Werckmeister's Septinarius temperament

Tom has a web page on this I'm trying to decipher:

http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html

It it he presents the deviations from JI fifths for his two
reconstructions of this temperament in terms of a new (to me) and
horrible notation for positive rational numbers, whereby p/q > 1 is
written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the
standard mathematical notation everyone learned in grade school?

Now that I've vented about that (sorry Tom, but you hit my pet peeve
button) I'll give the deviations for "D=175", starting from C-G and
working the circle of fifths around to F-C: 392/393, 524/525, 350/351,
1, 1, 416/417, 1, 1, 1, 1, 440/441, 1

Now, a problem with this is that the prodcut of this deviations isn't
524288/531441, so octaves are slightly tempered. Presuming this isn't
intentional, at least one of these ratios is off, and in fact G#-D# is
given as 496/496. If instead we make that 4448/4455, we get untempered
fifths. In the vague hope that this is more or less what is intented,
here's a first go at what this temperament would be:

! sep.scl
Septanarius scale?
12
!
1568/1485
28/25
196/165
49/39
4/3
196/139
196/131
784/495
196/117
98/55
49/26
2

> Could we PLEASE stick to the
> standard mathematical notation everyone learned in grade school?
//
> here's a first go at what this temperament would be:
>
> ! sep.scl
> Septanarius scale?
> 12
> !
> 1568/1485
> 28/25
> 196/165
> 49/39
> 4/3
> 196/139
> 196/131
> 784/495
> 196/117
> 98/55
> 49/26
> 2

Thank you, Gene.

-Carl

Hi gene

your scale has degrees of 1568-edl
you can see something about this EDl-based well temperament and 196-EDL and The Septenarius, Werckmeister's mythical tuning in :
http://240edo.googlepages.com/equaldivisionsoflength(edl)

it is interesting that 1568=196*7.

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web site?? ???? ????? ?????? <http://240edo.googlepages.com/>

My farsi page in Harmonytalk ???? ??????? ?? ??????? ??? <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia ????? ?????? ??????? ??????? ???? ???? <http://en.wikipedia.org/wiki/Shaahin_mohajeri>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Gene Ward Smith
Sent: Monday, February 12, 2007 11:47 PM
To: tuning@yahoogroups.com
Subject: [tuning] Werckmeister's Septinarius temperament

Tom has a web page on this I'm trying to decipher:

http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html <http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html>

It it he presents the deviations from JI fifths for his two
reconstructions of this temperament in terms of a new (to me) and
horrible notation for positive rational numbers, whereby p/q > 1 is
written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the
standard mathematical notation everyone learned in grade school?

Now that I've vented about that (sorry Tom, but you hit my pet peeve
button) I'll give the deviations for "D=175", starting from C-G and
working the circle of fifths around to F-C: 392/393, 524/525, 350/351,
1, 1, 416/417, 1, 1, 1, 1, 440/441, 1

Now, a problem with this is that the prodcut of this deviations isn't
524288/531441, so octaves are slightly tempered. Presuming this isn't
intentional, at least one of these ratios is off, and in fact G#-D# is
given as 496/496. If instead we make that 4448/4455, we get untempered
fifths. In the vague hope that this is more or less what is intented,
here's a first go at what this temperament would be:

! sep.scl
Septanarius scale?
12
!
1568/1485
28/25
196/165
49/39
4/3
196/139
196/131
784/495
196/117
98/55
49/26
2

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Could we PLEASE stick to the
> > standard mathematical notation everyone learned in grade school?
> //
> > here's a first go at what this temperament would be:
> >
> > ! sep.scl
> > Septanarius scale?
> > 12
> > !
> > 1568/1485
> > 28/25
> > 196/165
> > 49/39
> > 4/3
> > 196/139
> > 196/131
> > 784/495
> > 196/117
> > 98/55
> > 49/26
> > 2
>
> Thank you, Gene.
>
> -Carl
>

... Once more - this scale is NOT correct. Honestly, all you have to
do is read the webpage from start to finish to find out exactly what
Werckmeister said. I have no idea why people are unwilling to do this.

~~~T~~~

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> Tom has a web page on this I'm trying to decipher:
>
> http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
>
> It it he presents the deviations from JI fifths for his two
> reconstructions of this temperament in terms of a new (to me) and
> horrible notation for positive rational numbers, whereby p/q > 1 is
> written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the
> standard mathematical notation everyone learned in grade school?
>

The DEFINITION of the scale is in the monochord numbers, which are the
first table in the webpage. I was working on the assumption that
people would start at the beginning and read towards the end...

Therefore the scale is defined to be (apologies for malformed Scala)

! sep.scl
Septenarius scale (choose either value of D)
12
!
1
196/186 = 98/93
196/176 or 196/175
196/165
196/156 = 49/39
196/147 = 4/3
196/139
196/131
196/124 = 49/31
196/117
196/110 = 98/55
196/104 = 49/26

The notation Gene dislikes is not a notation for numbers; it is a
notation for tempering of fifths. It's actually the way Werckmeister
set out his fifths. It's certainly not the definition of the tuning.

Anyway, you are correct that G#-D# is a typo on my part. Try a wide
fifth tempered by 496/495. Where Gene got 4448/4455 from I can't tell.

392/393 * 524/525 * 350/351 * 416/417 * 278/279 * 496/495 * 440/441
*(3^12)/(2^19) = 1

No need for any integer exceeding 525!

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > Tom has a web page on this I'm trying to decipher:
> >
> > http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
> >
> > It it he presents the deviations from JI fifths for his two
> > reconstructions of this temperament in terms of a new (to me)
and
> > horrible notation for positive rational numbers, whereby p/q > 1
is
> > written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick
to the
> > standard mathematical notation everyone learned in grade school?
> >
>
> The DEFINITION of the scale is in the monochord numbers, which are
the
> first table in the webpage. I was working on the assumption that
> people would start at the beginning and read towards the end...
>
>
> Therefore the scale is defined to be (apologies for malformed
Scala)
>
> ! sep.scl
> Septenarius scale (choose either value of D)
> 12
> !
> 1
> 196/186 = 98/93
> 196/176 or 196/175
> 196/165
> 196/156 = 49/39
> 196/147 = 4/3
> 196/139
> 196/131
> 196/124 = 49/31
> 196/117
> 196/110 = 98/55
> 196/104 = 49/26
>
>
> The notation Gene dislikes is not a notation for numbers; it is a
> notation for tempering of fifths. It's actually the way
Werckmeister
> set out his fifths. It's certainly not the definition of the
tuning.
>
> Anyway, you are correct that G#-D# is a typo on my part. Try a wide
> fifth tempered by 496/495. Where Gene got 4448/4455 from I can't
tell.
>
> 392/393 * 524/525 * 350/351 * 416/417 * 278/279 * 496/495 * 440/441
> *(3^12)/(2^19) = 1
>
> No need for any integer exceeding 525!
>
> ~~~T~~~

Apparently you missed this earlier post of mine...

> 0: 1/1 0.000 unison, perfect prime
> 1: 98/93 90.661
> 2: 28/25 196.198 middle second
> 3: 196/165 298.065
> 4: 49/39 395.169
> 5: 4/3 498.045 perfect fourth
> 6: 196/139 594.923
> 7: 196/131 697.544
> 8: 49/31 792.616
> 9: 196/117 893.214
> 10: 98/55 1000.020 quasi-equal minor seventh
> 11: 49/26 1097.124
> 12: 2/1 1200.000 octave
>
> EDL is as old as the hills, isn't it?
>
> Cents schments, I'm digging the ratios,
>
> Hmmm....sounds pretty damn good!
>
> Thanks, Tom Dent!

> Therefore the scale is defined to be (apologies
> for malformed Scala)
>
> ! sep.scl
> Septenarius scale (choose either value of D)
> 12
> !
> 1
> 196/186 = 98/93
> 196/176 or 196/175
> 196/165
> 196/156 = 49/39
> 196/147 = 4/3
> 196/139
> 196/131
> 196/124 = 49/31
> 196/117
> 196/110 = 98/55
> 196/104 = 49/26

Believe it or not, Tom, I have about 2323 phone calls, 30949
e-mails, and 3909409 web pages to answer and read every day,
only a fraction of them about music. So when sharing a share
with this community, people like me really appreciate that
you use the standard way of communicating scales here, which
is giving a Scala file as you've done (finally). You can make
it well-formed simply by giving the period instead of the
unity, and by prefixing your comments on each line with a
bang, like this:

! septenarius.scl
Septenarius scale ('Werckmeister VI')
12
!
98/93 ! =196/186
196/176 ! or 196/175
196/165
49/39 ! =196/156
4/3 ! =196/147
196/139
196/131
49/31 ! =196/124
196/117
98/55 ! =196/110
49/26 ! =196/104
2
!

Now I'm going to scan through the boring historical
stuff on your page for an explanation as to why there
is a choice involving the second degree. Anything you
can do to speed that up would also be appreciated.

-Carl

> Now I'm going to scan through the boring historical
> stuff on your page for an explanation as to why there
> is a choice involving the second degree. Anything you
> can do to speed that up would also be appreciated.

! septenarius.scl
Septenarius scale ('Werckmeister VI')
12
!
98/93 ! =196/186
196/175 ! 196/176 apparently a typo
196/165
49/39 ! =196/156
4/3 ! =196/147
196/139
196/131
49/31 ! =196/124
196/117
98/55 ! =196/110
49/26 ! =196/104
2
!

-Carl

> ! septenarius.scl
> Septenarius scale ('Werckmeister VI')
> 12
> !
> 98/93 ! =196/186
> 196/175 ! 196/176 apparently a typo
> 196/165
> 49/39 ! =196/156
> 4/3 ! =196/147
> 196/139
> 196/131
> 49/31 ! =196/124
> 196/117
> 98/55 ! =196/110
> 49/26 ! =196/104
> 2
> !

It's got one sharp fifth, but no thirds sharper than
81/64 like Paul's Continuo tuning. It does have a fifth
even flatter than the 1/4-comma fifths he so decries in
that text.

-Carl

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > ! septenarius.scl
> > Septenarius scale ('Werckmeister VI')
> ...
>
> It's got one sharp fifth, but no thirds sharper than
> 81/64 like Paul's Continuo tuning. It does have a fifth
> even flatter than the 1/4-comma fifths he so decries in
> that text.
>
> -Carl

Indeed...

"the major thirds are perfectly consistent with Werckmeister's other
tunings [of 1691]: the purest lie at F-A and C-E, while the
little-used thirds at F#-Bb, C#-F and G#-C are all nearly a comma
sharp. The note D#, which must also do duty as Eb, is placed almost
equally between B (natural) and G, approaching Equal Temperament in
this triple of major thirds."

I think the 'Septenarius' is to be compared with the other organ
tunings of 1691 - including 'IV' which has 1/3 comma flat fifths (and
some thirds 1/3 comma wider than 81/64).

But by 1698, and discussing stringed keyboard instruments,
Werckmeister seems to have been singing a different tune.

It seems to me one has to decide whether or not one is interested in a
subject enough to spend time on it. If you're not really interested in
historical tuning, I think it's unrealistic to hope to understand
anything with five minutes' skimming. It's complicated and subtle and
can't be served up on a plate.

I decided not to include any cent values or fractions of a comma, if
at all possible, because that was not how Werckmeister conceived of
this tuning. (Logarithmic approaches have in the past led to incorrect
versions of it.) To understand what it is about - and 196-EDL is more
or less correct in that sense - you have to look at how it was worked
out. There is no short cut. It is possible to find out anything you
need to know about it with a bit of integer arithmetic.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> ... Once more - this scale is NOT correct. Honestly, all you have to
> do is read the webpage from start to finish to find out exactly what
> Werckmeister said. I have no idea why people are unwilling to do this.

Give the scale or don't give it, but please don't blame other people
for your failures.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> The notation Gene dislikes is not a notation for numbers; it is a
> notation for tempering of fifths.

Which are NUMBERS. This is, without doubt or question, a nonstandard
notation for positive rational numbers. Had they been given correctly
in this notation or a standard one I could have derived the scale from
them, as I tried to do. I liked that idea better, since I was, I
thought, on firm ground with a circle of fifths, whereas it wasn't as
clear to me what you were saying about the monochord. It sounded as if
it might be some kind of starting point and not final product, and when
I noticed the circle of fifths didn't come to an exact octave, that
raised more concerns.

> It's actually the way Werckmeister
> set out his fifths. It's certainly not the definition of the tuning.

If you want to define a scale of rational numbers most perspicuously,
give it as an ascending sequence of rational numbers. However, a circle
of fifths will in fact define the tuning.

And if Werckmeister uses a clumsy, nonstandard notation, do not follow
him. Make a note of it if you like, but don't *adopt* it. What's
commonly done in history of math or science when someone's work is
discussed is that the notation is modernized so that the substance can
be seen more clearly.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> Now I'm going to scan through the boring historical
> stuff on your page for an explanation as to why there
> is a choice involving the second degree. Anything you
> can do to speed that up would also be appreciated.

The alternative is Werckmeister's actual figures, but those give screwy
results, and assuming he made a writeo clears that up.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > ... Once more - this scale is NOT correct. Honestly, all you have to
> > do is read the webpage from start to finish to find out exactly what
> > Werckmeister said. I have no idea why people are unwilling to do this.
>
> Give the scale or don't give it, but please don't blame other people
> for your failures.

MY failures? What the hell do you mean by that?

The very first table in the webpage gives the scale with complete
clarity; for an unfathomable reason, some people ignored it. I made
ONE easily identifiable typo in a subsequent table: 496:496 instead of
496:495. The correct ratio could be deduced using the table of
monochord numbers and some elementary arithmetic. For some reason some
people managed to screw that up and get a nonsensical ratio of
4-figure integers instead. I'm not responsible for that.

How about we talk about your failures too? And everyone else's
failures? What a NICE discussion that would make.

~~~T~~~

> It seems to me one has to decide whether or not one is
> interested in a subject enough to spend time on it. If you're
> not really interested in historical tuning, I think it's
> unrealistic to hope to understand anything with five minutes'
> skimming. It's complicated and subtle and can't be served up
> on a plate.

The dynamic range of WTs is quite narrow, even if you do
things like allow sharp fifths. Listening tests here suggest
that it's harder to distinguish them than theoretical
discourse implies. The music sounds just fine in any number
of WTs. I personally don't like any thirds over about 404
cents in keys I plan to use. So this tuning isn't of much
interest to me. I'd never use a tuning because it's
"historical", because as I say, I don't think we have the
records of high enough quality to make that term very
meaningful. Case in point, there's a serious typo/error
in the present text, and this doesn't seem unusual for
these texts. The study of tunings is today a niche, and
clearly it was even less of one in the baroque or there'd
be more/better records.

> I decided not to include any cent values or fractions of a
> comma, if at all possible, because that was not how
> Werckmeister conceived of this tuning.

People of antiquity did all kinds of crazy things, like
wear wigs. Do you put one on before listening?

-Carl

Carl Lumma wrote:

> > I personally don't like any thirds over about 404
> cents in keys I plan to use. So this tuning isn't of much
> interest to me. I'd never use a tuning because it's
> "historical", because as I say, I don't think we have the
> records of high enough quality to make that term very
> meaningful. Case in point, there's a serious typo/error
> in the present text, and this doesn't seem unusual for
> these texts. The study of tunings is today a niche, and
> clearly it was even less of one in the baroque or there'd
> be more/better records.
> Oh get off it, Carl! There are a lot of serious people doing a lot of very good and solid research in this field. There's also a lot of good, solid evidence about what was done in the past. If you don't like the results, then just say you don't like it; don't go inventing problems which don't exist. You sound like a Bush administration lackey talking about global warming, or a cigarette manufacturer claiming there's no link to lung cancer. If you want to know just how incredibly uninformed you seem by making the above statement, first go learn German, then read at least Mark Lindley's Stimmung und Temperatur, if not the original sources themselves.

Ciao,

Paul

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Listening tests here suggest
> that it's harder to distinguish [WTs] than theoretical
> discourse implies.

That's mainly because most of the ones that have been tested here
aren't very unequal, and/or because of acoustic conditions (timbre and
sustain). Try temperament ordinaire with a bright pipe organ timbre
and you'll notice pretty soon. Or Kirnberger II ... not that I'm
recommending it, just that it should be obvious.

> I don't think we have the
> records of high enough quality to make that term very
> meaningful.

This particular Werckmeister effort is a 'worst case' scenario, in
that he chose an innovative way of constructing a temperament and
slipped up a bit in labeling his monochord. Almost all historical
sources that people take seriously are a lot *less* problematic than
the 'Septenarius'... But you won't discover that if you don't look at any.

> People of antiquity did all kinds of crazy things, like
> wear wigs.

... and use just intonation and microtonal scales. How can we possibly
distinguish between wig-wearing and unequal tuning? It's all Stuff of
Antiquity, innit?

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> That's mainly because most of the ones that have been tested here
> aren't very unequal, and/or because of acoustic conditions (timbre and
> sustain). Try temperament ordinaire with a bright pipe organ timbre
> and you'll notice pretty soon. Or Kirnberger II ... not that I'm
> recommending it, just that it should be obvious.

When in the history of circulating temperament did anyone *ever* use a
temperament as unequal as grail--which tested out very nicely?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > That's mainly because most of the ones that have been tested here
> > aren't very unequal, and/or because of acoustic conditions (timbre and
> > sustain). Try temperament ordinaire with a bright pipe organ timbre
> > and you'll notice pretty soon. Or Kirnberger II ... not that I'm
> > recommending it, just that it should be obvious.
>
> When in the history of circulating temperament did anyone *ever* use a
> temperament as unequal as grail--which tested out very nicely?
>

Rameau's 1726 advice asks for a run of 7 quarter-comma fifths
(starting on either C or Bb) then gradual widening over the remaining
5, the last two being probably the widest to reach the starting-point
again.

If we start on Bb then Bb-D is pure and D-F# is nearly pure; therefore
F#-Bb is likely to be about 423 cents. Indeed, Grail is a lot more
unequal than that.

I rather suspect Grail sounded good because the piece fitted it...

What about putting together a piece in B minor, which (among familiar
keys) should be the reverse of G minor? BWV544?

If one wanted to make a synthesized comparison to Rameau I would take
his tuning starting from C, with C#-G# pure and the remaining four
fifths sharing the leftover wideness. This will be (7*3 - 13)/12 = 2/3
comma, so the four sharp fifths can have 1/6 comma each.

~~~T~~~

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> Listening tests here suggest
> that it's harder to distinguish them than theoretical
> discourse implies.

If we were to have a color test with things like cold orange-reds
and a warm red-oranges, would you say that the colors are hard to
distinguish based on a general failure of the testees to name the
correct Photoshop hex-codes?

We don't really know how differently the WTs sounded to the testees:
there were several cases where people noted that they could hardly
tell the difference between x and y (somewhere there were two that I
couldn't distinguish at all, for example) but otherwise, most of the
responses described specific impressions. And these tests were often
handicapped by questionable Soundfont tuning accuracy. The tests
showed only that in these conditions, the testees couldn't name the
specific WT scheme consistently.

Of course, you could say that the testees were simply imagining
differences. This could only be tested by sneaking in doubles, and
even then, just because a person imagines a difference in the case
of two identical examples doesn't mean that he's imagining
differences between things that actually are different.

Not to mention the fundamental difference between hearing things in
the very emotional experience of listening to music and the this
clinical, repetitive (and probably even stressful for some)
experience, and you just can't make such a general statement based
on these informal tests.

-Cameron Bobro

> > Listening tests here suggest
> > that it's harder to distinguish them than theoretical
> > discourse implies.
>
> If we were to have a color test with things like cold orange-reds
> and a warm red-oranges, would you say that the colors are hard to
> distinguish based on a general failure of the testees to name the
> correct Photoshop hex-codes?

No, but I couldn't even tell the difference between all of
the scales. And as I said, I wasn't trying to identify
them, either; just rate their consonance and match this up
with the name of the temperament with the most consonant
most consonant key. Without extensive listening to all of
the scales in all keys, no testee could hope to identify them,
and few claimed to be trying.

> And these tests were often
> handicapped by questionable Soundfont tuning accuracy.

Not questionable at all.

> Not to mention the fundamental difference between hearing things in
> the very emotional experience of listening to music and the this
> clinical, repetitive (and probably even stressful for some)

Yes, all of this has been mentioned by the testees.

> experience, and you just can't make such a general statement based
> on these informal tests.

None of this applies to the soundness of my statement.
Maybe try reading it again (at the top).

-Carl

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote/discussed in:
>
>http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html

W's modified stringlengths can be interpreted as absolute frequencies
of an 5ths circle:

a'440Hz a220 A110
e'330 e165
b' 495
1485/1484 f#"742 f#'371
1113/1112 c#"556 c#'278 c#139
417/416 g#208 G#104
eb'312 eb156 Eb78
bb'468 bb234 Bb117
f'351
1053/1052 c"526 c'263/262 131
g'393/392 196 98 49=7*7
d147
441/a'440 cps

on the keys:

_____________________
| c' 263 middle-C
+---------|278 c#'=db'
| d' 294
+---------|294 eb'=d#'
| e' 330
+---------------------
| f' 351
+---------|371 f#'=gb'
| g' 393
+---------|416 g#'=ab'
| a' 440 Hz
+---------|468 bb'=a#'
| b' 468
+--------------------
| c" 526
+---------| c#"
&ct...

!septenarius440Hz.scl
!
sparschuh's septenarius @ middle c'=263Hz or a'=440Hz
!
12
!
278/263 ! C#
294/263 ! D
312/263 ! Eb
330/263 ! E
351/263 ! F
351/263 ! F#
393/263 ! G
416/263 ! G#
440/263 ! A
468/263 ! Bb
495/263 ! B
2/1

Just try it out to play in that yourself!

http://www.strukturbildung.de/Andreas.Sparschuh

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

> !septenarius440Hz.scl
> !
> sparschuh's septenarius @ middle c'=263Hz or a'=440Hz
> !
> 12
> !
> 278/263 ! C#
> 294/263 ! D
> 312/263 ! Eb
> 330/263 ! E
> 351/263 ! F
> 351/263 ! F#
> 393/263 ! G
> 416/263 ! G#
> 440/263 ! A
> 468/263 ! Bb
> 495/263 ! B
> 2/1

F and F# are the same note.

Comments below!

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> >http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
>
> W's modified stringlengths can be interpreted as absolute
frequencies
> of an 5ths circle:
>
> a'440Hz a220 A110
> e'330 e165
> b'495
> 1485/1484 f#"742 f#'371

Instead of 496 and 372. This avoids having a wide e-b fifth and is
better for G major and D major.

> 1113/1112 c#"556 c#'278 c#139
> 417/416 g#208 G#104
> eb'312 eb156 Eb78
> bb'468 bb234 Bb117
> f'351
> 1053/1052 c"526 c'263/262 131
> g'393/392 196 98 49=7*7

I don't think this is so good, you have C-G tempered by 262/263 which
is about 1/3 comma...

Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?

Then C-G is pure and

> d147

rather 1179/1176 d''588 d'294

> 441/a'440 cps
>
> on the keys:
[edited]
> _____________________
> | c' [262] middle-C
> +---------|278 c#'=db'
> | d' 294
> +---------|[312] eb'=d#'
> | e' 330
> +---------------------
> | f' [350]
> +---------|371 f#'=gb'
> | g' 393
> +---------|416 g#'=ab'
> | a' 440 Hz
> +---------|468 bb'=a#'
> | b' [495]
> +--------------------
> | c" [524]
> +---------| c#"
> &ct...
>
> !septenarius440Hz.scl
> !
> sparschuh's septenarius @ middle c'=263Hz or a'=440Hz
> !
> 12
> !
> 278/263 ! C#
> 294/263 ! D
> 312/263 ! Eb
> 330/263 ! E
> 351/263 ! F
> 351/263 ! F# [should be 271]
> 393/263 ! G
> 416/263 ! G#
> 440/263 ! A
> 468/263 ! Bb
> 495/263 ! B
> 2/1
>

or:

!septenarius440Hzmk2.scl
!
TD's septenarius @ middle c'=262Hz or a'=440Hz
!
12
!
278/262 ! C#
294/262 ! D
312/262 ! Eb
330/262 ! E
350/262 ! F
371/262 ! F#
393/262 ! G
416/262 ! G#
440/262 ! A
468/262 ! Bb
495/262 ! B
2/1

(cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2,
370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...)

Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' near-
equal tuning?
eg Eb-G = 150:189 ...
via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189

then continue:
... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848)
F#356=178 C#267 (801) G#400 Eb300

seems to work nicely at late Baroque pitch levels - only three pure
fifths between Eb-Bb, G#-Eb, F#-C#.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> Comments below!
>
> > a'440Hz a220 A110
> > e'330 e165
> > b'495
> > 1485/1484 f#"742 f#'371
>
> Instead of 496 and 372. This avoids having a wide e-b fifth and is
> better for G major and D major.

right, in order to get rid of the oversharp wolfs in W's original
stringlength numbers, alike in his famous #3 the 'quaternarius'
has only 4 flattend and 8 pure 5ths:
A E B>F# C# G# Eb Bb F C>G>D>A

>
> > 1113/1112 c#"556 c#'278 c#139
> > 417/416 g#208 G#104
> > eb'312 eb156 Eb78
> > bb'468 bb234 Bb117
> > f'351
> > 1053/1052 c"526 c'263/262 131
> > g'393/392 196 98 49=7*7
>
> I don't think this is so good, you have C-G tempered by 262/263
> which is about 1/3 comma...
>
...flattend than a pure 5th: 3/2.
So far about that strongest detuned 5th C>G.
Consider accordingly the belonging 3rd C>E in the major-chord C>E>G:
with compareable sharpness:
e165 e'330 e"660 e'"1330/1315=5*c"263
shortening by common factor 5 yiels
a tempering of C>E about 264/265,
so that the C-major chord consists in

C(1/1)-E*(265/264)-G*(262/263)
or
4:(5*(265/264)):(6*262/263)
instead barely 4:5:6 without the idea of tempering
3rds and 5ths in the same range of magintude.

Many 'experts' do recommend to sharpen the 3rds about the amount in
amplitude alike the corresponding 5ths become flattened, so that the
beatings of 3rds and 5ths beat almost the same in reverse directions.
Skilled organ-builders use that effect in order to demonstrate
that their robust organs survive even such impressive resonances.
It appears that already
http://en.wikipedia.org/wiki/Arnolt_Schlick
knew that old well-known tuning-trick/method in his instructions.

> Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?
Good idea, if you intend to stay nearer to W's original version,
when somehow aiming to approximate "ET" however.

>
> Then C-G is pure and
makes that only sense according the above demands
when C-E is also pure chosen?

>
> >
Correction of: !septenarius440Hz.scl

> > 351/263 ! F# [should be 271] for Gene: 371 is the correct pitch.

> or:
>
> !septenarius440Hzmk2.scl
> !
> TD's septenarius @ middle c'=262Hz or a'=440Hz
> !
> 12
> !
> 278/262 ! C# short 138/131
> 294/262 ! D short 147/131
> 312/262 ! Eb short 156/131
> 330/262 ! E short 165/131
> 350/262 ! F short 175/131
> 371/262 ! F#
> 393/262 ! G
> 416/262 ! G# 208/131
> 440/262 ! A 220/131
> 468/262 ! Bb 234/131
> 495/262 ! B
> 2/1

The pure middle c' in reference to a'=440Hz becomes in the just case
440Hz*3/5 = 264Hz. Hence i do prefer the nearer 263Hz instead
yours lower 262Hz, which appears just a little bit to flat lowered
in my personal taste, especially when having just intonation
in mind or ear instead the virtual "et".
Most professional and skilled tuners do to keep
the frequent keys with few accidentials somehow purer than
the less used keys with many accidentials, so that the
strange keys got more pythagorean 3rds, by purer
or even just pure 5ths inbetween them.

>
> (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2,
> 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...)
those irrational numbers are far to complicated for solving the
problem. Who needs the advanced precision of 4 decimal digits
in the octave from c' to c"?

But if you want to approach "et" whatsoever, then 262 would be the
better choice for converging "et" as the above approximation suggest.

>
> Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near-
> equal tuning?
I.m.o: the nearer one draws to approach "et",
the most frequently used 3rds get to much worse detuned

Luckily nobody can tune irrational intervals in practice,
so that the worsest case: "et" remains barely a theoretically fiction,
excluded from real implementation on a real sounding instrument.
Simply try out how well can you reproduce by yours ears:
on the one hand:
a pure 5th ratio 3:2=1.5 ~702cents
and on the other hand:
sqrt(2) = 600 Cent "et"-tritous, that's geometrically interpreted:
http://mathworld.wolfram.com/PythagorassConstant.html

Experimental result:
There is no psychoacusitcally evidence for departening
the ratio of a 5th 3:2 for the benefit of sqrt(2) ET-tritone.
Quoting Herrmann Helmholtz: "The ear prefers simple ratios."

> eg Eb-G = 150:189 ...
> via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189

But by that procedure one does also loose to much of the 'Baroque'
key-characteristics.
http://www.societymusictheory.org/mto/issues/mto.95.1.4/mto.95.1.4.code.html
http://de.wikipedia.org/wiki/Tonartencharakter
>
> then continue:
> ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848)
> F#356=178 C#267 (801) G#400 Eb300
>
> seems to work nicely at late Baroque pitch levels - only three pure
> fifths between Eb-Bb, G#-Eb, F#-C#.
>
hmm, is it still apt to call that 'baroque'-style?
As far as i understood late "Baroque" post-meantone instructions:
There i found a tendency to keep the 5ths inbetween the accidentials
F#-C#-G#-Eb-Bb
purer more pure (within the upper black keys on the piano)
than in the ordinary F-C-G-D-A-E-B: that got generally more tempering.

In extreme form i do start from my prototype model.
The procedure consists in a chain of 11 almost pure 5ths,
that contains the JI pitches, but also schismic Pythagorean-enharmonics:

Here the chain
F-A-C-E-G-B-D
are all 3 pure major 4:5:6 chords
in exact beatless just proportions:

A 440. 220 110 55
E 165
B 495.
F# 1485
4455 C# 4454 2227
6681 G# 6680 3340 1670 835
2505 Eb 2504 1252 626 313.
Bb 939
2817 F 2816 1408 704 352. 176 88 44 22 11
C 33
G 99
D 297. / 296 148 74 37
111 A 110 55

A E B F#~C#~G#~Eb Bb~F C G D~~~~~~~~~~~~~~~~~~~~~~~~~~~~A

The schisma 32805/32768=5*3^8/2^15=
(4455/4454)(6681/6680)(2505/2504)(2817/2816)
is tempered out by the subdivsion in to that
product of 4 superparticular factors.
Respectively the
SC=81/80=(297/296)(111/110) into 2 parts at one @ D>A 40:27

Rearranging same pitches in ascending order yields:

C' 264Hz middle-C
C# 278.375
D' 297
Eb 313
E' 330
F' 352
F# 371.25
G' 396
G# 417.5
A' 440Hz reference pitch
Bb 469.5
B' 495
C" 528

so far about the 11 other frquencies that i percieve instantly
also in mind immediatley when hearing a 440Hz tuning-fork by ear,
or simpy when imagening that pitch-levels enwraped when reading
musical scores in any fitting tuning.

schismatic_just440Hz.scl
!
sparschuh's-schisma-subdivision(4455/4454)(6681/6680)(2505/2504)(2817/2816)
!
2227/2112 ! C#
9/8 ! D
313/264 ! Eb
5/4 ! E
4/3 ! F
45/32 ! F#
3/2 ! G
835/528 ! G#
5/3 ! A=440Hz
313/176 ! Bb
15/8 ! B
2/1

But, how about that almost similar alternative one at the moment on my
piano?

A 440. 220 110
330 E 329.
B 987
2961 F# 2960 1480 740 370. 185
C# 555
1665 G# 1664 832 416. 208 104 52 26 13
Eb 39
Bb 117
F 351.
1053 C 1052 526 263.
789 G 788 394. 197
591 D 590 285./284 147
441 A 440. (or 3*285=885 A 880 440.)

with strongest tempering @ D>A: 885/880=(285/284)(441/440)=177/176
but still less than SC^(1/2) ~161/160 or ~162/161,
hence rather tolerable than
the ancient Erlangen-monochord or Kirnberger#1,
that charge a full SC on D>A alike the above 'schismatic_just.scl'.

So far my reccomendation for those who prefer to stay nearer at JI
than to the i.m.o. over-detuned "ET", that i do meanwhile consider as
outdated intuneable fiction.

sparschuh_gothic_style440Hz.scl
!
12
!
555/526 ! C# 277.5 Hz
285/263 ! D
312/263 ! Eb
329/263 ! E
351/263 ! F
370/263 ! F#
394/263 ! G
416/263 ! G#
440/263 ! A reference-pitch 440Hz
468/263 ! Bb
987/526 ! B 493.5 Hz
2/1

on the keys

+-----------
| C 263 middle-C
+--|277.5=C#
| D 285
+--|312=Eb
| E 329
+-----------
| F 351
+--|370=F#
| G 394
+--|416=G#
| A 440
+--|468=Bb
| B 493.5
+-----------
| C'526
&ct.

If you dont't like any of that, it's up to you to create yours own
personal version, according yours private preferences.

A.S.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> > 1113/1112 c#"556 c#'278 c#139
> > 417/416 g#208 G#104
> > eb'312 eb156 Eb78
> > bb'468 bb234 Bb117
> > f'351
> > 1053/1052 c"526 c'263/262 131
> > g'393/392 196 98 49=7*7
>
> I don't think this is so good, you have C-G tempered by 262/263
> which
> is about 1/3 comma...
agreed, hence i do return to W's original 131.
>
> Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?
>
also right, hence so the resulting ratios get even more simple:
The calculations benefit from that by less computational0 overhead.

Follow the classical way:
start traditional @ pitch-class GAMMA=G alike in:

http://www.celestialmonochord.org/log/images/celestial_monochord.jpg

GG 49 := 7*7 (GAMMA-ut, the empty string in the picture)
D 147
3D441 > a'440 a220 A110 AA55Hz=the AA-string of a double-bass
e 165
3e495 > b'494 b247
3b741 > f#"740 f#'370 f#185
c# "555
3c#"1665 > g#"'1664 g#"832 g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13
EEb 39
Bb 117
3Bb351 > f'350 f175
3f525 > c"524 c'263 c131
3c393 > g'392 g196 G98 GG49=7^2 returned

!septenarius_GG49Hz.scl
sparschuh's version @ middle-c'=262Hz or a'=440Hz
12
!absolute pitches relativ to c=131 Hz
555/524 ! c# 138.75 Hz
147/131 ! d
156/131 ! eb
165/131 ! e
175/131 ! f
185/131 ! f#
196/131 ! g
208/131 ! g#
220/131 ! a 440Hz/2
234/131 ! bb
247/131 ! b
2/1

That results on my old piano in the first/lowest octave:

AAA 27.5 Hz lowest pitch, on the first white key on the left side
BBBb29.25 next upper black key
BBB 30.875 http://en.wikipedia.org/wiki/Double_bass "at~30.87 hertz"..
CC_ 32.875
CC# 34.6875 := c"555/16
DD_ 36.75
EEb 39 := GGGG#13Hz*3
EE_ 41.25 ..."E1 (on standard four-string basses) at ~41.20 Hz
FF_ 43.75
FF# 46.25
GG_ 49 := 7*7 Werckmeister's/Scheibler's initial septimal choice
GG# 52 = GGGG#13Hz*4
AA_ 55 = 440Hz/8 ; 3 octaves below Scheibler's choice

http://mmd.foxtail.com/Tech/jorgensen.html
#133: "Johann Heinrich Scheibler's metronome method of 1836"

http://www.41hz.com
"41 Hz is the frequency of the low E string on a double bass or an
electric bass." if it has none additional 5th string
for midi(B0)=BBB 30.875 Hz an 2nd above AAA 27.5 Hz,
the lowest A on the piano, without attending or even careing
"string-imharmonicty"
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN0000760000S1000S22000004&idtype=cvips&gifs=yes

Consider the 3rds qualities in violins empty sting G-D-A-E block order:

1: G > B > Eb> G
2: D > F#> Bb> D
3: A > C#> F > A
4: E > G#> C > E

1: G > B > Eb > G.
absolute analysis:
GG 49.
5*49 = 245 < b'247 < 248 124 62 31
5*31 = 155 < eb156 Eb78 EEb39
5*39 = 195 < g196 G98 GG7*7.
relative diesis 128/125 subpartition:
G 123.5/122.5 B 96/95 Eb 196/195 G
G ~14.1 cents B ~18.1 Eb ~ 8.86c G

2: D > F# > Bb > D.
abs:
d147. < 148 74 37
5*37 = f#185 < 186 93
5*31*3=3*155 < 156*3 78*3 39*3 = Bb117
5*39*3=3*195 < 196*3 98*3 49*3 = d147.
rel:
D 148/147 F# (1/9+85)/(84+1/9) Bb 196/195 D
D ~ 11.4c F# ~ 20.5 cents Bb ~ 8.86cents D

3: A > C# > F > A.
abs:
AA55. A110 < 111 = 37*3
5*111= c#"555 = 5*111 < 112*5 56*5 28*5 14*5 7*5
5*35 = f175 < 176 88 44 22 11
5*11 = AA55.
rel:
A 111/110 C# 112/111 F 176/175 A ; with all 3 factors superparticular
A ~ 15.7c C# ~ 15.5c F ~ 9.86c A

4: E > G#> C > E.
abs:
e165. < 166 83
5*83 = 415 < g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13 §§
5*13 = 65 130 < c131 < 132 66 33
5*33 = e165.
rel:
E (6/7+118)/(117+6/7) G# 131/130 C 132/131 E
E ~ 14.6 cents ~ G# ~ 13.3 cents C ~ 13.2c E
§§ GGGG# 13 Hz has negative "midi"-index G#_-1,
which midi-keyboard supports negative key indices?

Summary:
3rds martix in "Cents"
5ths in top>down order
3rds in left>right direction respectively:

1: G 14.1 B_ 18.1 Eb 8.86 G
2: D 11.4 F# 20.5 Bb 8.86 D
3: A 15.7 C# 15.5 F_ 9.86 A
4: E 14.6 G# 13.3 C_ 13.2 E

Conversely "ET" detunes all 3rds about the same amount:
(128/125)^(1/3) = ~13.7Cents or ~127/126,
Attend that: the "septenarius" fits therefore better than ET to
horns and trumpets in Eb,Bb & F, with inherent natural
3rds Eb>G, Bb>D, & F>A that turn out less than 10Cents out of tune
in the septenarius case.

> (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2,
> 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...)

In the "ET" case, it is difficult to resolve the
septenarian root-factors above alike:
11,13,31,37 & 83 below the well known 3,5 & 7 limits.
>
> Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near-
> equal tuning?
on the one hand is:
(63/50)(4/5)=126/125
but but on the other side
the diesis 128/125=(128/127)(127/126)(126/125)
contains 3 factors.
hence:
(63/50)^3 = 2.000376... > 2/1
overstretched octave
or as superparticular ratio:
((63/50)^3)/2= 250047/250000=(7/47+5320)/(5319+7/47)
~1/2 per mille
but a better approximation of the octave delivers 127/126
the factor in the middle:
((5 / 4) * (127 / 126))^3 = ~1.99999802............

> eg Eb-G = 150:189 ...
> via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189
>
> then continue:
> ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848)
> F#356=178 C#267 (801) G#400 Eb300
>
> seems to work nicely at late Baroque pitch levels - only three pure
> fifths between Eb-Bb, G#-Eb, F#-C#.
>
that is expanded:
A 424 212
3A636 > E635 > 634 317
B951 > 950 475
3*475=1425 > F#1424 712 356 178 89
C# 267
801 > G# 800 400 200 100 50 25 Werckmeister's tief-Cammerthon 400Hz
Eb 75
Bb 225
675 > F 674 337
1011 > C 1050 505 > 504 252 126 63
G 189
567 > D 566 283
849 > A 848 424

recombining that 5ths-circle in ascending ordered pitches yields:
C 252.5 Hz middle_C
C#267
D 283
Eb300
E 317.5 or better 317?
F 337
F#356
G 378
G#400 Beekman's, Descartes's, Mersenne's & Sauveur's standard-pitch
A 424
Bb450
B 475.5 or better 475?
C'505

T.D. remarked already in his numbers inbetween:

>B951 (2853/2848)
> F#356

that there appears an unsatisfactory irregular gap:
of 2853/2848 = 570.6/569.6 = (951/950)(1425/1424)
induced by the choice of
E 635 > 634 317 instead
635 > E 634 317
That results in the above none-integral superparticular ratio bug.

Hence i do suggest to replace
by the tiny changes
1: E 635-->>>634
2: B 951-->>>950
in order to fix the bug.

so that now all 5th-tempering steps become integral superparticular
ratios, without any exception:

A 424 212 106
3A=318 E 317 instead formerly 317.5
3E=951 B 950 475 instead formerly 951 475.5
3B=1425 F# 1424 712 356 178 89
&ct.
the rest of the circle remains unchanged.
Is that ok?

Analysis of the:
3rds sharpness, -how much wider than 5/4-
per diesis subpartition into superparticular factors,
so that the product of 3 tempered 5ths results an octave
in each of the 4 blocks:

1: G > B > Eb > G.
abs:
G378. 189 > 190 95
5*95 = B 475 < 480 240 120 60 30 15
5*15 = Eb75
5*75 = 3*125 < 126*3 = G378.
rel. 2^7/5^3=128/125=
G 190/189 B 160/159 Eb 126/125 G remember (63/50)(4/5)=126/125
?or formerly in the original version:
?G378. 189 > 190 95
?5*95 = 475 950 < B951 < 960 480 240 120 60 30 15
?......
?rel. 2^7/5^3=128/125=
?G (190/189)(951/950) B 320/317=(2/3+106)/(105+2/3) Eb 126/125 G
That appears i.m.o. much more complicated than my suggested change.

2: D > F# > Bb > D.
abs:
D283. < 284 142 71
5* 71 = 355 < F#356 178 89 < 90 45
5* 45 = Bb225 < 226 113
5*113 = 565 < D556 283.
rel. 128/125=
D (284/283)(356/355) F# 90/89 Bb (226/225)(556/565) D

3: A > C# > F > A.
abs:
A424. 212 106 53
5* 53 = 265 < C#267 < 268 134 67
5* 67 = 335 < F 337 < 338 169
5*169 = 845 < A 848 424.
rel: 128/125=
A 133.5/123.5 C# (268/267)(168.5/166.5) F >>>
>>> F (338/337)((2/3+282)/(281+2/3)) A

4: E > G# > C > E.
E 317. < 320 160 80 40 20
5* 20 = G#100 < 101
5*101 = C 505 < 506 253
5*253 = 1265 < E 1268 634 317.
rel: 128/125=
E (2/3+106)/(105+2/3) G# 101/100 C (506/505)/(422.666.../421.666...) E
?or formerly
?E 635? > 640 320 .....
?....&ct. alike above...
?....
?5*253 =1265 < 1270 635?

try to find out similar improvements in order to reduce the ratios
to less complicated proportions

have a lot of fun in whatever tuning you do prefer
A.S.