Tuning-Math Digests messages 11375 - 11399

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Message: 11375

Date: Tue, 13 Jul 2004 18:32:48

Subject: Re: 50

From: monz

hi Gene,

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > 
> > have you ever seen this?
> > 
> > http://tonalsoft.com/monzo/woolhouse/essay.htm#temp *
> 
> you quote Woolhouse as saying:
> 
> [Woolhouse 1835, p 46:]
> 
>     This system is precisely the same as that which
> Dr. Smith, in his _Treatise on harmonics_ [Smith 1759],
> calls the scale of equal harmony.
> It is decidedly the most perfect of any systems in
> which the tones are all alike. 
> 
> Is Smith's tuning 50-equal?


yes.

but i haven't read his book.


> 
> I'd also cut Woolhouse some slack on 5-limit vs 5-odd-limit. The 
place
> where he discusses that you *might* interpret to say that 2 and 3/2
> are consonances, but 4, 3 or 6/5 are not, but I would presume his
> assumption of octave equivalence would be clear from elsewhere.
> 
> I also find this comment interesting:
> 
> He then analyzes the resources of a 53-EDO 'enharmonic organ', built
> by J. Robson and Son, St. Martin's-lane, but says that the number of
> keys is too much to be practicable, and settles again on 19-EDO.
> 
> Is this the first time someone built a 53-edo instrument?


i don't think so, but can't cite anything concrete.

Mercator studied 53edo and recommended it for instruments,
but i don't know what was/wasn't built.

 
> Incidentally, convergents to the Woolhouse fifth go 7/12,
> 11/19, 18/31, 29/50, 76/131, 257/443 ... . It would be
> interesting to dig up someone who advocated 131-equal for
> meantone!



hmmm ... there's a blank space in my equal-temeperaments
table, just waiting for that!  maybe Gene will be the first.




-monz


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Message: 11376

Date: Tue, 13 Jul 2004 18:52:06

Subject: Re: 50

From: monz

hi Paul,


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> wrote:
> 
> > but you quote
> > Woolhouse as saying:
> > 
> > [Woolhouse 1835, p 46:]
> > 
> >     This system is precisely the same as that which
> > Dr. Smith, in his Treatise on harmonics [Smith 1759], calls
> > the scale of equal harmony.
> > It is decidedly the most perfect of any systems in which
> > the tones are all alike. 
> > 
> > Is Smith's tuning 50-equal?
> 
> It's close, but it's much closer to 5/18-comma meantone
> than to 50-equal. Search the tuning list for more info ;)
> Also see Jorgenson.



hmm ... Woolhouse *did* write exactly what Gene quoted
from my webpage.


it's too much of a pain to search the list archives.
can you give us some details?  exactly how did Smith
measure his tuning?


here's a very interesting webpage about Smith's 
"Equal Harmony" tuning (described as 50et), and its
application to harpsichords.

http://216.239.41.104/search?q=cache:daufeki--acj:www.music.ed.ac *.
uk/russell/conference/robertsmithkirckman.html+%22robert+smith%22+%
2250%22+tuning%22&hl=en

or

http://tinyurl.com/6hbwu *




-monz


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Message: 11379

Date: Tue, 13 Jul 2004 23:54:30

Subject: Extreme precison (Olympian) Sagittal

From: Dave Keenan

Hi Gene,

Sorry for gaving you a hard time about jargon on "tuning". But I 
think you'd rather get it from me, now, than have it build up until 
you get attacked by a serious mob of maurauding math-phobes. :-)

If you're not too pissed off with me to still be interested, here's 
one way of approaching the extreme-precision Sagittal question.

The raw data you need is I think contained in Table 1 and Figure 3 
of the XH paper.

http://users.bigpond.net.au/d.keenan/sagittal/Sagittal.pdf *

In table 1, you can ignore everything but the first comma listed for 
each symbol (not necessarily the bold one, but the topmost one, the 
one on the same line as the symbol). All the other roles for a 
symbol are allowed to break, and be taken over by another (possibly 
accented) symbol in the extreme-precision set.

And ignore the second row of symbols in Figure 3 since their 
definitions are purely as apotome complements of those on the first 
line no matter what set we are using.

The idea then is to treat each of the 9 flags (left and right of 
barb, arc, scroll, boathook; plus accent) as a generator and find 
the optimum value in cents for each so as to minimise the maximum 
error over all the symbol/comma relationships in table 1.

There may be some other important symbol-comma relationships not 
shown in table 1, that we don't want to break, but I'd like to see 
what happens when we just base it on those first.

Due to the apotome complementarity we have one constraint on the 
flag values initially, so we are down to 8 degrees of freedom. That 
is that /| + |\ + (| + |) = apotome. Although you could ignore that 
and see what happens.

Then once you have the value of each flag you can calculate the 
value of each symbol in the top row of figure 3 and the same with up 
accents and the same with down accents. And maybe figure out what 23-
limit comma might be assigned as the primary role of each.
Commas which notate more popular (simpler or lower-prime-limit) 
ratios are to be preferred, as are commas which notate ratios 
without having to go too far up or down the chain of fifths.


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Message: 11385

Date: Wed, 14 Jul 2004 20:11:22

Subject: Re: Beep and ennealimmal

From: Carl Lumma

>"Beep" was supposed to bring to mind "BP" (as in
>"Bohlen-Pierce"); 27/25 is a frequent step size of
>the untempered BP scale. But it's as if we were to
>call father "diatonic" because it tempers out the
>diatonic semitone. "Bug" appears to be the older
>name, and I don't know why it was dropped, unless
>the name was just forgotten. So I vote for "bug".

I too prefer bug, but Joseph Pehrson has done a piece
called Beepy, so I think we should stick with that name.

-Carl


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Message: 11391

Date: Thu, 15 Jul 2004 03:31:45

Subject: Re: Beep and ennealimmal

From: Carl Lumma

>X-eGroups-Return: perlich@xxx.xxxx.xxx
>Date: Thu, 15 Jul 2004 03:20:28 -0000
>From: "Paul Erlich" <perlich@xxx.xxxx.xxx>
>To: Carl Lumma <ekin@xxxxx.xxx>
>Subject: Re: Beep and ennealimmal
>User-Agent: eGroups-EW/0.82
>X-Mailer: Yahoo Groups Message Poster
>X-Originating-IP: 199.103.208.200
>
>--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> >"Beep" was supposed to bring to mind "BP" (as in
>> >"Bohlen-Pierce"); 27/25 is a frequent step size of
>> >the untempered BP scale. But it's as if we were to
>> >call father "diatonic" because it tempers out the
>> >diatonic semitone. "Bug" appears to be the older
>> >name, and I don't know why it was dropped, unless
>> >the name was just forgotten. So I vote for "bug".
>> 
>> I too prefer bug, but Joseph Pehrson has done a piece
>> called Beepy, so I think we should stick with that name.
>> 
>> -Carl
>
>Joseph's piece wasn't in beep/bug; it was in BP!

Eek- sorry!

-C.


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Message: 11395

Date: Fri, 16 Jul 2004 07:42:39

Subject: Re: Naming temperaments

From: Graham Breed

Herman Miller wrote:

> To be more precise, since 27-ET isn't 11-limit consistent, you could 
> call it 11/(27,43,63,76,93). But unless there turns out to be a lot of 
> useful-looking temperaments that can't be notated any other way (which 
> could very well be the case once we start dealing with the 11-limit), it 
> would be simpler just to assume that 11/27 denotes a temperament based 
> on the best mapping of each of the primes to degrees of 27-EDO, which is 
> (27,43,63,76,93,100,110,115,122,131,134,...).

It depends on what you think is useful.  I don't find the inconsistent 
search finds anything I'm interested in in the 11-limit.  But you may 
prefer less accurate temperaments.   When you get to the 17-limit, 
inconsistently composed temperaments score high.  At least by my 
criteria that try to keep the number of notes manageable.

You can test all this with the online script.


                      Graham


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Message: 11396

Date: Fri, 16 Jul 2004 00:04:17

Subject: Re: Naming temperaments

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> I've been thinking about names for linear temperaments and ways of 
> classifying them; there's probably nothing much new here, but a 
summary 
> might have some use.

Yes. This is good. Thanks Herman.

> For most of the better temperaments, if you just give the mapping 
by 
> steps of the octave (which is equivalent to a combination of two 
> octave-based ET's), you can fill in the others by taking the best 
> approximation of the primes in each of the ET's. So we could call 
this 
> the 5&22 temperament.

Well I'd prefer to call it the 22&27-LT. You can get the 5 by 
subtraction and you can get an even better ET for it (49) by 
addition. i.e. the two ETs should be two extremes of the generator 
value such that you would actually consider using those ETs for the 
LT. You wouldn't really use 5. The 5 just tells you that pentatonic 
scales make sense in this LT. And as I say, you can get that by 
subtraction.

> But there's another alternative that looks somewhat interesting. 
Notice 
> that the first thing in Graham Breed's temperament finder results 
is a 
> fraction of an octave: in this case:
> 
> 11/27, 489.3 cent generator

Right. But why _is_ this 11/27 and not 20/49?

> One drawback is that you also need to specify the period if 
> it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for 
pajara.
> 

That's a pretty serious drawback. At least with the two-ETs method 
you can see if they are both even and know immediately that the 
period is a half octave.

But you could just call Pajara "twin 4/46" or is it "twin 19/46", 
or "twin 27/46", or should the denominator be some other ET?

> And of course the other way of naming these things is by giving 
them 
> actual names. The big advantage is recognizability: if you've 
heard of 
> "superpyth", you take one look at the name and say "Ah yes, *that* 
> temperament". You might not immediately recognize "7/31" 
or "9&22", but 

Why wouldn't you immediately recognise one of these, if that's what 
you've been used to seeing it called.

Superpyth isn't a good example of the problem, since it at least 
suggests something related to pythagorean. The real problem is with 
names like "keenan" or "porcupine" or "orwell".

> probably anyone who's familiar with it knows the name "orwell". 

That's a tautology.

> The 
> disadvantage, of course, is that if you haven't heard of it, it 
doesn't 
> mean anything.

Exactly.

> So you might as well use the name in combination with one 
> or more of the other methods, especially if it's a less familiar 
one 
> like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.

So who needs the "lemba". It adds absolutely nothing, for me. For 
some reason it suggests "unleavened bread" to me. Huh?


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Message: 11398

Date: Fri, 16 Jul 2004 00:53:26

Subject: Re: Naming temperaments

From: Dave Keenan

Good summary Herman. But you left out my favourite method which 
unlike some of the others does not treat ETs as if they were more 
fundamental than LTs or JI, but instead relates to what this is all 
about -- approximating JI.

The name starts with a word for the number of periods per octave, if 
more than one: twin, triple, quadruple, quintuple, 6-fold, 7-
fold, ....

And then the generator is described in terms of the simplest n-odd-
limit consonance (from the diamond) (or its octave inversion or 
extension, as required). That is the one that takes the fewest 
generators to approximate according to the LTs mapping. 

I use the following words if there is more than one generator to the 
consonance: semi, tri, quarter, 5-part, 6-part, ....

Followed by the ratio or words for the consonance as given here:
http://users.bigpond.net.au/d.keenan/Music/Miracle/MiracleIntervalNam *
ing.txt

e.g. Miracle is "semi 7:8's" or "semi supermajor seconds". 

This is used up to some point where the LT is so complex you just 
describe the generator in cents. e.g. What used to be called 
Aritoxenean is the 12-fold 15 cent LT.

This at least works up to 11-limit.


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