gneralizing Werckmeister's septenarius Newton's 53, was: Re:Buxtehude
- --- In firstname.lastname@example.org, "Mike Battaglia" <battaglia01@...> wrote:
>Dear Mike & all others,
> Or, if anyone has any recommendations for recordings of classical
> works done in 53-tet or other temperaments, or even perhaps JI...
for Newton's "horogramm" in 53:
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
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:
Sparschuh's 53 generalization of Werckmeister's septenarius
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
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
That small interval was also historically also used for coin-making:
"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."
- --- In email@example.com, "Andreas Sparschuh" <a_sparschuh@...>
> Newton's "horogramm" in 53:mto.93.0.3.lindley7.gif
Can someone help these lazy people?
- Clark wrote:
> > Newton's "horogramm" in 53:Why do you call them that? The above is already
> Can someone help these lazy people?
linked in their discussion.
- --- In firstname.lastname@example.org, "Carl Lumma" <carl@...> wrote:
>I know some howlers where historians just relied on a
> Clark 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
> Why do you call them that? The above is already
> linked in their discussion.
picture or worse instead of reading the accompanying text
and at least one of those editors didn't bother to look it
up in Lindley's book.
- --- In email@example.com, "rick_ballan" <rick_ballan@...> wrote:
>(I've re-ordered Helmholtz' sensation of tone but it takeshttp://books.google.com/books?id=x_A5AAAAIAAJ
> months to get here to Australia).
> >(I've re-ordered Helmholtz' sensation of tone but it takesWe've discovered that certain Google books that are available
> > months to get here to Australia).
in the U.S. are not available down under. However, the second
link, to the internet archive, looks like a winner!
- --- In firstname.lastname@example.org, "threesixesinarow" <CACCOLA@...> wrote:
> > Newton's "horogramm" in 53:Dear CACCOLA & all others that want to understand N's draft sketch,
> Can someone help these lazy people?
here comes some more work for "lazy people" beyond
my earlier interpretations of Newton's layout:
Newton's autograph drawing is dated November 1665:
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:
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 :-(
Lindley's scholar article reproduces also an similar
even earlier "horogramm"
-appearently an forerunning predecessor-
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."
R. Descartes reports about that encounter when meeting Beekman personally:
" 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:
or if you prefer the same pic in somewhat higher resolution:
Attend God's divine hand out of the clouds that tunes the string.
'Il temperamento di Dio' == ???"God's temperament"???
including an Italian foro di propagare:
Parlare a vuoto di un agrumento putativo inprofessionale:
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.
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
s and an major-tone 9:8 intbetween both of them:
Yilding an C-major
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:
2. dominant hard(sharp)-hexachord G-major-Mixolydian 7-tone scale
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:
3. subdominant moll(soft)-hexachord F-major-Mixolydian 7-tone scale:
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:
Bb=44ut, 52re, 8mi, 13fa, 22sol, 30=la, 35--, 44ut,=Bb
respectively inside the interiorst cycle represents
double-dominant-sharpend D-major mixolydian hepatonics:
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:
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:
when applied to the good-old song,
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
(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:
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:
Was du ererbt von Deinen Vätern hast, erwirb es, um es zu besitzen."
"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)."
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
- --- In email@example.com, "Andreas Sparschuh" <a_sparschuh@...>
>Do you think you can amend the statement on Wikipedia
> [Lindley] 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
so it more accurately reflects how he refers to Newton's
treatment of 53 equal in this article?
> Lindley's scholar article reproduces also an similarNeat, and there's more than one diagram.
> even earlier "horogramm"
> -appearently an forerunning predecessor-
> made by [René Descartes]
> that appeared in his famous:
> "1618. Compendium Musicae.
Musicae compendium / Renati Des Cartes (1695)
Excellent compendium of musick with necessary and
judicious animadversions there upon : by a person of
honour / Renatus Descartes (1653)
- --- In firstname.lastname@example.org, "threesixesinarow" <CACCOLA@...> wrote:
>Dear Clark and all others that seek a deeper understanding of Newton,
> 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?
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:
"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:
"Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was
briefly in use during the late medieval period."
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
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
I really don't know if N. overtook the 2^(1:53) concept from
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.
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
>Lindley's copy originates exactly from the horogramm there in that.
> > Renï¿½ Descartes, 1618. Compendium Musicae.
> Neat, and there's more than one diagram.
> Musicae compendium / Renati Des Cartes (1695)
"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
- - --- In email@example.com, "threesixesinarow" <CACCOLA@> wrote:
Hi all lovers of old slogans,
> J.W.Goethereferred to D's & N's
in reference to n's quote:
"Pigmaei gigantum humeris impositi plusquam ipsi gigantes vident"
'If I have seen a little further it is by standing on the shoulders of
That sounds in G's version:
"Was du ererbt von Deinen Vaetern hast, erwirb es, um es zu besitzen."
'Retrieve yours fahters inheritance,
for possessing that as internalized own property'
G's dictum can i.m.h.o. also be referred well to D's & N's tunings