(Originally posted to the Onelist Tuning List, appearing in
Onelist
Tuning Digest # 446, message 6.
Revised and updated for this webpage.)
From: Joe Monzo
There has been discussion here over the past few days about
the theories of Wesley S. B. Woolhouse. Here's a summary
of his book.
Recent references to Woolhouse on the Tuning List
[Dave Hill, TD 439.10:]
At present, although most of us see the belief in the
absolute supremacy of integers as applied to music to be
naive, it also seems, as Paul Erlich has said with regard
to simultaneous notes, that integer ratios or very near
integer ratios (closer than called for by 12-EQT) correspond
to a psychological reality as regards our perception of music.
I believe that Mr. Woolhouse, in stating that equal
temperament is rightly the long established basis of tuning,
really went too far and overlooked the fact of experience
that the deviations of equal temperament from the just ratios
are so large that they really do have an appreciable effect
on the sound of music performed in equal temperament.
Some of Mr. Woolhouse's contemporaries strongly disagreed
with him, too.
[Paul Erlich, TD 439.11:]
Second and more importantly, I think Dave Hill has missed
the importance of the last quoted statement (from Woolhouse)
above. Many if not most common-practice musical passages
performed in just intonation would result in contradictory
tunings for the same written pitch. Whether or not Mr.
Woolhouse went "too far", Mr. Hill has failed to address
Woolhouse's point here. Furthermore, far from overlooking
the errors from just intonation, Woolhouse sought the best
way to reduce them while preserving the musical meaning
of the notes in the Western tradition.
To: tuning@onelist.com
Subject: Woolhouse's Essay on Musical Intervals...
Contents
Woolhouse's book
Introductory
My conclusions
Sound
Musical intervals
5-limit JI
Harmonics
The 'mean semitone'
The basic JI intervals for scale construction
730-EDO as a basic unit of measurement
Problems with JI
Temperament
12-EDO
Beats of Imperfect Concords
The optimal temperament: 7/26-comma meantone
50-EDO
31-EDO
19-EDO
53-EDO
Miscellaneous Additions
References
Recent references to Woolhouse on the Tuning List
A spokesman for the new view is the musical mathematician
W. S. B. Woolhouse, who wrote in the 19th century: "It is
very misleading to suppose that the necessity of temperament
applies only to instruments which have fixed tones. Singers
and performers on perfect instruments must all temper their
intervals, or they could not keep in tune with each other,
or even with themselves; an on arriving at the same notes
by different routes, would be continually finding a want
of agreement. The scale of equal temperament obviates all
such inconveniences, and continues to be universally accepted
with unqualified satisfaction by the most eminent vocalists;
and equally so by the most renowned and accomplished performers
on stringed instruments, although these instruments are capable
of an indefinite variety of intonation. The high development
of modern instrumental music would not have been possible,
and could not have been acquired, without the manifold
advantages of tempered intonation by equal semitones, and
it has, in consequence, long become the established basis of
tuning."
Note that Wesley Woolhouse was a prominent advocate of
19 tone equal temperament.
First of all, my knowledge of Woolhouse's theories suggests
that he viewed some form of meantone temperament to be ideal.
According to Mandelbaum, Woolhouse derived an optimal meantone
tuning (I believe it was the squared-error optimal tuning for
the three 5-limit consonances
, namely 7/26-comma meantone),
and decided that 19-tone equal tempermant was a close enough
approximation, and one which gave to the musician the desirable
properties of a closed system which were giving 12-equal its
rise to prominence at the time. In fact, 31- or 50-tone equal
temperaments would be better approximations (though not as
convenient from a practical point of view), and neither of
those tunings commits any errors larger than 6 cents in any
of the classic (5-limit) consonant intervals. 19-equal has
major thirds and perfect fifths that are over 7 cents off.
12-equal, of course, has major thirds 14 cents off and minor
thirds 16 cents off.
[Dave Hill, TD 440.1:]
I took the quote from Woolhouse from the book: "Piano Tuning" by J. Cree Fischer. The book was originally published in 1907. I have the Dover edition of 1975. The author, Fischer, introduces the quote from Woolhouse as follows (p. 144 of the Dover edition):"That the equal temperament is the only practical temperament, is confidently affirmed by Mr. W. S. B. Woolhouse, an eminent authority on musical mathematics, who says:- 'It is very misleading to suppose that the necessity of temperament applies only to instruments which have fixed tones...' - rest of longish quote -
Apart from the quotation in Fischer, I know nothing about W. S. B. Woolhouse excepting that he lived in the 19th century and that he is mentioned in Ellis' translation of Helmholtz in connection with a 19 tone equal temperament. In posting the quote to the tuning list, I assumed Mr. Fischer's reliability in conveying Mr. Woolhouse's point of view. Is it possible that Mr. Woolhouse had done studies on 19 tone equal temperament, but nevertheless emphatically advocated the more usual 12 equal temperament for practical musical performance?
[Carl Lumma, TD 440.7:]
[Paul Erlich wrote...]Whether or not Mr. Woolhouse went "too far", Mr. Hill has failed to address Woolhouse's point here. Furthermore, far from overlooking the errors from just intonation, Woolhouse sought the best way to reduce them while preserving the musical meaning of the notes in the Western tradition.This would unquestionably involve meantone. But there are two points I'd like to make here: [etc...snip]
[Paul Erlich, TD 441.16:]
Dave Hill wrote,Is it possible that Mr. Woolhouse had done studies on 19 tone equal temperament, but nevertheless emphatically advocated the more usual 12 equal temperament for practical musical performance?Quite possible, given what composers of his time were trying to do (by the 19th century, most composers thought of G# and Ab, etc., as entirely interchageable and used such enharmonic equivalency to modulate around fractions or the entirety of the circle of fifths). However, his advocacy of 19-tone equal tempermant, though it would not allow such practices, clearly had a very "practical" component, since the "optimal" meantone he derived (from 16th-18th century musical considerations) was essentially 50-tone equal temperament, and the only possible reason for suggesting 19 instead of 50 would be a practical one of actually getting all those notes onto instruments.
The references here to Woolhouse are all from secondary sources: Dave Hill quoting Fisher 1907, and Paul Erlich summarizing Mandelbaum 1961.
I just got myself a copy of Woolhouse's little book yesterday, and as I seem to be the only one on the List who has it, thought it would be good to give a brief summary of the book.
Here's the full citation:
Woolhouse, W. S. B. 1835. Essay on Musical Intervals,
Harmonics, and the Temperament of the Musical Scale, &c..
J. Souter, London. xii + 84 p.
This small book is divided into 48 'articles', which are grouped into 6 chapters:
Sound
Musical Intervals
Harmonics
Temperament
Beats of Imperfect Concords
Miscellaneous Additions
As most people on this List are familiar with basic ideas
in tuning theory, I will concentrate mainly on Woolhouse's
2nd and 4th chapters, giving a detailed examination of his
thoughts concerning intervals and temperament.
The first chapter (only p 1-2) sets out the basic concepts
concerning sound.
The second chapter begins [p 3-7] with definitions of important
terms regarding vibrations, sonance, and measurement of intervals,
explains [p 8-9] how to measure the string-lengths of the
'important consonances' of the basic 5-limit JI scale.
He makes a statement about his prime- or odd-limit
(he doesn't specify which interpretation of 'limit'):
[Woolhouse 1835, p 8:]
4/5, 3/4, 2/3, 3/5, 1/2
[p 10] Woolhouse devises the following pentatonic scale from
the basic 5-limit JI ratios:
Woolhouse then observes that 240/180 = 360/270 (= 4/3)
and divides this scale into 2 'tetrachords' (even tho
they only have 3 notes at this point) which have dissimilar
spacing, then adds 'D' and 'B' to the scale to 'equalize'
each tetrachord (i.e., give them identical intervallic structure),
using the following ratios:
which produces the usual 5-limit JI 'major' scale.
Adding these two notes to our lattice gives us:
[p 12] Woolhouse defines 'mean semitone' as 2^(1/12). He
shows the mean semitone values of the JI intervals, almost
exactly the same way I use Semitones (the exception being
that he carries the decimal part further than 2 places).
He explains [p 11-16] how to calculate string-lengths of the
12-EDO scale for comparison to the just ratios.
Then he defines [p 17] the following basic intervals (I have
added the tilde [~] to indicate that a value is only approximate):
(Note that this definition of limma is quite different
from the usual Pythagorean one, where
it is equal to
256/243 [= ~0.90 Semitone]; Woolhouse's limma is in fact much closer to
the larger Pythagorean semitone, the apotome [= ~1.12
Semitones]. See my Tuning Dictionary.)
Towards the end of this chapter [p 18-19] comes Woolhouse's most
original contribution: 730-tET, or as I prefer
to call it, 730-EDO 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:]
Note the emphasis on his 5-limit JI conception here.
He examines some other notably accurate divisions of the 'octave':
[Woolhouse 1835, p 19:]
[p 20]
Woolhouse mentions likening the octave to a circle, and
using degrees, minutes, and seconds to measure the intervals.
This amounts to 1296000-EDO (yes, that's over a million degrees:
360 * 60 * 60), which Woolhouse dismisses as 'of no advantage
in musical computations'.
Then he mentions 301-EDO of Sir J. Herschel [paper on 'Sound',
vol 2 of mixed sciences, Encyclopedia Metropolitana], and
presents a table comparing the errors from JI for 301-, 53-,
and 730-EDO, which shows that
[Woolhouse 1835, p 20:]
Woolhouse chose this division precisely because he wished to
avoid using decimals or fractions, and the integer values
of 730-EDO designating the three basic intervals of 15:16, 9:10,
and 8:9 are so close to the actual JI values:
Woolhouse gives the more exact value (which I placed in the
last column of the first table), then notes how closely the integer
values (which I put in the second table) approximate them.
He will use the abbreviations and Greek letters in his math equations.
The conclusion of this second chapter:
[p 23] Woolhouse refers us back to where he selected an
arbitrary measurement for 'D' and 'B', the '2nd' and '7th'
of the scale, respectively, to 'equalize' the two tetrachords.
Now he emphasizes the arbitrariness of that choice, and says
that the '2nd' my be tuned either to 9/8, to fit it
into the harmonic series over the 'keynote', or 10/9, which
makes the intervallic structure of the tetrachords identical.
[p 24] Then follows Woolhouse's crucial statement concerning
JI versus temperament, referring to, but not mentioning, commatic drift:
[Woolhouse 1835, p 24:]
The third chapter [p 25-35] is an explanation of harmonics,
about which most here on the List know plenty already. Woolhouse gives
tables [p 30-31] showing harmonics up to the 20th, and the
'error' of the harmonics from the 'true diatonic' (i.e.,
5-limit JI).
Without making any claims for their use, he gives measurements
[p 33-35] which show how to get the set of 'natural harmonics'
on a violin, thus encouraging the reader to experiment.
The fourth chapter is Woolhouse's exploration of temperament,
which really forms the climax of the book.
He starts out [p 36] with an emphasis on 'a very approximate
diatonic series on the assumption of any one of them as a
key-note', whose 'practical formation' demands temperament.
First is an examination of 12-EDO, noting that its 'greatest
imperfections are those of the 3rd and 6th', with a maximum
of 9&1/2 'degrees' deviation
[(6/5) / 2^(3/12) {or 2^(9/12) / (5/3)}
= ~15.641 cents = 2^(~9.515/730)],
and after tabulating all the 'errors' from JI, he says that
[Woolhouse 1835, p 38:]
(2^(13/730) = ~21.370 cents;
comma = (81/64) / (5/4) = ~21.506 cents = 2^(~13.083/730).
By 'considerably less' Woolhouse is referring to
the ~5.865 cent difference between the syntonic comma
and the ~15.641 cents maximum deviation of 12-EDO from JI cited above;
in other words, the maximum deviations in 12-EDO do not
distort melodies as badly as the commatic deviations that
would be encountered in true JI performance.)
Note that here he emphasizes the perception of melody and
not of harmony. He also says [p 38-39] that any human tuner
is bound to commit small errors because of the imperfections
of his hearing, and that because of this, it is futile to
argue that 12-EDO produces a blandness which the unequal
temperaments don't have. (Sounds like Johnny Reinhard
saying 'people aren't perfect...12-equal is microtonal!...')
He explains very briefly [p 40-41] that 12-EDO can be
tuned by tempering the '5ths' 'about one degree [= 2^(1/730)
= ~1.64 cents] flat', because 'each fifth must contain
425&5/6 degrees':
(730 degrees / 12 semitones) * 7 semitones [ a '5th'] = 425&5/6 degrees.
3/2 = 2^(~427.023/730)
2^(7/12) = 2^(425&5/6 / 730)
The exact figure to flatten the '5th' is
(3/2) / 2^(7/12) = ~1.955 cents =
This may be done either with an ascending series of '5ths'
(C to A#, with the break in notation between A# and F, from
where the cycle repeats) or a descending series (which Woolhouse
again, this time strangely, notates as sharps), the latter
being the usual method used.
Finally on 12-EDO he concludes:
[Woolhouse 1835, p 41:]
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-EDO. But there it is.
I think examining that paragraph alone like this is a perfect
example of taking something out of context.
It seems to me at first like he's making a great praise that
ultimately turns out to be pretty empty. In what instance
must a keyboard 'necessarily have but one sound for both a
sharp and the flat of the next upper note'?
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. And indeed,
the rest of the book will pretty much bear this out.
He immediately continues by saying that a system of temperament
that gives two different sizes of semitone provides a much
better approximation to JI. Note that he wants only one size
for the tone, so that the comma will vanish, thus preventing
commatic drift.
[p 42] With echoes of
Marchetto, Woolhouse discusses the
'major' or 'diatonic semitone', the 'minor' or 'chromatic
semitone', and the 'enharmonic diesis', the latter being
the difference between the two former. Their relationships
can be summarized thus:
diatonic semitone > (9/8)^(1/2)
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:
(This is the same method Paul used to independently discover the
same 7/26-comma
meantone tuning.)
Putting that on a lattice for a geometric view of the situation:
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.
Then [p 43-45] he uses algebra to find the difference between
JI and this ideal temperament with τ (the Greek letter tau)
to represent the tempered tone and σ (sigma) to represent the
tempered diatonic semitone, and the other abbreviations presented
earlier for the JI intervals; these will make up the six intervals
on my lattice.
The result is:
[Woolhouse 1835, p 45:]
σ = θ + (9/26)c = 72&1/2 degrees
Or in English and math:
He notes that the relationship between these two is 'very
nearly in the ratio of 8 to 5'. More precisely:
Woolhouse then presents a simplified tuning which is a good
approximation to this:
[Woolhouse 1835, p 45:]
Woolhouse is assuming a diatonic scale with a basic interval
mapping of 5L,2s; that is, 5 large 'steps' and 2 small ones
(see L&s mapping in my Tuning Dictionary).
Solving the simple algebraic equation 5L + 2s = 50 gives
L = 2s - 2
s = L/2 + 1
L = 8 and s = 5
This produces a basic scale of 21-out-of-50 tones per octave,
which Woolhouse illustrates as follows with the number of 50-EDO
degrees between both the Diatonic and Chromatic scale members
(I have added ratios and Semitones, and for comparison, the
cents-values for his optimal 7/26-comma meantone, which I
had previously called "6/13-&-9/26-comma meantone"):
With deviations all under 2 cents, it is apparent that 50-EDO
is indeed a very close approximation to his optimal meantone.
[Woolhouse 1835, p 46:]
Based on this scale, Woolhouse gives a list [p 47] of
'all the major-keys which are necessary in music', which
can be summarized as:
and he proposes [p 48] to eliminate the keys of F#, C# and Cb major
as superfluous.
Woolhouse then gives [p 49] the same kind of description of
Huygens's 31-EDO, about which he says:
[Woolhouse 1835, p 49:]
Next he considers 19-EDO:
[Woolhouse 1835, p 50:]
and he gives a list of all the keys and a diagram of how it
could be produced on a keyboard.
Here is a table which provides the cents-values for 19-EDO
and the deviation of it from Woolhouse's optimal meantone tuning:
It is clear from this table that 50-EDO provides a much closer
approximation to the optimal meantone than 19-EDO, which indicates
that Woolhouse advocated this temperament purely for the practical
reason of instrument-construction limitations.
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.
Woolhouse concludes this chapter [p 56] with a table providing
exact monochord-string measurements for 12-, 19-, 31-, 50-, and 53-EDO.
The last regular chapter in the book, 'Beats of Imperfect
Concords', is a detailed examination of beats arising from
the 5 equal-temperaments listed in the table.
After elaborating on the most scrupulous methods for
experimenting with the tuning of a stretched string, including
laying it vertically rather than horizontally so the pressure
of the string on the bridge will not affect the tuning,
and putting the effect of the earth's gravity into his equation,
he says:
[Woolhouse 1835, p 64:]
So Woolhouse expresses his limit of accuracy here in Hz as
about 1/4 percent, which is about 4 cents in this frequency range.
He gives a table [p 65] showing the beat divisors for 12-,
19-, 31-, 50-, and 53-EDO, and at the end of this chapter
[p 66-68] gives the beats for all '5ths' in the 12- and
19-EDO systems.
The final section of the book, 'Miscellaneous Additions',
gives some specific information regarding various instruments,
from tuning forks to strings to winds, with observations
on how the latter two apply to pianos & harpsichords, and
organs, respectively.
I know nothing of any of Woolhouse's other work, but according
to this book, he seems to have only been resigned to accepting
the error involved in 12-EDO because of its ubiquity and
practicality.
In fact, looking thru the Table of Contents again, it seems to
me now like he ingeniously praised 12-EDO for its practicality,
and then demonstrated just that in his tuning guide, for a few
pages, just to 'hook' the reader into something familiar,
before his short but complex foray into optimum meantone and
then (with a quick segue) into temperaments that are a good
approximation of that tuning.
He certainly noted the disadvantages of strict 5-limit JI, yet
he sought ultimately a temperament which would approximate its
basic consonant intervals better than 12-EDO.
Knowing from Fischer 1907 only that quote posted by Dave,
I would venture to say that Fischer approved of 12-EDO
and quoted Woolhouse a bit out of context to make him
appear a champion of that tuning. [...confirm, Dave?]
But based on a reading of Woolhouse's Essay, it is clear
that he preferred several other temperaments which approximated
5-limit JI better than the 12-EDO system is capable of, with
19-EDO giving the 'most bang for the buck' in terms of the
mutually exclusive aims of better approximation to JI and
practicality on instruments, especially those of fixed pitch.
Woolhouse doesn't talk specifically about notation much,
but he mentions enough about those of 19- and 50-EDO to
offer that as another advantage over the more cumbersome
possibilities.
I had only scanned thru Mandelbaum 1961 once at Johnny
Reinhard's, reading certain small parts of it more deeply
because they piqued my interest. Most of what I remember
was my first encounter with Fokker's work, but if I did
read about Woolhouse and don't remember, I can say now
that based on what Paul wrote in TDs
439.12
and
441.16,
Mandelbaum gives an excellent summary of Woolhouse's book.
(If Paul did the summarizing, then kudos to you Paul!)
Citing Mandelbaum on Woolhouse, these statements are
absolutely correct:
[Paul Erlich, TD 439.12]
Absolutely.
Indeed it was - Woolhouse's description led me to believe
originally that I'd be discussing two different tunings right
here...
: ) (more on that below)
In a private email he's given me permission to share, Paul
said:
I was going to say: it's still 'no cigar', but I'd say it's
closer than an eyelash away. [Well, it turns
out that they are the same - see below.]
Take a look at this analysis:
The total difference between the sum of ratios in JI and those
in the meantone is 3 commas. This must be spread over all
notes in the scale to make the 'octave' be 1:2, which is
specified by Woolhouse.
The amounts added to the JI 'tone' and 'semitone' are
respectively 6/13- [= 12/26] and 9/26-comma.
3 commas, each divided into 26 parts = (3*26)/26
= 78/26 commas.
What I find most interesting about Woolhouse's approach
is that it is more along the lines of what I've just recently
gotten very involved in, via Dan Stearns: L&s mapping.
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', which are respectively the L and s intervals
in a '5L,2s' mapping, applying different amounts of tempering
to the two different intervals:
I saw that the total scale was:
(5*(10/9)*((81/80)^(6/13))) + (2*(16/15)*((81/80)^(9/26)))
Focusing on the addition of just the commatic parts
shows how Woolhouse divided the 3-comma discrepancy:
which (I don't know if there's any significance) can be
further reduced to:
(Paul can manipulate these numbers better, and perhaps show
some new insight into what Woolhouse is doing.)
But the 5L,2s mapping puts the '7' in there... don't know
if that means anything either, in regard to 7/26-comma meantone.
Now, the reason why I wrote earlier that 'I was going to say...':
Paul sent me a table of cents-values of his 7/26-comma
meantone diatonic 'major' scale carried out to three decimal
places, and it was exactly the same as Woolhouse's temperament
at that accuracy level:
I was intrigued by this and decided to do the calculation
myself and see how far I had to take the decimal place
to find a difference. After trying 12 decimal places,
I finally found a discrepancy of one digit at the end of
only one of the numbers, which I think was probably due
to rounding in my spreadsheet anyway.
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!
[Paul explained Woolhouse's derivation
of the meantone here, and an
explanation of the flattening of the '5th' is
here.]
[back to Paul]
Absolutely.
Again, 100% true.
In fact, 31- or 50-tone equal temperaments would be better
approximations (though not as convenient from a practical point
of view)
Both points made by Woolhouse.
[Paul, TD 441.16]
As far as I recall, Woolhouse doesn't specifically mention
anything about repertoire.
This is all absolutely true, except that I would say his
optimal tuning was the 7/26-comma meantone, and
even advocating 50-EDO was a bit of backsliding; due to
what, I don't know: he could have given beat-counting charts
for the meantone as well as the ETs. Perhaps this gives
a bit of weight to Dave Hill's hypothesis that Woolhouse
was more willing to accept the tuning status-quo than the
book makes him seem. But Paul (or Mandelbaum) is correct
in that 50-EDO is aurally virtually identical to the optimal
meantone.
A very intriguing little treatise. I'll be making a webpage
out of this in the next few days, and can probably make the
whole book into a set of webpages without too much work.
I'm pretty sure there's no copyright restriction on it by now.
Stay 'tuned' for updates.
Another tidbit: in his Introduction to Bosanquet 1987
[p 40 and 45], Rasch shows that earlier versions of Bosanquet's
book included summaries of Woolhouse's, which were deleted
from the published edition, thus proving that Bosanquet was
quite familiar with Woolhouse 1835.
And here's a challenge to Paul or any others of the
mathematically-inclined out there: since we've all heard
so much about how well 53-EDO approximates both 3- and
5-limit JI and relatively little about 50-EDO, how about
showing us exactly why 50-EDO is better. Or if it's not
better, then tell us what descriptions best characterize
the comparison of these two temperaments. The most important
difference I can discern is that the syntonic comma vanishes
in 50-EDO, while 53-EDO is the smallest EDO that clearly
differentiates it.
Smith, Robert. 1749.
Herschel, J. W. F. ?date?
Woolhouse, W. S. B. 1835.
Fischer, J. Cree. 1907.
Mandelbaum, M. Joel. 1961.
Bosanquet, R. H. M. 1876.
[Paul Erlich,
Onelist
Tuning Digest # 447, message: 17]
Bless you, Joe, for digging this up!
[Monzo:]
[Woolhouse 1835, p 8]
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.
[Monzo:]
[Woolhouse 1835, p 24]
Commatic drift or shift would be two ways of addressing these difficulties.
Joe, it's not too hard to understand what Woolhouse is saying. Read it
again.
[Monzo:]
If it has only 12 notes per octave!
[Monzo:]
So you do understand? But what's slyly indignant about it?
[Monzo:]
[Woolhouse 1835, p 45]
Right.
[Monzo:]
You could just as easily have depicted a minor triad instead.
[Monzo:]
Not sure what it proves.
[Monzo:]
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
and the 16/15 is tempered by
5*(7/26) - 1 comma
So what Woolhouse discovered was simply 7/26-comma meantone. I'm glad he and
I agree!
[Woolhouse 1835, p 46]
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.
[Monzo:]
53-tET would also be subject to the same objection he levelled on JI, while
19-, 31-, and 50-tET would not.
[Monzo:]
And one that would eliminate the comma, which 12-, 19-, 31-, 43-, 50-, and
55-tET do, but 53-tET does not.
[Monzo:]
It of course amounts to the same thing.
[Monzo:]
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]
and the 16/15 is tempered by
5*(7/26) - 1 comma [because the 16/15 is five fifths up and a comma down]
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 Erlich,
Onelist
Tuning Digest # 447, message: 24]
Joe Monzo wrote,
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.
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.
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 . . .
(posted by me to the Tuning List by me, monz, answering Paul:)
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.
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:
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.
[p 24] Then follows Woolhouse's crucial statement concerning
JI versus temperament, without mentioning commatic drift:
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.
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:
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.
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.
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.
... 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.
Putting that on a lattice for a geometric view of the
situation:
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.
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.
It proves what he said in the part I quoted just before 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
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, and no others,
in his derivation of the 7/26-comma meantone.
Tell you what... I'll make a separate post quoting his
mathematics for this derivation.
his optimal 6/13-&-9/26-comma
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:
A system of sounds may, however, be formed, in which all the
keys are more nearly approximate than that of equal semitones
['equal semitones' means 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!
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?
He's referring here to 50-tET (which I'd rather call 50-EDO),
as 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.
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!!
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.
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.
. . .
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...
Hey Paul,
I've written out all the steps in the derivation of
Woolhouse's optimal meantone. He skips most of these
in the book. With my limited knowledge of algebra,
I think it's remarkable that I've accomplished as much
as I did. But...
I need your help with one section, clearly marked,
about 2/3 of the way down. Thanks.
After you explain this to me, I'll post it to the List.
==================================================
FOR THE TUNING LIST:
First, here are the variables used in Woolhouse's equation.
I'm substituting Roman letters for Woolhouse's Greek ones,
which are lacking in ASCII, to keep everything as simple as
possible.
Some of the just relations can be expressed:
Woolhouse states that the 5L,2s 'octave' must = 1:2 ratio
thus, 'octave' == unison.
So let's examine the composition of the 'octave' in terms
of these basic-interval variables.
The 'octave' is broken down into these basic steps:
Therefore,
Subtracting the just from the tempered:
We can eliminate the variables S and s by solving for (S - s).
We get:
Now let's look at our three basic intervals.
Subracting the JI minor 3rd from the tempered minor 3rd,
we find the 'error':
Substituting what we found above for (S - s):
Subracting the JI major 3rd from the tempered major 3rd,
we find the 'error':
This has no need of further simplification.
Subracting the JI 4th from the tempered 4th,
we find the 'error':
Substituting what we found above for (S - s):
Now all three errors are in the same form:
Now to find to
RMS
[= 'root mean square'] total error of
the three intervals,
To determine the value of (T - t,), so that the
sum of the squares of these three errors shall be
the least possible, multiply them by the respective
coefficients -(3/2), +2, -(1/2), and the sum of
the products will be
(13/2)(T - t,) - 3c.
By putting this = 0, we find
(T - t,) = (6/13)c,
which used in the above value of (S - s), gives also
(S - s) = (9/26)c
Paul Erlich explained this crucial step to me in more detail:
You can't just use algebra to solve an optimization
problem You need calculus.
Let's look at the expression for the errors:
Let's simplify the notation by calling (T - t,) "x":
The sum of the squared error is thus:
To minimize this sum-of-squared errors as a function of x,
you set its "derivative" or rate of change with respect to x
equal to zero. That means that the rate at which the
sum-of-squared error is changing when x changes is zero,
which can only happen at a local minimum or a local maximum.
In this case we know it will be a local minimum.
The derivative of the sum-of-squared error with respect to x is:
Since we are setting this equal to 0, we can eliminate the
three "2*"'s:
Simplifying:
(9/4)*x - (3/4)*c + 4*x - 2*c + (1/4)*x - (1/4)*c = 0
or
(13/2)*x - 3*c = 0
= (13/2) * (T - t,) - 3c = 0
[Paul Erlich, private communication]
What the equation above actually represents is not
the sum of the squared errors, but a relationship
between the amount of tempering and the comma when
the amount of tempering is such as to minimize the
sum of the squares of the errors. And that relationship
comes from setting the derivative (or rate of change)
of the sum-of-squared error, with respect to the amount
of tempering, equal to zero, because that is a condition
that only occurs at the minimum.
Now we can calculate the amount of temperament for the tone.
Solving for (T - t,) gives:
There's our error, or the amount of tempering, for the tone.
So the tempered tone is:
T = (t, + ((6/13) * c))
For the semitone:
Solving for (S - s) gives:
There's our error, or the amount of tempering, for the semitone.
S = (s + ((9/26) * c))
In addition to knowing the tempering from the 'minor-tone' as
calculated above, it will be convenient to know the amount of
tempering for the tone from the 'major-tone':
t = (t, + ((13/13) * c))
tempered tone = just major-tone - 7/13 comma
Now, proving that this temperament is indeed Paul Erlich's
7/26-comma meantone, let's calculate the tempered '5th':
Which in numbers, of course, is:
(3/2) / ( (81/80)^(7/26) ) = ~696.165 cents
There it is.
I asked Paul Erlich to factor that as far as
could be done, and his answer was:
Comparing this with the "5th" of the 50-EDO recommended by
Woolhouse, we find a deviation of:
-monz
Updated: 2001.7.8, 2001.5.30, 2000.2.6, 2000.2.4, 1999.12.19
I welcome
feedback about this webpage:
Sound
Musical intervals
5-limit JI
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 -
note interval ratio string-length
C octave 2/1 180/360
A sixth-major 5/3 216/360
G fifth 3/2 240/360
F fourth 4/3 270/360
E third-major 5/4 288/360
C first/keynote 1/1 360/360
which can be illustrated
by this lattice diagram:
A---E
/ \ / \
F---C---G
note interval ratio string-length
B seventh 15/8 192/360
C second 9/8 320/360
A---E---B
/ \ / \ / \
F---C---G---D
The 'mean semitone'
The basic JI intervals for scale construction
interval ratio mean semitones
major-tone 9/8 ~2.0391
minor-tone 10/9 ~1.8240
limma 16/15 ~1.1173
730-EDO as a basic unit
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.
It has been proposed by some to divide the octave into 53
divisions, taking 9 of them for the major-tone, 8 for the
minor-tone, and 5 for the limma [the 15:16 semitone], which
furnishes a pretty accurate scale.
the last, which has been found by various trials, is that
which differs less than any other from the true series,
unless we ascend to very high numbers; and is the one which
is therefore most to be recommended'.
interval ratio mean semitones abbrev. log(ratio)*(730/log(2))
JI:
major-tone 9/8 ~2.0391 t ~124.045
minor-tone 10/9 ~1.8240 t, ~110.962
limma 16/15 ~1.1173 θ ~ 67.970
comma 81/80 ~0.2151 c ~ 13.083
730-EDO:
major-tone 2^(124/730) ~2.0384 t 124.000
minor-tone 2^(111/730) ~1.8247 t, 111.000
limma 2^( 68/730) ~1.1178 θ 68.000
comma 2^( 13/730) ~0.2137 c 13.000
(θ is the Greek letter theta.)
Problems with JI
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.
Harmonics
Temperament
12-EDO
These deviations, however, which are considerably less
than a comma or 13 degrees,
are too small to affect, in a very sensible degree, the
melody of the intervals.
2^(~1.189/730),
or quite close to 1&1/5 (and even closer to 1&3/16) 730-EDO degrees.
This scale [12-EDO] 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.
The optimal temperament: 7/26-comma meantone
chromatic semitone < (9/8)^(1/2)
enharmonic diesis = diatonic - [minus] chromatic semitones
... 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.
E
/ \
M3 m3
m6 M6
/ \
C -P4/P5- G
τ = t, + (6/13)c = 117 degrees
tempered tone = (10/9) * ((81/80)^(6/13)) = 2^(~117.001/730)
tempered semitone = (16/15) * ((81/80)^(9/26)) = 2^(~ 72.499/730)
~117.001 : ~72.449
= ~ 1.613832172 : 1
= ~ 8.06916086 : 5
50-EDO
We may therefore divide the octave into 50 equal divisions,
and appropriate 8 of them to the tone and 5 to the diatonic
semitone.
50-EDO 7/26-comma meantone
cents difference
ratio cents cents of 50-EDO
/ C \ 2^(50/50) 1200 1200.000 0.000
| 2
| B# < 2^(48/50) 1152 ~1153.978 1.978
5 1
| Cb < 2^(47/50) 1128 ~1126.846 -1.154
| 2
> B < 2^(45/50) 1080 ~1080.824 0.824
| 3
| Bb < 2^(42/50) 1008 ~1007.670 -0.330
8 2
| A# < 2^(40/50) 960 ~961.648 1.648
| 3
> A < 2^(37/50) 888 ~888.495 0.495
| 3
| Ab < 2^(34/50) 816 ~815.341 -0.659
8 2
| G# < 2^(32/50) 768 ~769.319 1.319
| 3
> G < 2^(29/50) 696 ~696.165 0.165
| 3
| Gb < 2^(26/50) 624 ~623.011 -0.989
8 2
| F# < 2^(24/50) 576 ~576.989 0.989
| 3
> F < 2^(21/50) 504 ~503.835 -0.165
| 2
| E# < 2^(19/50) 456 ~457.813 1.813
5 1
| Fb < 2^(18/50) 432 ~430.681 -1.319
| 2
> E < 2^(16/50) 384 ~384.659 0.659
| 3
| Eb < 2^(13/50) 312 ~311.505 -0.495
8 2
| D# < 2^(11/50) 264 ~265.484 1.484
| 3
> D < 2^( 8/50) 192 ~192.330 0.330
| 3
| Db < 2^( 5/50) 120 ~119.176 -0.824
8 2
| C# < 2^( 3/50) 72 ~ 73.154 1.154
| 3
\ C / 2^( 0/50) 0 0.000 0.000
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.
2^(x/50)
/ (C)
5
> B
3
> Bb
5
> A
3
> Ab
5
> G
3
> Gb
5
> F
5
> E
3
> Eb
5
> D
3
> Db
5
\ C
31-EDO
This scale, therefore, has the greatest temperament in the
minor-third, and its inversion the major-sixth, which is the
principal objection to it, as it is known that these concords
are most readily put out of tune, and consequently should have
the least temperament. However, taking all into account, it
must be acknowledged to be a very good scale.
19-EDO
For the practical tuning of a keyed instrument, such as the
organ, in which the full enharmonic scale is to be introduced,
perhaps the best method after all would be to divide the octave
into 19 equal intervals by 20 keys.
degree cents cents deviation from optimal meantone
19 1200.000 0.000
18 ~1136.842 ~17.136
18 ~1136.842 -~ 9.996
17 ~1073.684 ~ 7.140
16 ~1010.526 -~ 2.856
15 ~ 947.368 ~14.280
14 ~ 884.211 ~ 4.284
13 ~ 821.053 -~ 5.712
12 ~ 757.895 ~11.424
11 ~ 694.737 ~ 1.428
10 ~ 631.579 -~ 8.568
9 ~ 568.421 ~ 8.568
8 ~ 505.263 -~ 1.428
7 ~ 442.105 ~15.708
7 ~ 442.105 -~11.424
6 ~ 378.947 ~ 5.712
5 ~ 315.789 -~ 4.284
4 ~ 252.632 ~12.852
3 ~ 189.474 ~ 2.856
2 ~ 126.316 -~ 7.140
1 ~ 63.158 ~ 9.996
0 0.000 0.000
53-EDO
Beats of Imperfect Concords
the pitch-note A vibrates about 424 times in one second.
This may differ one or two vibrations from the truth, on
account of the unavoidable small defects of the materials
used in the experiment.
Miscellaneous Additions
My conclusions
First of all, my knowledge of Woolhouse's theories suggests
that he viewed some form of meantone temperament to be ideal.
According to Mandelbaum, Woolhouse derived an optimal meantone
tuning (I believe it was the squared-error optimal tuning for
the three 5-limit consonances,
namely 7/26-comma meantone),
It seems he was about an eyelash away from discovering my
cherished 7/26-comma meantone.
ratio Semitones
tempered tone = (10/ 9)*((81/80)^(6/13)) = ~1.92
tempered semitone = (16/15)*((81/80)^(9/26)) = ~1.19
3/1 commas
= 78/26
= 60/26 + 18/26
= 30/13 + 18/26
= (5*6/13) + (2*9/26) commas
= 5*(3*4) + 2*(3*3) / 26
= ((2^2)*3*5) + (2*(3^2)) / (2*13) commas
50-EDO 7/26-comma meantone
C 1200 1200.000
B 1080 ~1080.824
A 888 ~ 888.495
G 696 ~ 696.165
F 504 ~ 503.835
E 384 ~ 384.659
D 192 ~ 192.330
C 0 0.000
and decided that 19-tone equal tempermant was a close
enough approximation,
and one which gave to the musician the desirable properties
of a closed system which were giving 12-equal its rise to
prominence at the time.
the "optimal" meantone he derived (from 16th-18th century
musical considerations)
was essentially 50-tone equal temperament, and the only
possible reason for suggesting 19 instead of 50 would be
a practical one of actually getting all those notes onto
instruments.
REFERENCES
Harmonics, or the Philosophy of Musical Sounds.
Cambridge.
[2nd edition: 1759, London.]
'Sound', in Encyclopedia Metropolitana.
[Rasch gives 1845, vol 4, p 747-825, London. But obviously
for Woolhouse to refer to it, it had to have appeared earlier.
Woolhouse's citation: 'vol ii, mixed sciences, p 794.]
Essay on Musical Intervals, Harmonics,
and the Temperament of the Musical Scale, &c..
J. Souter, London. xii + 84 p.
Piano Tuning.
reprint: Dover 1975.
Multiple Division of the Octave
and the Tonal Resources of the 19-Tone Equal Temperament.
PhD dissertation, University of Indiana. (unpublished)
An Elementary Treatise on Musical Intervals and Temperament.
Macmillan & Co., London.
[Reprint: 1987. Ed. Rudolf Rasch. ]
[With introduction, bibliography and index.]
[Diapason Press, Utrecht. ]
Further dialog on this from the Tuning List forum:
He makes a statement about his prime- or odd-limit
(he doesn't specify which interpretation of 'limit'):
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 -
[p 24] Then follows Woolhouse's crucial statement concerning
JI versus temperament, without mentioning commatic drift:
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.
[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.
In what instance
must a keyboard 'necessarily have but one sound for both a
sharp and the flat of the next upper note'?
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.
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:
... 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.)
Putting that on a lattice for a geometric view of the situation:
E
/ \
M3 m3
m6 M6
/ \
C -P4/P5- G
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.
(I have added ratios and Semitones, and for comparison, the
cents-values for his optimal 6/13-&-9/26-comma):
50-EDO 6/13-&-9/26-comma meantone
= 1 - 7/13 comma
= 6/13 comma
= 35/26 - 1 comma
= 9/26 comma
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.
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.
He certainly noted the disadvantages of strict 5-limit JI, yet
he sought ultimately a temperament which would approximate it
better than 12-tET.
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',
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...
= 1 - 7/13 comma
= 6/13 comma
= 35/26 - 1 comma
= 9/26 comma
[Paul Erlich, in private email:]
the "optimal" meantone he derived (from 16th-18th century
musical considerations)
[Monzo:]
[Monzo:]
[Monzo:]
730-tET as a basic unit
[Woolhouse 1835, p 18]
Note the emphasis on his 5-limit JI conception here.
[Paul Erlich, TD 447.17, in his initial response to the
question of what type of limit Woolhouse had in mind]
[Paul Erlich, TD 447.23]
[me, monz, TD 446.6]
[Paul, TD 447.17]
[Woolhouse 1835, p 24]
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.
[Paul, TD 447.17]
[Woolhouse 1835, p 41]
[me, monz]
[me, monz]
[Paul, TD 447.17]
E
/ \
M3 m3
m6 M6
/ \
C -P4/P5- G
[me, monz]
[Paul, TD 447.17]
[Woolhouse 1835, p 45]
[me, monz]
[Paul, TD 447.17]
[Woolhouse 1835, p 41]
[Woolhouse 1835, p 46]
[Paul, TD 447.17]
[Paul, TD 447.17]
[me, monz]
[Paul, TD 447.17]
Woolhouse's derivation of 7/26-comma 'optimal' meantone
Woolhouse Monzo name ratio
variables not known
tau T tempered tone ?
sigma S tempered semitone ?
variables known
t t just major-tone 9/8
t, t, just minor-tone 10/9
theta s just diatonic semitone 16/15
c c syntonic comma 81/80
t = t, + c major-tone = minor-tone + comma
t, = t - c minor-tone = major-tone + comma
c = t - t, comma = major-tone - minor-tone
degree: I II III IV V VI VII (I)
\ / \ / \ / \ / \ / \ / \ /
tempered T T S T T T S
just t t, s t t, t s
'octave'
= 5T + 2S tempered intervals
= 3t + 2t, + 2s just intervals
= 3(t, + c) + 2t, + 2s just intervals
= 5t, + 2s + 3c just intervals
5T + 2s tempered intervals
- 5t, + 2s + 3c just intervals
----------------------------
= 5(T - t,) + 2(S - s) - 3c
= 0
5(T - t,) + 2(S - s) - 3c = 0
= 2(S - s) - 3c = -5(T - t,)
= 2(S - s) = -5(T - t,) + 3c
= (S - s) = (-(5/2)*(T-t,))+((3/2)*c)
Minor 3rd
tempered minor 3rd = T + S
just minor 3rd = t + s
= (t, + c) + s
= t, + s + c
T + S tempered minor 3rd
- t, + s + c just minor 3rd
-------------------------
= (T - t,) + (S - s) - c minor 3rd error
(T - t,) + (S - s) - c
= (T - t,) + (-(5/2)*(T-t,))+((3/2)*c) - c
= (-(3/2) * (T - t,)) + ((1/2) * c)
Major 3rd
tempered major 3rd = 2T
just major 3rd = t + t,
= (t, + c) + t,
= 2t, + c
2T tempered major 3rd
- 2t, + c just major 3rd
----------------
= 2(T - t,) - c major 3rd error
4th
tempered 4th = 2T + S
just 4th = t + t, + s
= (t, + c) + t, + s
= 2t, + s + c
2T + S tempered 4th
- 2t, + s + c just 4th
--------------------------
= 2(T - t,) + (S - s) - c 4th error
2(T - t,) + (S - s) - c
= 2(T - t,) + (-(5/2)*(T-t,))+((3/2)*c) - c
= (-(1/2) * (T - t,)) + ((1/2) * c)
minor 3rd error = (-(3/2) * (T - t,)) + ((1/2) * c)
major 3rd error = 2 * (T - t,) - c
4th error = (-(1/2) * (T - t,)) + ((1/2) * c)
[Woolhouse 1835, p 44]
[Paul Erlich, private communication]
minor 3rd error = (-(3/2) * (T - t,)) + ((1/2) * c)
major 3rd error = 2 * (T - t,) - c
4th error = (-(1/2) * (T - t,)) + ((1/2) * c)
minor 3rd error = (-(3/2) * x) + ((1/2) * c)
major 3rd error = 2 * x - c
4th error = (-(1/2) * x) + ((1/2) * c)
((-(3/2) * x) + ((1/2) * c))^2
+ (2 * x - c )^2
+ ((-(1/2) * x) + ((1/2) * c))^2
2*((-(3/2) * x) + ((1/2) * c))*(-(3/2))
+ 2*(2 * x - c )*2
+ 2*((-(1/2) * x) + ((1/2) * c))*(-(1/2))
((-(3/2) * x) + ((1/2) * c))*(-(3/2))
+ (2 * x - c )*2
+ ((-(1/2) * x) + ((1/2) * c))*(-(1/2))
= 0
(13/2) * (T - t,) - 3c = 0
(13/2) * (T - t,) = 3c
13 * (T - t,) = 6c
(T - t,) = (6/13) * c
tempered tone = just minor-tone + 6/13 comma
(S - s)
= (-(5/2) * (T - t,) ) + ((3/2) * c)
= (-(5/2) * ((6/13) * c)) + ((3/2) * c)
= (-(15/13) * c ) + ((3/2) * c)
= (-(30/26) * c ) + ((39/26) * c)
= (9/26) * c
tempered semitone = just diatonic semitone + 9/26 comma
T = (t, + (( 6/13) * c))
T = (t - (( 7/13) * c))
'5th'
= 3T + s
= 2(t - ( 7/13) * c) + (t, + ( 6/13) * c) + (s + (9/26) * c)
= 2(t - (14/26) * c) + (t, + (12/26) * c) + (s + (9/26) * c)
= 2t - ((28/26) * c) + t, + (12/26) * c) + s + (9/26) * c)
= 2t - t, + s - ((7/26) * c)
= just '5th' - ((7/26) * c)
= just '5th' - 7/26 comma
((2^-1)*(3^1)) / (((2^-4)*(3^4)*(5^-1))^(7/26))
= ((2^-1)*(3^1)) / (((2^-(28/26))*(3^(28/26))*(5^-(7/26))))
= 2^(2/26)*3^(-2/26)*5^(7/26)
= 2^(1/13)*3^(-1/13)*5^(7/26)
2^ 3^ 5^
| 1/13 -1/13 7/26 | 7/26-comma "5th"
- | 29/50 0 0 | 50-EDO "5th"
--------------------------
|-327/650 -1/13 7/26 | difference
= ~27 midipus = ~1/6 cent.
Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------
By Joe Monzo
or try some definitions.
corrections, improvements, good links.
Let me know if you don't understand something.
