Tuning-Math Digests messages 6852 - 6876

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Message: 6852

Date: Wed, 04 Jun 2003 17:22:03

Subject: Re: ...continued...

From: Carl Lumma

>>Can someone give the vals for 5-limit meantone?
>
>Various choices are possible:
>
>(1) Eytan, consisting of h12 = [12, 19, 28] and h7 = [7, 11, 16]
>
>(2) Octave and fifth, consisting of oct = [1, 1, 0] and 
>fif = [0, 1, 4]
>
>(3) Nominals and accidentals, consisting of h7 = [7, 11, 16] and
>h5 = [5, 8, 12]
>
>(4) Very meantone, h31 = [31, 49, 72] and h19 = [19, 30, 44]
>
>And so forth. I suggest you try sticking each of these pairs into
>my Maple routine "a5val" and seeing what comes forth.

If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back
the input.

>If we have linearity, then if 9 is as consonant as 5, 3 will have to
>be twice as consonant.

Why?

-Carl


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Message: 6854

Date: Wed, 04 Jun 2003 20:49:08

Subject: Re: ...continued...

From: Carl Lumma

>>>And so forth. I suggest you try sticking each of these pairs
>>>into my Maple routine "a5val" and seeing what comes forth.
>> 
>>If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back
>>the input.
>
>a5val is a Maple function, so you need to enter
>
>a5val([19, 30, 44], [31, 49, 72]);

Ah, I'd tried that without the comma between the two lists.

Great.  So they all give the comma for meantone.  And I'm
beginning to understand vals.


>>>>>>Finally, note that I'm still confused about prime- vs. odd-limit
>>>>>>as regards pi(p)-1.  Obviously I'm assuming prime-limit here,
>>>>>>but should pi(p) be changed to ceiling(p/2)?  That is, how many
>>>>>>commas does a 9-limit linear temperament require?  Paul?
>>>>>
>>>>>... linear independence.  So the 5-limit (2-3-5) requires one
>>>>>comma. So does the 2-3-7 limit.  And so would a system composed
>>>>>of octaves, fifths, and 7:5 tritones, although it uses 4 prime
>>>>>numbers.
>>>>
>>>>Ok, but what about stuff like (2-3-5-9) where we don't have linear
>>>>independence but wish to consider 9 as consonant as 3 or 5?  How
>>>>does visualization in terms of blocks work on a lattice with a
>>>>9-axis?
>>>
>>>If we have linearity, then if 9 is as consonant as 5, 3 will have
>>>to be twice as consonant.
>> 
>>Why?
>
>Because if a is the approximation to 3, a^2 will be the approximation
>to 9. Then the error log(a^2/9) will be twice that of log(a/3).

I'm not talking about error, but the assumption brought to the table,
namely that ratios of 9 are no less consonant than ratios of 3 in a
9-limit tuning, so that whatever errors exist should not be weighted
differently when optimizing the tuning.

Have you been including errors in ratios of 9 in optimizations?

I notice you never talk of the 9-limit.  Only the 7- and 11- limits.
At the 7-limit, the 3-error is counted once.  At the 11-limit, it
should be counted thrice.

-Carl


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Message: 6857

Date: Fri, 06 Jun 2003 00:35:48

Subject: Re: Interval Database Experiences

From: Dave Keenan

Hi Alex,

Welcome to tuning-math. Here's what Gene was talking about,
implemented in an Excel spreadsheet. 

The page cannot be found *

You enter an interval in cents and it gives a (two-dimensional) series
of ever-better ratios with their errors in cents.

I remember when I first learned how to calculate convergents and
semi-convergents (which I don't think Gene mentioned), to get ratios
from cents, it seemed like magic.

On the question of how much error is acceptable in just intonation:

In my humble opinion, many folks find +-3 cent errors entirely
acceptable for most music intended to be played in just intonation,
and indeed this is a typical intonation error for many fixed-pitch
acoustic intruments, but for others you'd have to be within +-0.5
cents (and not tell them about it) before they would be happy. But
there is little point in accepting more than about 3 cents error, or
in striving for less than about 0.5 cents. But of course there are
extreme and unusual contexts in which these do make sense. There are
no sharp cutoffs.

Others might be amused to learn that George Secor and I, in our
deliberations on a universal microtonal notation system, have coined
the terms "mortal" and "olympian" (as both adjective and noun) for
these two extremes of accuracy. 

Obviously a notation for JI can have fewer symbols if greater errors
can be tolerated. The general approach is to define symbols only for
the most common ratios and to reuse these symbols for less common
ratios which are sufficiently close.

But we'll never make everyone happy with one such notation. The
olympians would complain about the errors or ambiguities in a notation
designed for mere mortals, while the mortals would complain about the
number and complexity of the symbols in a system designed for olympians. 

A system about midway between mortal and olympian we call "herculean".
A herculean notation will of course have too many symbols for mortals
and too much ambiguity for olympians. So we've decided to specify
three systems. The trick is in making the transition from one to the
other as seamless as possible; to have as much in common between the
three as possible.

I'm afraid we're progressing rather slowly (but surely) on this at
present because we have both had (and are still having) long periods
where (heaven forbid) we had to concentrate on our jobs and families.
We are also working with Manuel Op de Coul on a Scala implementation
of the notation system.

Regards,
-- Dave Keenan


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Message: 6859

Date: Fri, 06 Jun 2003 14:21:05

Subject: Need help finding unusual linear temperaments

From: David C Keenan

Hi guys,

George Secor and I could use some help with the sagittal notation project.

We're approaching the notation of ratios from a new direction. We expect it 
will give the same results as before at the level typically used by mere 
mortals, but we're now going totally off the deep end and attempting a JI 
notation suitable for use by the gods on Mount Olympus. ;-)

It will have a maximum error of around half a cent (say 0.4 to 0.9 c). It 
will not be based on an equal division of the octave.

What we're looking at are planar temperaments. Specifically planar 
temperaments where one generator is fixed at the just fifth (and of course 
the octave is fixed too). It seems to me that this is equivalent to a 
_linear_ temperament where the "interval of equivalence" is the apotome 
3^7/2^11 (~ 113.685 cents). In any case, what we really want to know is how 
best to divide up the apotome in a relatively even (Rothenberg proper?) 
manner, from minus half an apotome to plus half an apotome, so as to 
closely approximate many commas for many popular ratios.

We've found 3 such temperaments by ad hoc means, but we'd really like to 
know that we haven't missed any really good ones. What we need is a list of 
(minimax) generators and periods to examine (where the period is either the 
apotome or an integer fraction of the apotome, and the generator is smaller 
than half the period. For example, one such generator is, perhaps 
unsurprisingly, an approximate syntonic comma (of about 21.57 cents) with a 
whole apotome period. Another is an approximate minor diesis of about 41.1 
cents in a whole apotome, and another is an approximate syntonic comma in a 
1/3-apotome period.

It is reasonably safe to assume a 23 prime-limit, although 31 would 
probably be better. And you can assume an odd (actually 
factor-of-2-and-3-free) limit of 625, although 1225 would be better. If 625 
is too hard I'd settle for 385.

Ideally I'd supply you with a list of commas, giving each a popularity 
rating but you can probably come up with a reasonable surrogate 
mathematical function for this yourself. For example a weighted sum of 
prime exponents such that these ratios are approximately equally popular
175/1
19/1
245/1
13/7
625/1
23/1
49/25

The problem is, while we have stats on the popularity of ratios (from the 
Scala archive), what we really need is the popularity of their commas.

Take any 2,3-free ratio and add factors of 2 and 3 in such a way as to 
produce a ratio in the range -apotome/2 to +apotome/2 and you have a comma 
for the original ratio. There are an infinite number of possible commas for 
any ratio, however the relative popularity of the different commas will be 
strongly inversely related to their exponent of 3. Most commas with an 
absolute 3-exponent greater than 12 can be ignored. All commas with a 
3-exponent greater than 17 can be ignored. There are rarely more than 3 
commas of significance for any ratio.

A good temperament for our purpose then is one that gets close to many 
popular commas without having to iterate the generator too many times.

We're not interested in chains of more than 160 generators, (meaning N 
chains of 160/N generators). And there is unlikely to be anything useful 
with fewer than 45 generators. Minimax errors should be less than 0.9 cents 
for the X most popular commas where X is greater than 20, and less than 0.5 
cents for the Y most popular commas where Y is greater than 10, and we are 
interested in what these values of X and Y are for each temperament.

Can anyone help? Do you need more info?

Regards,


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Message: 6862

Date: Fri, 06 Jun 2003 23:58:45

Subject: Re: Need help finding unusual linear temperaments

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <D.KEENAN@U...> 
> wrote:
> 
> > You have misunderstood me. I'm sorry I wasn't clearer. I hope the
> > above helps. Octaves are not important.
> 
> this is going to be a bizarre notation indeed! without octave 
> equivalence???

Oh dear. I almost despair of communicating my intentions here if _you_
are not getting it Paul. But this certainly is not the kind of "linear
temperament" any of us have ever considered before. So I shouldn't be
surprised.

Let me reassure you that the notation has octave equivalence. It is
the same sagittal notation whose essential details were worked out on
this list over a year ago and have not changed since. The letter-names
and sharps and flats are based on octaves and fifths as you would expect.

What I am investigating here is what happens _after_ you have chosen a
letter and sharps or flats for a ratio, and deals only with the comma
inflections.

For this purpose, I am looking only at the region within half an
apotome either side of any 3-limit pitch and looking at dividing up
this space by considering the proper moments-of-symmetry of a
comma-sized generator iterated modulo this apotome.

> > The single most popular comma in the entire universe (for notational
> > purposes such as ours) is of course the syntonic comma 81/80. Then 
> the
> > septimal comma 64/63. Then the double syntonic comma 6561/6400.
> 
> how did that leapfrog over the diaschisma 2048/2025??

It seems to me that folks prefer to notate ratios of 25 as stacked
ratios of 5, e.g. 1:5:25 as C E\ G#\\ rather than C E\ Ab'\ where \ is
syntonic comma down and '\ is diaschisma down. (By the way, the
apostrophe ' can also be read, when on its own, as schisma up).

But lets not get bogged down in details like this. I don't mind if you
want to rank the diaschisma as more popular (for comma-inflected
notation) than the double-syntonic comma. It is still likely to
benefit from the same kind of apotome-MOS or
linear-temperament-of-the-apotome.

Regards,
-- Dave Keenan


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Message: 6864

Date: Fri, 06 Jun 2003 07:01:32

Subject: Re: Need help finding unusual linear temperaments

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
>  It seems to me that this is equivalent to a 
> > _linear_ temperament where the "interval of equivalence" is the
apotome 
> > 3^7/2^11 (~ 113.685 cents). 
> 
> This makes zero sense to me. Should I give examples of planar
> temperaments?

OK. Forget the planar temperament stuff. The things I'm interested in
are mathematically like a _linear_ temperament, but instead of
repeating at the octave they repeat at the apotome, and instead of
approximating simple-ratio JI harmonies they approximate popular
notational commas.

This is for the purpose of notating JI by means of comma-inflection
symbols applied to a chain of Pythagorean fifths. We want maximum
accuracy for the maximum number of popular commas with the minimum
number of symbols.

We are just concentrating on the region within +-1/2-apotome of a
single note in the chain of fifths.

> > We've found 3 such temperaments by ad hoc means, but we'd really
> like to 
> > know that we haven't missed any really good ones. What we need is a
> list of 
> > (minimax) generators and periods to examine (where the period is
> either the 
> > apotome or an integer fraction of the apotome, and the generator is
> smaller 
> > than half the period.
> 
> If you are using exact octaves, the period can only be an approximate 
> apitome. However, it is certainly possible to look for such beasts.

You have misunderstood me. I'm sorry I wasn't clearer. I hope the
above helps. Octaves are not important. Exact apotomes are. We could
divide the apotome into equal pieces and call the resulting object an
EDA (equal division of the apotome), but equal divisions are not
necessary, all we need is an approximately even (proper) division, and
we can probably get greater accuracy that way.

> > The problem is, while we have stats on the popularity of ratios
> (from the 
> > Scala archive), what we really need is the popularity of their commas.
> 
> Why not simply take lists of commas which pass a goodness test instead?

I'm certainly open to suggestions. Goodness for what? Actually I don't
need to know what. Just post a list and I'll tell you if I think they
are roughly in popularity order.

The single most popular comma in the entire universe (for notational
purposes such as ours) is of course the syntonic comma 81/80. Then the
septimal comma 64/63. Then the double syntonic comma 6561/6400. After
that it starts to get a little hazy with regard to the commas, but
decreasing popularity of the ratios which the commas are meant to
notate start off like this (with factors of 2 and 3 removed).

1/1
5/1
7/1
25/1
7/5
11/1
35/1
125/1
49/1
13/1
11/5
11/7
17/1
25/7
49/5
13/5
175/1
19/1
245/1
13/7
625/1
23/1
49/25
55/1
77/1
17/5
19/5
35/11
13/11
31/1
343/1
29/1
125/7
55/7
17/11
77/5
19/7
385/1
55/49
17/7
1225/1
37/1
121/1
23/5
19/13
17/13
23/7
25/11

Another reason for looking at these linear temperaments of the apotome
is that they may provide us with a series of proper divisions of the
apotome that increase in resolution but belong to a common "family".
This is so that we can provide a number of notations that trade off
accuracy for number-of-symbols, in different ways, while making it
easy to move between them. Each notation would correspond to a proper
MOS of the same LTA (linear temperament of the apotome).

I note that the first 7 ratios above account for 71% of all ocurrences
of ratios in the Scala archive (when factors of 2 and 3 are removed).
So ideally their commas would all be represented in a reasonably short
chain of generators (say 20 or less).


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Message: 6866

Date: Fri, 6 Jun 2003 16:39:28

Subject: Re: Interval Database Experiences

From: Manuel Op de Coul

Alex, see this:
Stichting Huygens-Fokker: List of intervals *
There's a link on the bottom to another page of Dave
for more info.

Manuel


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Message: 6867

Date: Mon, 09 Jun 2003 17:08:31

Subject: equal-beating temperaments

From: Carl Lumma

Gene,

So what's the deal with the synchronous versions of
temperaments?  Can you give a rundown of of some of
them -- metameantone, wreckmeister (?), wendell nat.
sync well, sync. vallotti... are they all based on
the beat ratio btwn. maj and min 3rds?  Have you been
able to hear a difference?  I think metameantone is
the best-sounding meantone I've heard, but I haven't
given a close listen to other sync. temperaments yet.

Maybe we should have a midi file comparing triads
with various beat ratios?

-Carl


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Message: 6868

Date: Wed, 11 Jun 2003 14:27:34

Subject: Re: Interval Database Experiences

From: Manuel Op de Coul

Alex wrote:
>Now what's your oppinion, should the terminology be redefined and 
>form a new tradition, like Dave is working on, or should one bother 
>on the collection of historycal names that are available out there?

No opinion about what should be done, but I felt more for making a
list of historical names, and Dave for making a consistent terminology.

>and that other names 
>belonging to not that important terminologies are irrelevent to you 
>people, is that right? 

Well there's an infinite amount of interval ratios, there's no point in
trying to find a name for all of them.

Manuel


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Message: 6870

Date: Wed, 11 Jun 2003 13:52:33

Subject: Re: Transformation

From: monz

hi Ben,


> From: "bdenckla" <bdenckla@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, June 11, 2003 12:58 PM
> Subject: [tuning-math] Re: Transformation
>
>
> If you're interested in Regener, my Master's thesis 
> presents his work, among other things:
> 
> No Title *
> 
> I've struggled to get a small subset of this work,
> in particular my streamlined presentation of Regener's
> work, published with no luck.



wow, thanks!  i'll enjoy reading this!
good luck with publication ... keep trying.



-monz


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Message: 6872

Date: Fri, 13 Jun 2003 18:17:54

Subject: Re: Interval Database Experiences

From: Manuel Op de Coul

>but I still wonder if that 
>fokker list is complete for this matter, any clues?

I'm not aware of missing important intervals, and open
to any corrections.

Manuel


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