Newton/tunings

I've been reading "Temperament," by Stuart Isacoff, and came across some references to Isaac Newton, and his interest in tunings, which I had no idea existed. There's not a lot of detail, but Isacoff says Newton investigated tunings, and seemed to be opposed to temperaments as being against Natural laws. Isacoff says Newton wrote "perhaps the whole frame of nature may be nothing but various contextures of some certain ethereal spirits," and Isacoff goes on to say that Newton "created a standard unit of measure for determining the sizes of musical intervals, and calculated the precise values of pitches found in various scales. And he proposed tuning schemes of his own."
He also says that Newton "adopted the pseudonym Jeova Sanctus Unus-linking his own name with the declaration that Jehovah is the one holy God." (And, Newton was also interested in alchemy). So here's one of the great scientists of all time, who also was a super serious Christian, and also very interested in what many might call a mystical pursuit. He obviously saw no contradiction between these various ways of viewing life, which I think is super cool. Does anybody have any info on his tuning pursuits, what sorts of scales he may have proposed, and if any music ever came from them? Interesting stuff...best...HHH
microstick.net

At 03:49 PM 1/10/2006, you wrote:
> I've been reading "Temperament," by Stuart Isacoff, and came across some
>references to Isaac Newton, and his interest in tunings, which I had no idea
>existed. There's not a lot of detail, but Isacoff says Newton investigated
>tunings, and seemed to be opposed to temperaments as being against Natural
>laws. Isacoff says Newton wrote "perhaps the whole frame of nature may be
>nothing but various contextures of some certain ethereal spirits," and
>Isacoff goes on to say that Newton "created a standard unit of measure for
>determining the sizes of musical intervals, and calculated the precise
>values of pitches found in various scales. And he proposed tuning schemes of
>his own."
> He also says that Newton "adopted the pseudonym Jeova Sanctus
>Unus-linking his own name with the declaration that Jehovah is the one holy
>God." (And, Newton was also interested in alchemy). So here's one of the
>great scientists of all time, who also was a super serious Christian, and
>also very interested in what many might call a mystical pursuit. He
>obviously saw no contradiction between these various ways of viewing life,
>which I think is super cool. Does anybody have any info on his tuning
>pursuits, what sorts of scales he may have proposed, and if any music ever
>came from them? Interesting stuff...best...HHH
>microstick.net

Erv mentioned Newton's interest in tunings (and mysticism) to me,
but didn't give any details...

-Carl

--- In MakeMicroMusic@yahoogroups.com, "Neil Haverstick"
<microstick@m...> wrote:

> I've been reading "Temperament," by Stuart Isacoff, and came
across some
> references to Isaac Newton, and his interest in tunings, which I had
no idea
> existed. There's not a lot of detail, but Isacoff says Newton
investigated
> tunings, and seemed to be opposed to temperaments as being against
Natural
> laws.

Indeed he did, and I even gave Newton a mention when I was doing the
53-et article. Newton not only knew about the 5-limit (not just
3-limit) excellence of 53, he was aware of 612-et.

> So here's one of the
> great scientists of all time, who also was a super serious
Christian, and
> also very interested in what many might call a mystical pursuit. He
> obviously saw no contradiction between these various ways of viewing
life,
> which I think is super cool. Does anybody have any info on his tuning
> pursuits, what sorts of scales he may have proposed, and if any
music ever
> came from them? Interesting stuff...best...HHH

I have "Music, science, and natural magic in seventeenth-century
England" by Penelope Gouk on order, so I hope to know more soon.

> So here's one of the
> great scientists of all time, who also was a super serious Christian, and
> also very interested in what many might call a mystical pursuit. He
> obviously saw no contradiction between these various ways of viewing life,
> which I think is super cool.

Actually, the connection of religion to mysticism to natural science
was completely natural through most of history. Specifically,
medieval philosophers believed that everything was ordered as part of
God's creation, so the idea that "good math" could create "good music"
made perfect sense. That philosophy can be traced all the way back to
Plato, I suppose. It's one that I try to take seriously, even though
it flies in the face of much contemporary thought, which (in my
understanding) proposes that such things as "good math" are only an
illusion. I'll leave off here, since to continue I would have to move
to meta. This is what I do while csound is rendering...

:)

--
"... free speech is meaningless if the commercial cacaphony has risen
to the point that no one can hear you." -Naomi Klein

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Neil Haverstick"
> <microstick@m...> wrote:

> Indeed he did, and I even gave Newton a mention when I was doing the
> 53-et article. Newton not only knew about the 5-limit (not just
> 3-limit) excellence of 53:

http://www.societymusictheory.org/mto/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

i.m.o: The middle hexachord of the 5 ones in the pic. means:

0=53 ut 1/1 todays "do"
8 re 10/9
17 mi 5/4
22 fa 4/3
31 so 3/2
39 la 5/3

the next circle inwards is shifted by an 5th=31 comma steps:
31,39,48,53,9,17
the innerst circle another 5th: makes an 2nd=9(=31*2-53)
9,17,26,31,40,48
........
respectively in outwards direction by 4ths -31=+22
22,30,39,44,53,8
the outest by an diminshed 7th -9=+44:
44,52,8,13,22,30
........
so that the steps 8>9,30>31,52>53 amount only an SC=81/80.
2^(1/53)/SC ~= 1.00065594... ~1.13521984...Cents
out of 53edo.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> 53-et article.
What had Newton to do with 53 et?
He preferd the true pure ratios of the intervals,
prefereably the Pythagorean ones.

> Newton not only knew about the 5-limit (not just
> > 3-limit) excellence of 53:
His 53 drawing can be interpreted as well in 3-limit
as in ?questionable? 5-limit too:
>
http://www.societymusictheory.org/mto/issues/mto.93.0.3/mto.93.0.3.lindley7.gif
>
53=0: |__|solut|fa|--| 3^0=1
____: |__|__|__|__|__| 3^...
8___: |mi|la|re|__|__| 3^-10 ~ ?10/9 an schisma 32805/32768 to sharp?
9___: |__|__|__|solut| 3^2=9/8
____: |__|__|__|__|__| 3^...
13__: |fa|__|__|__|__| 3^-3=32/27
____: |__|__|__|__|__| 3^...
17__: |__|__|mi|la|re| 3^-8 ~ ?5/4?
____: |__|__|__|__|__| 3^...
22__: |solut|fa|--|__| 3^-1=4/3 an 4th
____: |__|__|__|__|__| 3^...
26__: |__|__|__|__|mi| 3^-6=1024/729
____: |__|__|__|__|__| 3^...
30__: |la|re|__|__|__| 3^-11 ~ ?40/27?
31__: |__|__|solut|fa| 3/2 an 5th
____: |__|__|__|__|__| 3^...
35__: |--|__|__|__|__| 3^-4=128/81
____: |__|__|__|__|__| 3^...
39__: |__|mi|la|re|__| 3^-9 ~ ?5/3?
40__: |__|__|__|__|sol 3^3=27/16
____: |__|__|__|__|__| 3^...
44__: |ut|fa|--|__|__| 3^-2=16/9
____: |__|__|__|__|__| 3^...
48__: |__|__|__|mi|la| 3^-7 ~ ?15/8?
____: |__|__|__|__|__| 3^...
52__: |re|__|__|__|__| 3^-12=2/PC ~ ?160/81=2/SC doubtful!
53=0: |__|solut|fa|--| 3^0=1 modulo (any power of) 2

Apply Occams razor: Exclude all 5-intervals from 53!
The theorists of mediaevial Hexachords, like Newton too,
knew well why they kept their concept within
the traditional Pythagorean 3-limit.

Retune your keyboard-instrument and try out yourself
my 12-tone subset scale out of 53N!

C___0: 1/1____ ut or today "do"
d___8: 3^-10__ re_Newtonian
D___9: 9/8____ RE_Gothic
e__17: 3^-8___ mi_Newtonian
E__18: 81/64__ mi_Gothic
F__22: 4/3____ fa
G__31: 3/2____ sol
a__39: 3^-9___ la_Newtonian
A__40: 27/16__ LA_Gothic
Bb_44: 16/9___ |--|_Newtonian, in german the note:"B"
b__48: 3^-7___ (ti)_NeoGothic, the later german:"h"
B__49: 243/128 (TI)_NeoGothic, the later german:"H"
C'_53: 2/1____ ut'

Retune your keyboard-instrument and try out yourself,
my preferred layout, that i do reccomend:

----------------------
8 key C
---------|9=key=C#=Db|
17 key D
---------|18=key=Eb=D#
22 key E
----------------------
31 key F
---------|39=key=F#=Gb|
40 key G
---------|44=key=G#=Ab| ~ 415 Hz
48 key A ~ 440 Hz
---------|49=key=Bb=A#|
53=0 key B
-----------------------
8 key C'
---------|key C# &ct.

but any other distribution or starting pitch frequency
can be chosen too arbitrarily.

Have a lot of fun with that!

--- In MakeMicroMusic@yahoogroups.com, "a_sparschuh"
<a_sparschuh@y...> wrote:

> Apply Occams razor: Exclude all 5-intervals from 53!
> The theorists of mediaevial Hexachords, like Newton too,
> knew well why they kept their concept within
> the traditional Pythagorean 3-limit.

But Newton didn't. You can get to 53-et by doing that, you won't get
to 612-et.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > The theorists of mediaevial Hexachords, like Newton too,
> > knew well why they kept their concept within
> > the traditional Pythagorean 3-limit.
>
> But Newton didn't.
Dear Gene,
Only the young N. considered 5-limit,
but in his later years he restricted himself to Pyth. 3-limit.

> You can get to 53-et by doing that,
error(53):=
(1 200 * ln((3^53) / (2^84))) / ln(2) =~ 3.61504587...Cents

> you won't get to 612-et.
error(612):=
(1 200 * ln((3^612) / (2^970))) / ln(2) =~ -3.53947038...Cents
Both errors 53 & 612 are well clear audible, even for layman.

Only their sum 665=612+53 becomes scarcely audible: ~1/13 Cents,
meanwhile indistinguishable at least for my ears.

So 665et consisting of arbitrarily chosen
Delfi-Units DU:=2^(1/665)
has to be considered as the lowest adaequate passable et
to approximate Pythagorean proportions somehow tolerable however.
cf:
http://www.byzantine-musics.com/mathematics_and_music.htm
or
http://www.xs4all.nl/~huygensf/doc/measures.html

Below 665 exist i.m.o. no reasonable et in a good sense
of well acceptable approximation.
But:
Why replacing excact Pyth. ratios 3^... by barely crude
irrational et approximations?
The fundamental flaw in all et is general inevitable:
Lacking back-compatibility,
as conversely common inherent in all Pyth. integral-powers-of-3
systems, beyond the above mentioned errorness of all ets.
To much disadvantages!

Final conclusion:
I see no need to approximate exact 3^.../2^... pure ratios
any crude transcendent error deviations.

et= tears for my ears

--- In MakeMicroMusic@yahoogroups.com, "a_sparschuh"
<a_sparschuh@y...> wrote:

> > You can get to 53-et by doing that,
> error(53):=
> (1 200 * ln((3^53) / (2^84))) / ln(2) =~ 3.61504587...Cents

If by "error" you refer to closing the circle of pure fifths, correct.

> > you won't get to 612-et.
> error(612):=
> (1 200 * ln((3^612) / (2^970))) / ln(2) =~ -3.53947038...Cents
> Both errors 53 & 612 are well clear audible, even for layman.

But this is wrong. 612 inherits its fifth from 306, this why why you
simply will not come up with it on 3-limit grounds.

Correct is half of the above: 2^485 / 3^306, which comes in at 1.77
cents. Like 53, 306 is a (denominator of a) convergent for log2(3), so
this comma is smaller than the previous one.

> So 665et consisting of arbitrarily chosen
> Delfi-Units DU:=2^(1/665)
> has to be considered as the lowest adaequate passable et
> to approximate Pythagorean proportions somehow tolerable however.

Not hardly. Take 53, and make the circle close exactly, and you are in
business.

> Below 665 exist i.m.o. no reasonable et in a good sense
> of well acceptable approximation.

You have a bizarre definition of acceptable. With 53, you can't hear
the difference, since it is only 0.068 cents flat. Not being able to
hear the difference ought to be acceptble.

> Why replacing excact Pyth. ratios 3^... by barely crude
> irrational et approximations?

Why would anyone care to make the Pythagorean ratios exact and ignore
all others? Moreover, 53 as we have seen is not "barely crude"; you,
personally, do not have ears good enough to hear any difference, so
this is almost numerology.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > error(612):=
> > (1 200 * ln((3^612) / (2^970))) / ln(2) =~ -3.53947038...Cents
>well clear audible, even for layman.
>
> But this is wrong. 612 inherits its fifth from 306, this why why you
> simply will not come up with it on 3-limit grounds.
>
> Correct is half of the above: 2^485 / 3^306, which comes in at 1.77
> cents.
error(612)=2*error(306)=~-2*1.77...cent=~-3.54...cent, twice err(306)
if you consider an spiral chain of 612 consecutive pure 5ths,
53 less than 655, you have to consider the double of 306.

Consider the inherently distuned et-5ths series:
2^(7/12)=~1.4983..... clearly percievable an PC^(1/12) wrong flat
2^(24/41)=~1.50042... well audiable faulty ~1/2 cent wrong sharp
2^(31/53)=~1.499941...sounds after few such ill 5ths simply to flat
2^(179/306)=~1.50000501...same error sharp as 358 steps in 612 too
2^(389/665)=~1.49999902...first passable flat irrational approximation
......
....
...
..
.
> Like 53, 306 is a (denominator of a) convergent for log2(3), so
> this comma is smaller than the previous one.
Look further coefficients of log2(3) continued-fraction-expansion in:
http://www.ericr.nl/wondrous/cycles.html
but: Who needs such big as 15601, 31867, 79335 or even 111202,... et.
> > So 665et consisting of arbitrarily chosen
> > Delfi-Units DU:=2^(1/665)
> > has to be considered as the lowest adaequate passable et
> > to approximate Pythagorean proportions somehow tolerable however.
>
> Not hardly.
I.m.o. any et below 665 remains incompetent amateur work.
Sorry, with all respect for your opinion,
but the interplay of my mind and my ears demand such
an high-fidelty internal precision.

> Take 53, and make the circle close exactly, and you are in
> business.
simply tune in practice a few (2^(31/53))^n artificial created 5ths
and hear yourself how the errors accumulate:
The intermediate results become more and more of of tune
by each step arising detuned
versus the correct chain of just (3/2)^n pure 5ths.
Retrace that by experiment in order to comprehend that acoustical too.
Then you have a clue of my own acustics experiments.
>
> > Below 665 exist i.m.o. no reasonable et in a good sense
> > of well acceptable approximation.
>
> You have a bizarre definition of acceptable. With 53, you can't hear
> the difference, since it is only 0.068 cents flat.
makes already after an dozen steps an ~0.068cents * 12 = ~0.816cents
audiable flat distuned PC=3^12, more than 3 times
as my Werckmeister step of ~1/4cents, see downwards.
Learn to distingusih the different commas one from the other:
3^12/2^19 ~23.4 cents 13 steps in 665 (~~12 in 612)
2^65/3^41 ~19.8 cents 11 steps in 665 (~~10 in 612)
instead of assuming them fuzzily equal,
!665/53=~12.54...that's not fish, not beef in 665!
but 665 delfi-units shold be kept integral.
Newtons octave division are 612/53=~11.54...incompatible
inter se too.

i'm fully aware in respecting Jing-Fangs comma discrimination
3^53/2^84 ~3.6 cents ~2 steps in 665 (or 612 too)
as mental in mind as acustical by ears.
Already Moritz Wilhelm Drobisch gained
the same accuracy of discrimination in the 19. century
http://de.wikipedia.org/wiki/Moritz_Wilhelm_Drobisch
theoretical and practical.
I try to attempt in following his methods.
> Not being able to
> hear the difference ought to be acceptble.
It's only an question of training getting aware of
the finer resolution if one wants: It's up to you.
If you alreay have grasped 612,
why not the better choice?
Superior 665!
>
>
> Why would anyone care to make the Pythagorean ratios exact and >ignore
> all others?
5 arises from 3^-8, 7 from 3^-14, 11 from 3^23 ...........
already in 53P.

> Moreover, 53 as we have seen is not "barely crude";
2^(1/53) has no good approximation in 665 like above
3^12 or 3^-41, as shown above.
The 612/12=51 trick of Newton has to fail in 665,
due to its 665=19*7*5 prime-factor decomposition,
containing 5,7,19,35,95 & 133et subsets.
> you,
> personally, do not have ears good enough to hear any difference, so
> this is almost numerology.
Are you doubting about my championship of 665?
Counterexample:
Just try to tune my Werckmeister concept in an real acustic organ:
@ A4=456 Hz as exact by ear as you are able, as your ears can:

C 2173 Hz
G (6561)6560,3280,1640,820,410,205(204,102,51)
D 153(152,76,38,19)
A 57
E 171
B 513(512,..,1)
F# 3
C# 9
G# 27
Eb 81
Bb 243
F 729
C 2173

using the same algorithm like my 1998 Bach WTC-tuning interpretation:
http://www.strukturbildung.de/Andreas.Sparschuh/

the crucial fire test in that Werckmeister
version will be for your hearing nerves
flattening the most prominent 5th C8>G8 preciesely the tiny amount of
just 6561/6560 down by exactly one singele Hertzian flat,
counting beats by watch or seconds pendulum like Werckmeister
or Bach themselfs.
Observe the 41-limit:
3^8/(41*5*32) ~0.264...cents or ~8.1... TUs
cf: 1 TemperamentUnit:=PC^(1/720)
http://www.xs4all.nl/~huygensf/doc/measures.html

i wish you good luck and success in retuning that
A.S.

--- In MakeMicroMusic@yahoogroups.com, "a_sparschuh"
<a_sparschuh@y...> wrote:

I've replied to this on the main tuning list, tuning@....

Hi Neil, just wondering -- why did you post this here? It seems to me
like a post much more appropriate for the tuning list, but you don't
seem to have posted it there . . .

--- In MakeMicroMusic@yahoogroups.com, "Neil Haverstick"
<microstick@m...> wrote:
>
> I've been reading "Temperament," by Stuart Isacoff, and came
across some
> references to Isaac Newton, and his interest in tunings, which I
had no idea
> existed. There's not a lot of detail, but Isacoff says Newton
investigated
> tunings, and seemed to be opposed to temperaments as being against
Natural
> laws. Isacoff says Newton wrote "perhaps the whole frame of nature
may be
> nothing but various contextures of some certain ethereal spirits,"
and
> Isacoff goes on to say that Newton "created a standard unit of
measure for
> determining the sizes of musical intervals, and calculated the
precise
> values of pitches found in various scales. And he proposed tuning
schemes of
> his own."
> He also says that Newton "adopted the pseudonym Jeova Sanctus
> Unus-linking his own name with the declaration that Jehovah is the
one holy
> God." (And, Newton was also interested in alchemy). So here's one
of the
> great scientists of all time, who also was a super serious
Christian, and
> also very interested in what many might call a mystical pursuit. He
> obviously saw no contradiction between these various ways of
viewing life,
> which I think is super cool. Does anybody have any info on his
tuning
> pursuits, what sorts of scales he may have proposed, and if any
music ever
> came from them? Interesting stuff...best...HHH
> microstick.net
>

--- In MakeMicroMusic@yahoogroups.com, "a_sparschuh"
<a_sparschuh@y...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...> wrote:
> > > The theorists of mediaevial Hexachords, like Newton too,
> > > knew well why they kept their concept within
> > > the traditional Pythagorean 3-limit.
> >
> > But Newton didn't.
> Dear Gene,
> Only the young N. considered 5-limit,
> but in his later years he restricted himself to Pyth. 3-limit.
>
> > You can get to 53-et by doing that,
> error(53):=
> (1 200 * ln((3^53) / (2^84))) / ln(2) =~ 3.61504587...Cents

This calculation has nothing to do with how big the audible
deviations of 53-equal from JI are. I see Gene has moved this
converation to the tuning list and posted a sensible reply there.

On Thu, 12 Jan 2006, a_sparschuh wrote:

(possibly the quote of the century for lovers
of JI)

> et= tears for my ears

Yahya

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I think this subject came up once before, but I don't recall the outcome, thought I'd try again. I'm aware that Isaac Newton did some tuning research, but was wondering if his thoughts/ideas were ever written down and available to read? If anybody has any info, love to hear about it...post here or at microstick@...... best from the library...Hstick

www.microstick.net

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[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, Neil Haverstick <microstick@...> wrote:
>
>
> I think this subject came up once before, but I don't recall the outcome, thought I'd try again. I'm aware that Isaac Newton did some tuning research, but was wondering if his thoughts/ideas were ever written down and available to read? If anybody has any info, love to hear about it...post here or at microstick@... best from the library...Hstick
>
> www.microstick.net
>
> _________________________________________________________________
> Hotmail® goes with you.
> http://windowslive.com/Tutorial/Hotmail/Mobile?ocid=TXT_TAGLM_WL_HM_Tutorial_Mobile1_052009
>
> [Non-text portions of this message have been removed]
>
From Robert. Try
http://home.vicnet.net.au/~colmusic/opticks1.htm