outer scales.
N. also extends by the note '--' the classial Hexachords
into to the mixolydian scale, with distances in commatas:
C +8 D\ +9 E\ +5 F +9 G +8 A\ +5 Bb +9 C'
instead of the todays common usual 'Ionian'-C-major scale:
C +9 D +8 E\ +5 F +9 G +8 A\ +9 B\ +5 C'

N's outest circle exterior corresponds analogous to the
double-subdominant-flattend Bb-major mixolydian heptatonic scale:
Legenda:
Bb=44ut, 52re, 8mi, 13fa, 22sol, 30=la, 35--, 44ut,=Bb

And finally:
respectively inside the interiorst cycle represents
double-dominant-sharpend D-major mixolydian hepatonics:
Legenda:
D=9UT' 17RE' 26MI' 31FA' 40SOL' 48LA' 53=0--' 9UT'=D
in comleting all 5 cases.

hope that helps to illustrate and understand
N's division of the octave into 53.
when transferred into modern terminology.
from traditionally medivial hexachordian terminology
over N's personal Bb-F-C-G-D "Mixolydian" concept
over of Rameau's "Ionian" major 'triad' F-C-G

Rameau's terms for an change of the actual key:

C=tonique 1:1
with its 2 transpositons:
F=sous-dominant by an 4th 4:3, labeled by an b-sign(molle)in the score
C=super-dominant by an 5th 3:2, corresponds an #-sign(durum)in score

resulting finally in the point of:
http://en.wikipedia.org/wiki/Hugo_Riemann
's
http://en.wikipedia.org/wiki/Tonnetz

when applied to the good-old song,
http://en.wikipedia.org/wiki/Ut_queant_laxis
then N's drawing represents simply
the transpositions of that chant through
the pentard of keys:
outside = Bb > F > C > G > D = inside

For HIP experts:
Attend N. starts in his original considerations at GAMMA-UT = 0 == 53
http://tede.ibict.br/tde_busca/processaArquivo.php?codArquivo=354
http://jrma.oxfordjournals.org/cgi/reprint/115/2/145.pdf
(Warning: That both do deliver exhausting depletive informations!)

That can be easier expressed in todays modern terms by the
simplifications over the last few centuries:
http://en.wikipedia.org/wiki/Solf%C3%A8ge

Here ends my historically excursus,
into the conservative N's old mediaevel world,
his Baroque coevals considered his way of treating
the subject as an antediluvian-reactionar-fossil
relict of the middle-age, but still worth to read
for the purpose of inspiration of 53 refinements...

...just let me conclude in quoteing an prince of poets laureats:
J.W.Goethe
http://www.zitate-online.de/literaturzitate/allgemein/16754/was-du-ererbt-von-deinen-vaetern-hast-erwirb.html
Was du ererbt von Deinen Vätern hast, erwirb es, um es zu besitzen."
Translation:
"What you have (once) inherit from Yours (late) fathers,
retrieve it (yourself again anew), in order to possess it
(henceforward as yours own personal belongings)."

Quest:
Which native english speaker in that group here can offer
for us an more appropriate and accurate translation,
that is more apt, than my humble attempt?

With sorry for beeing so much verbose
Yours Sincerely
A.S.

🔗threesixesinarow <CACCOLA@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@...> wrote:
>
> Do you think you can amend the statement on Wikipedia
> so it more accurately reflects how he refers to Newton's
> treatment of 53 equal in this article?
>
Dear Clark and all others that seek a deeper understanding of Newton,

sorry, i'm afraid,
all i know about N's53 stems barely from one source alone:

Mark Lindley critizises Newton for an supposed neglect of the
schismatic 5*3^8:2^15 subdivision,
which consequently would result in an double allocation
for the 3rd at step 17 out of 53 into:

http://en.wikipedia.org/wiki/Interval_(music)
"A schismic major third is a schisma different from a just major
third, eight fifths down and five octaves up, Fâ in C."

L. assumes that N. had no idea of discerning properly inbetween:
1.) 2^13:3^8 = 8192:6561 ~384Cents an Pythagoren diminshed 4th
2.) 5:4 ~386Cents JI 3rd
in presuming that N. wasn't clear aware of the schismatic concept:
http://en.wikipedia.org/wiki/Schismatic_temperament
"Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was
briefly in use during the late medieval period."
Fully agreed.

For reassessing the uncertainty about N's view,
i simply need more historically checkable facts for verification,
that i do suspect in N's unpublished autograph.

For an apt review i would have to study N's own concrete ratios
myself:
How do they fit to N's drawing?

In order to stay honest and fair against N.
Without such an verification in reference to N's original
i do hesitate to repeat again L's complains
of finding fault in N's considerations,
that i want accept preliminary only tentative with reservation,
barely with a tiny grain of salt.

But unfortunately L. presents there no concrete numerical-values
in his (supposeable overly hypercritically?) evaluation:

At the moment my situation consists still
in a gap in my knowledge about N's real data:
Simply i know to less about it.

All i can say about it hitherto is precious few:
From L. discussion arises the question,
if N. actually refers to 53-EDO at all,
or if he had something complete rational in his mind,
when he penned down his 5 intersecting mixolydian hexachords?

There remain some open questions:

Weather meant N. really Holder's 53-EDO?:

The schisma of ~2Cents refers to an finer resolution than
the more coarse 2^(1:53) ~22.6415...Cents
http://en.wikipedia.org/wiki/Holdrian_comma

I really don't know if N. overtook the 2^(1:53) concept from
http://en.wikipedia.org/wiki/William_Holder
or vice versa just or was it just the other way around?

Whom of that invented 53-EDO?

Hence, please understand my caution about careless
overhasty jumps into premature rash half-cocked conclusions:
Without having read N's original myself,
i'm not able to decide whether M. Lindley's objections
against N's representation is justified or not?

But if Mark says so, his review has to be taken serious.

My conclusion:
All i can report at the moment:
My lacking personal insight into N's autograph and my respcet
for his genius forbids me to invent disputable hypothesises about
his work:
http://en.wikipedia.org/wiki/Hypotheses_non_fingo
>
> > Renï¿½ Descartes, 1618. Compendium Musicae.
>
> Neat, and there's more than one diagram.

> Musicae compendium / Renati Des Cartes (1695)
> http://gallica.bnf.fr/notice?N=FRBNF37240052
> http://gallica.bnf.fr/notice?N=FRBNF37240054

Lindley's copy originates exactly from the horogramm there in that.
>
But
http://en.wikipedia.org/wiki/Edmond_de_Coussemaker
's
"Scriptores de musica medii aevi (4 delen) (1864-1876)"
contains appearently an other one even older
more archaic-looking "horogramm" forerunner model of D's & N's.

in staying tuned what N. really wrote or may-be even not
time will show that
Yours Sincerely
A.S.

Dieterich Buxtehude and the Mean-Tone Organ

🔗Mike Battaglia <battaglia01@...>

🔗Charles Lucy <lucy@...>

🔗Mike Battaglia <battaglia01@...>

🔗rick_ballan <rick_ballan@...>

🔗Mike Battaglia <battaglia01@...>

🔗rick_ballan <rick_ballan@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗threesixesinarow <CACCOLA@...>

🔗Carl Lumma <carl@...>

🔗threesixesinarow <CACCOLA@...>

🔗threesixesinarow <CACCOLA@...>

🔗Carl Lumma <carl@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗threesixesinarow <CACCOLA@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Andreas Sparschuh <a_sparschuh@...>