Yahoo Tuning Groups Ultimate Backup tuning RE: summary of Woolhouse's _Essay on Musical Intervals.. .

Bless you, Joe, for digging this up!

>He makes a statement about his prime- or odd-limit
>(he doesn't specify which interpretation of 'limit'):

>> [Woolhouse 1835, p 8]
>>
>> It has been found by experience, that proportions exceeding
>> the number 5 are generally discordant, as the coincidences
>> become then so very seldom. Our consonances will thus be
>> limited to the proportions -
>>
>> 4/5, 3/4, 2/3, 3/5, 1/2

Joe, clearly Woolhouse means an _integer_ limit of 5, not a prime- _or_
odd-limit.

>[p 24] Then follows Woolhouse's crucial statement concerning
>JI versus temperament, without mentioning commatic drift:

>> [Woolhouse 1835, p 24]
>>
>> This difference in the note D, which in theory is a comma,
>> is entirely done away with in practice, as the harmonic
>> advantages which could be derived from the true theoretical
>> scale, as directed by nature, would by no means compensate
>> the difficulties of its performance.

Commatic drift or shift would be two ways of addressing these difficulties.

>> [Woolhouse 1835, p 41]
>
>> This scale [12-tET] is, without doubt, the best one for
>> such instruments as the common pianoforte, organ, &c. which
>> must necessarily have but one sound for both a sharp and the
>> flat of the next upper note.

>I don't really understand why he says that, because all thru
>the rest of the book he stresses that the goal of his work
>is to find a good tempered approximation to 5-limit JI which
>will give a practicable closed system, and he will go on to
>choose a meantone and several ETs which all fulfill these
>wishes better than 12-tET. But there it is.

Joe, it's not too hard to understand what Woolhouse is saying. Read it
again.

(a) most meantones are not closed systems
(b) ETs other than 12-tET do not have "but one sound for both a sharp and
the flat of the next upper note". In other ETs, Ab is different from G#, for
example.

>In what instance
>must a keyboard 'necessarily have but one sound for both a
>sharp and the flat of the next upper note'?

If it has only 12 notes per octave!

>An emphasis on that sentence seems to me to betray a slyly
>indignant way of implying that if one chooses to have an
>instrument specially built with more than 12 keys, then
>there are other temperaments that are better.

So you do understand? But what's slyly indignant about it?

>And here he utters the most important sentence in the book
>(and the basis of Paul's admiration of his work), where he
>sets out:

>> [Woolhouse 1835, p 45]
>>
>> ... to ascertain the particular values which must be assigned
>> to the *tone* and *diatonic semitone*, so that all the concords
>> shall be affected with the least possible imperfections; and
>> this we shall effect by the principle of least squares. We
>> must first observe, that the *third-minor*, *third-major* and
>> *fourth*, are the only concords necessary to be considered,
>> because the others are merely the inversions of these, and
>> we know that any error which may increase or diminish a concord,
>> will have precisely the same effect in decreasing or increasing
>> its inversion, as the octave, which is composed of them both,
>> is unchangeable.

>(This is the same method Paul used to discover his 7/26-comma
>meantone tuning.)

Right.

>Putting that on a lattice for a geometric view of the situation:

> E
> / \
> M3 m3
> m6 M6
> / \
> C -P4/P5- G

You could just as easily have depicted a minor triad instead.

>It should be obvious that the 5-limit lattice can be extended
>infinitely in both dimensions by adding on additional cells
>just like this one (or parts of it), thus proving Woolhouse's
>statement.

Not sure what it proves.

>(I have added ratios and Semitones, and for comparison, the
>cents-values for his optimal 6/13-&-9/26-comma):

> 50-tET 6/13-&-9/26-comma meantone

Joe, I don't know why you're using such a strange name for Woolhouse's
optimal temperament (is that what Woolhouse called it?). In 1/6-comma
meantone temperament, the 10/9 is tempered by 2/3-comma, and the 16/15 by
negative 1/6-comma. Do we call it "2/3-&-negative 1/6-comma meantone"? No,
we simply call it 1/6-comma meantone, after the amount by which the fifth is
tempered. Similarly, in 7/26-comma meantone, the 10/9 is tempered by

1 - 2*(7/26) comma
= 1 - 7/13 comma
= 6/13 comma

and the 16/15 is tempered by

5*(7/26) - 1 comma
= 35/26 - 1 comma
= 9/26 comma

So what Woolhouse discovered was simply 7/26-comma meantone. I'm glad he and
I agree!

> [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 Woolhouse here referring to 50-tET or 7/26-comma meantone? Smith's ideal
tuning was almost exactly 5/18-comma meantone, which is close to 50-tET but
on the other side of it relative to 7/26-comma meantone.

>He then analyzes the resources of a 53-tET '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-tET.

53-tET would also be subject to the same objection he levelled on JI, while
19-, 31-, and 50-tET would not.

>He certainly noted the disadvantages of strict 5-limit JI, yet
>he sought ultimately a temperament which would approximate it
>better than 12-tET.

And one that would eliminate the comma, which 12-, 19-, 31-, 43-, 50-, and
55-tET do, but 53-tET does not.

>Noticing that Woolhouse based his meantone not on a constant
>7/26-comma applied to all intervals as Paul did, but rather
>on the *two* basic intervals of the scale, the 'tone' and
>'semitone',

It of course amounts to the same thing.

>So it turns out that Woolhouse actually *did* describe
>Paul's 7/26-comma meantone, but not in that way. Woolhouse
>described it in terms of a '5L,2s' mapping, whereas Paul
>describes it based on a cycle of tempered 2:3s. Hmmm...

>Perhaps someone will bother to go thru the drudgery to
>show us exactly why it works out both ways; I'm certainly
>interested, especially in how 12/26 and 9/26 averages out
>to 7/26!

Again, in 7/26-comma meantone, the 10/9 is tempered by

1 - 2*(7/26) comma [because a 10/9 is two fifths up and a comma down, and
the direction of the tempering is opposite that of the fifth]
= 1 - 7/13 comma
= 6/13 comma

and the 16/15 is tempered by

5*(7/26) - 1 comma [because the 16/15 is five fifths up and a comma down]
= 35/26 - 1 comma
= 9/26 comma

>> the "optimal" meantone he derived (from 16th-18th century
>> musical considerations)

>As far as I recall, Woolhouse doesn't specifically mention
>anything about repertoire.

The construction of the diatonic scale, and the statement about the
difficulty in practice introduced by the two theoretical values of the note
D, are both considerations highly specific to Western commom practice
repertoire (including, I would say, most pop music).

>since we've all heard
>so much about how well 53-tET approximates both 3- and
>5-limit JI and relatively little about 50-tET, how about
>showing us exactly why 50-tET is better.

Again, it's the diatonic scale, and the fact that 53-tET (like 15-, 22-,
27-, 34-, and 41-tET) has two different versions of "D" while 50-tET (like
12-, ) does not. So everything Woolhouse says to admonish JI applies to
53-tET as well.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Joe Monzo wrote,

>730-tET as a basic unit
>.......................

>Towards the end of this chapter [p 18-19] comes Woolhouse's most
>original contribution: 730-tET, using 2^(1/730) as the basic unit
>of measurement for the comparison of different intervals - a
>precursor to Ellis's use of 2^(1/1200) = 1 cent in his 1875
>translation of Helmholtz.

>> [Woolhouse 1835, p 18]
>>
>> It will be useful to divide the octave into such a number
>> of equal divisions that each interval of the scale may comprise
>> an integral number of them ... such as will render the major
>> and minor-tones and limma whole numbers, since all other
>> intervals result from the various combinations of these
>> elemental ones.

>Note the emphasis on his 5-limit JI conception here.

730-tET can be found in Paul Hahn's
http://library.wustl.edu/~manynote/consist2.txt
(This chart only shows those ETs which have a higher consistency level at
some harmonic limit than all lower-numbered ETs, and goes up to 10000TET.)

In Paul Hahn's terminology, 730-tET is consistent at the 5-limit to level
22, meaning that any interval constructed of any combination of up to 22
consonant 5-limit intervals (minor thirds, major thirds, perfect fourths)
will be represented the same way whether the interval is first computed in
JI and then rounded to 730-tET, or if the consonant intervals are rounded to
730-tET first and then combined. By comparison, 12-tET is consistent at the
5-limit only to level 3: the ratio 648:625 (often called the greater
diesis) can be constructed from four 6:5 minor thirds; while four minor
thirds (minus an octave) add up to a unison in 12-tET, the actual size of
648:625 in JI is 62.565 cents, so it rounds up to one step in 12-tET. 53-tET
is consistent at the 5-limit to level 8.

Bosanquet's 612-tET is almost as good, as it is consistent at the 5-limit to
level 21. In order to improve on the consistency level of 730-tET in the
5-limit, one would have to go up to 1783-tET, which is consistent to level
42 (heh, I just said level 42), and then to improve on that, 4296-tET is
5-limit consistent to level 119. Note that 4296 is a multiple of 12. So if
you wanted to get a 12-tET-based synthesizer to provide ridiculously exact
JI (5-limit consonances within 0.002 cents of just), you could design it to
divide the semitone into 4296/12 = 358 equal parts . . . Being a little more
realistic, note that 612 is also divisible by 12 and gives 5-limit
consonances within 0.093 cents of just . . .

🔗Herman Miller <hmiller@xx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

Thanks to Dave Hill and Paul Erlich for expressing
their appreciation to me here for doing the summary on
Woolhouse. I'm certainly very glad I went thru the
trouble of getting a copy of this book - it's a small but
very valuable addition to my library!

And thanks very much to Paul for his valuable clarifications
and additions on what I had to say about Woolhouse. Following
are some further comments.

> [Paul Erlich, TD 447.17, in his initial response to the
> question of what type of limit Woolhouse had in mind]
>
> Joe, clearly Woolhouse means an _integer_ limit of 5, not
> a prime- _or_ odd-limit.

My initial response to that was:

Thanks for clearing that up, Paul. I think that's an
important point in his theory. Woolhouse blithely skims
over many things that your or I (or lots of others) would
probably elaborate much more fully.

But after reading Paul's subsequent observation, where
Woolhouse is about to set out his 'optimal meantone' tuning:

> [Paul Erlich, TD 447.23]
> I'd have to say that here (which is where it matters),
> Woolhouse is thinking of 5 as the _odd_ limit, since in an
> odd limit all intervals are considered equivalently to their
> inversions.

I'll say that I have to agree. This is clearly a 5-odd-limit.
Hmmm... then again, it *could* be a 5-prime-limit too,
couldn't it?... But in any case, it's *not* a 5-integer-limit.

>> [me, monz, TD 446.6]
>> [p 24] Then follows Woolhouse's crucial statement concerning
>> JI versus temperament, without mentioning commatic drift:
>>
>>> [Woolhouse 1835, p 24]
>>>
>>> This difference in the note D, which in theory is a comma,
>>> is entirely done away with in practice, as the harmonic
>>> advantages which could be derived from the true theoretical
>>> scale, as directed by nature, would by no means compensate
>>> the difficulties of its performance.
>
> [Paul, TD 447.17]
> Commatic drift or shift would be two ways of addressing these
> difficulties.

It's obvious that he's talking about commatic drift or shift
- I think the difference is negligible in Woolhouse's case,
since his point is to get rid of it -, and he does explain the
retunings that would be necessary on the second degree of the
scale (the 'D' in C major here) to clear up the problems, but,
at least in *this* book, he never actually comes right out and
*says* that commatic drift is the problem.

In the posting from Dave Hill that originally started all
this, where he quotes Fischer quoting Woolhouse, Woolhouse
says about JI:

> Singers and performers on perfect instruments (i.e., not
> fixed pitch: violins, etc.) ... on arriving at the same
> notes by different routes, would be continually finding
> a want of agreement.

So here he clearly expresses the musical movement that would
cause commatic drift. But this quote is not in the _Essay
on Musical Intervals..._ - at least I never found it there.
It must appear in something else Woolhouse wrote.

>>> [Woolhouse 1835, p 41]
>>>
>>> This scale [12-tET] is, without doubt, the best one for
>>> such instruments as the common pianoforte, organ, &c. which
>>> must necessarily have but one sound for both a sharp and the
>>> flat of the next upper note.
>>
>> [me, monz]
>> An emphasis on that sentence seems to me to betray a slyly
>> indignant way of implying that if one chooses to have an
>> instrument specially built with more than 12 keys, then
>> there are other temperaments that are better.
>
> [Paul, TD 447.17]
> ... what's slyly indignant about it?
>

Well... the whole point of Woolhouse's book is to present
and explain tunings that he feels are better than 12-tET
(or 12-EDO), and considering the fact that he goes thru all
the trouble of determining 7/26-comma meantone, and finding
the more practical 50-EDO close approximation to it, and then
finally accepts 19-EDO as more practical still, it seems
awfully strange that he uses such strong language when he
says 'This scale [12-tET] is, *WITHOUT DOUBT* [emphasis mine],
the best one for such instruments...'.

Sounds to me like, just for that one moment, he was succumbing
to the status-quo simply because of the improbability that anyone
would go thru the trouble to build an instrument to his
specifications. Maybe it's just the way I'm reading it.

But I do think that it points out one of the things Dave
was originally discussing: that this book emerged at a time
(1835) when 12-tET had still not fully 'conquered' the musical
scene. Woolhouse clearly felt the need to express a very
positive opinion about 12-tET *given certain conditions*,
even tho he knew there were other, better alternatives which
required different conditions.

In fact, 12-EDO may not yet have been the status-quo in
England. From what I've read, England seems to have been
the last country in Europe to generally adopt 12-EDO,
and so perhaps he was really making a legitimate push for
it here. But he certainly would have appreciated a more
inventive approach to building instruments, because he
clearly preferred 19-EDO.

>> [me, monz]
>> Putting that on a lattice for a geometric view of the
>> situation:
>>
>>
>> E
>> / \
>> M3 m3
>> m6 M6
>> / \
>> C -P4/P5- G
>>
>
> [Paul, TD 447.17]
> You could just as easily have depicted a minor triad instead.

Absolutely true - but all of Woolhouse's JI descriptions are
based on the 5-limit 'major' scale and 'major' chords. He
certainly implicity included 'minor', but doesn't *explicity*
say anything about it, except concerning the commatic problems
of the 'D' second degree. I was just presenting his own
explanations in diagrammatic form.

>> [me, monz]
>> It should be obvious that the 5-limit lattice can be extended
>> infinitely in both dimensions by adding on additional cells
>> just like this one (or parts of it), thus proving Woolhouse's
>> statement.
>
> [Paul, TD 447.17]
> Not sure what it proves.

It proves what he said in the part I quoted just before that:

> [Woolhouse 1835, p 45]
>
> the *third-minor*, *third-major* and *fourth*, are the only
> concords necessary to be considered, because the others are
> merely the inversions of these, and we know that any error
> which may increase or diminish a concord, will have precisely
> the same effect in decreasing or increasing its inversion

You can see plainly on the lattice that all these intervals
and their inversions can be added on infinitely in any direction
without involving any other intervals, which was his point.
He uses these three intervals in his derivation of the
7/26-comma meantone.

Tell you what... I'll make a separate post quoting his
mathematics for this derivation.

>> [me, monz]
>> his optimal 6/13-&-9/26-comma
>
> [Paul, TD 447.17]
> Joe, I don't know why you're using such a strange name for
> Woolhouse's optimal temperament (is that what Woolhouse
> called it?).

No - he actually didn't call it anything at all. He says,
by way of introducing it:

> [Woolhouse 1835, p 41]
> A system of sounds may, however, be formed, in which all the
> keys are more nearly approximate than that of equal semitones
> [= 12-EDO] ...

And then, after he's described the math for his 'optimal meantone'
(that's either your name or Mandelbaum's), he calls its close
cousin 50-EDO [p 46] 'decidedly the most perfect of any systems
in which the tones are all alike' [i.e., in which the 81/80
(syntonic comma) vanishes]. But he never really puts a
name on that meantone tuning.

Woolhouse segues *very* quickly after that into 50-EDO, and
never mentions that meantone again anywhere else in the book,
so I really don't think he ever actually envisioned its use
at all in practice. It's just a mathematically optimum
tuning, which he certainly thought could be well enough
represented by 50-EDO (with less than 2-cents error from
the meantone) if anyone actually wanted to try to build an
instrument in *that* tuning, and which he clearly felt
19-EDO (with up to ~16 cents error from the meantone) came
close enough to for practical purposes.

(I've added to my webpage a table of cents values and error
from the 7/26-comma meantone for 19-EDO.)

I didn't realize, until I carried the calculations of both
his scale and your 7/26-comma meantone out to 12 decimal
places, that they were indeed exactly the same tuning.
So that weird name is just my earlier description of it,
which I suppose I should change to '7/26-comma'. I certainly
appreciate your mathematical explanation that shows that
they are both the same. Thanks!

>> [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.
>
> [Paul, TD 447.17]
> Is Woolhouse here referring to 50-tET or 7/26-comma meantone?

He's referring here to 50-tET (which I'd rather call 50-EDO),
is Dr. Smith's 'scale of equal harmony'.

A side note: those who have looked at my webpage of this will
see that I've changed 'tET' to 'EDO' in every case. I did
that because Woolhouse specifically states that the 'octave'
is always to be a 1:2 ratio.

> [Paul, TD 447.17]
> Smith's ideal tuning was almost exactly 5/18-comma meantone,
> which is close to 50-tET but on the other side of it relative
> to 7/26-comma meantone.

How do you know that?! Do you have access to Smith's book?!
Tell us more!!

>> [me, monz]
>> He certainly noted the disadvantages of strict 5-limit JI,
>> yet he sought ultimately a temperament which would approximate
>> it better than 12-tET.
>
> [Paul, TD 447.17]
> And one that would eliminate the comma, which 12-, 19-, 31-,
> 43-, 50-, and 55-tET do, but 53-tET does not.

Yes, I realized after I put the posting together that Woolhouse's
objection to 53-tET would be the differentiation of the comma.

But still, he likes its close approximations to JI well enough
to keep including it (and 31-EDO) in his tables thru-out the
rest of the book.

The webpage is at:

http://www.ixpres.com/interval/monzo/woolhouse/essay.htm

I'll add this to it too. Paul, let me know if you mind my
adding your postings to the webpage - I think they're
valuable commentary. The more math, the better.

I'd love to be able to figure out an accurate lattice diagram
of the actual Woolhouse/Erlich 7/26-comma meantone, to accompany
the simple one of its 5-limit JI implications. Actually, my
lattice formula can probably already do it...

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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RE: summary of Woolhouse's _Essay on Musical Intervals.. ._

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Herman Miller <hmiller@xx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>