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

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.

Contents
--------

Recent references to Woolhouse on the Tuning List
Woolhouse's book
Introductory
Sound
Musical intervals
5-limit JI
The 'mean semitone'
The basic JI intervals for scale construction
730-tET as a basic unit of measurement
Problems with JI
Harmonics
Temperament
12-tET
The optimal temperament: 6/13- & 9/26-comma meantone
50-tET
31-tET
19-tET
53-tET
Beats of Imperfect Concords
Miscellaneous Additions
My conclusions
References

************************************************

Recent references to WoolWoolhouse on the Tuning List
=====================================================

> Dave Hill, TD 439.10]
>
> 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."
>
> 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]
>
> Note that Wesley Woolhouse was a prominent advocate of
> _19_ tone equal temperament.

> [Paul Erlich, TD 439.12]
>
> 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 l
rger 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.
>
> 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.

> [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.
>

Woolhouse's book
================

Introductory
------------

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 detail examination of his
thoughts concerning intervals and temperament.

Sound
-----

The first chapter (only p 1-2) sets out the basic concepts
concerning sound.

Musical intervals
-----------------

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.

5-limit JI
..........

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
>

[p 10] Woolhouse devises the following pentatonic scale from
the basic 5-limit JI ratios:

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

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, using the following ratios:

note interval ratio string-length

B seventh 15/5 192/360
C second 9/8 320/360

which produces the usual 5-limit JI 'major' scale:

A---E---B
/ \ / \ / \
F---C---G---D

The 'mean semitone'
...................

[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-tET scale for comparison to the just ratios.

The basic JI intervals for scale construction
.............................................

Then he defines [p 17] the following basic intervals (I have
added the tilde [~] to indicate that a value is only approximate):

interval ratio mean semitones

major-tone 9/8 ~2.0391
minor-tone 10/9 ~1.8240
limma 16/15 ~1.1173

(Note that this definition of _limma_ is quite different
from the usual Pythagorean one, which makes it equal to
256/243 [= ~0.90 Semitone]; it is in fact much closer to
the larger Pythagorean semitone, the _apotome_ [= ~1.12
Semitones]. See my Tuning Dictionary.)

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.

He examines some other notably accurate divisions of the 'octave':

> [Woolhouse 1835, p 19]
>
> 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.

[p 20]
Woolhouse mentions likening the octave to a circle, and
using degrees, minutes, and seconds to measure the intervals.
This amounts to 1296000-tET (yes, that's over a million degrees:
360 * 60 * 60), which Woolhouse dismisses as 'of no advantage
in musical computations'.

Then he mentions 301-tET 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-tET, which shows that

> [Woolhouse 1835, p ]
> 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'.

Woolhouse chose this division precisely because he wished to
avoid using decimals or fractions, and the integer values
of 730-tET designating the three basic intervals of 15:16, 9:10,
and 8:9 are so close to the actual JI values:

interval ratio mean semitones 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 _theta_ ~ 67.970
comma 81/80 ~0.2151 c ~ 13.083

730-tET:

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 _theta_ 68.000
comma 2^( 13/730) ~0.2137 c 13.000

(He uses the actual Greek letter _theta_ to represent the
_limma_.)

Woolhouse gives the more exact value (which I placed at the
end 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 in his math equations.

Problems with JI
................

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 give 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, 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.

Harmonics
---------

The third chapter [p 25-35] is an explanation of harmonics,
about which most here 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.

Temperament
-----------

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.

12-tET
......

First is an examination of 12-tET, noting that its 'greatest
imperfections are those of the 3rd and 6th', with a maximum
of 9&1/2 'degrees' [= ~15.64 cents] deviation, and after
tabulating all the 'errors' from JI, he says that

> [Woolhouse 1835, p 38]
>
> These deviations, however, which are considerably less
> than a comma or 13 degrees [= 2^(13/730) = ~21.37 cents],
> are too small to affect, in a very sensible degree, the
> melody of the intervals.

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-tET 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-tET 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'. The exact figure is obviously 1&1/6
degrees [= 2^((7/6)/730) = ~1.92 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-tET he concludes:

> [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.

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.

The optimal temperament: 6/13- & 9/26-comma meantone
....................................................

He immediately continues by saying that a system of temperament
that gives two different sizes of semitone provides a much
better approximation to JI.

[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)
chromatic semitone < (9/8)^(1/2)
enharmonic diesis = diatonic - [minus] chromatic semitones

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.)

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.

Then [p 43-45] he uses algebra to find the difference between
JI and this ideal temperament with _tau_ to represent the
tempered tone and _sigma_ to represent the diatonic semitone,
and the other abbreviations presented earlier for the JI
intervals, to make up the six intervals on my lattice.

The result is:

> [Woolhouse 1835, p 45]
>
> _tau_ = t, + (6/13)c = 117 degrees
>
> _sigma_ = _theta_ + (9/26)c = 72&1/2 degrees
>

Or in English and math:

tempered tone = (10/9) * ((81/80)^(6/13)) = 2^(~117.001/730)
tempered semitone = (16/15) * ((81/80)^(9/26)) = 2^(~ 72.499/730)

He notes that the relationship between these two is 'very
nearly in the ratio of 8 to 5'. More precise figures are
~1.613832172 : 1 = ~8.06916086 : 5. Thus,

50-tET
......

> [Woolhouse 1835, p 45]
>
> We may therefore divide the octave into 50 equal divisions,
> and appropriate 8 of them to the tone and 6 to the diatonic
> semitone.

Woolhouse is assuming a diatonic scale with a basic interval
mapping of 5L,2s; that is, 5 large 'steps' and 2 small ones.
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-tET
degrees between both the Diatonic and Chromatic scale members
(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

ratio cents cents difference of 50-tET

/ 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

> [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.

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:

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

and he proposes [p 48] to eliminate F#, C# and Cb major
as superfluous.

31-tET
......

Woolhouse then gives [p 49] the same kind of description of
Huygens's 31-tET, about which he says:

> [Woolhouse 1835, p 49]
> 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-tET
......

Next he considers 19-tET:

> [Woolhouse 1835, p 50]
> 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.

and he gives a list of all the keys and a diagram of how it
could be produced on a keyboard.

53-tET
......

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.

[p 56]

Woolhouse gives a table providing exact monochord-string
measurements for 12-, 19-, 31-, 50-, and 53-tET.

Beats of Imperfect Concords
---------------------------

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]
>
> 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.

So Woolhouse expresses his limit of accuracy here in Hz
as about 1/4 percent .

He gives a table [p 65] showing the beat divisors for 12-,
19-, 31-, 50-, and 53-tET, and at the end of this chapter
[p 66-68] gives the beats for all '5ths' in the 12- and
19-tET systems.

Miscellaneous Additions
-----------------------

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.

My conclusions
==============

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-tET 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-tET 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 approximated
them well.

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

Knowing from Fischer 1907 only that quote posted by Dave,
I would venture to say that Fischer approved of 12-tET
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-tET system is capable of, with
19-tET 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-tET 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 hey 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]
>
> First of all, my knowledge of Woolhouse's theories suggests
> that he viewed some form of meantone temperament to be ideal.

Absolutely.

> According to Mandelbaum, Woolhouse derived an optimal meantone
> tuning (I believe it was the squared-error optimal tuning for
> the three 5-limit consonances,

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)

> namely 7/26-comma meantone),

In a private email he's given me permission to share, Paul
said:

> It seems he was about an eyelash away from discovering my
> cherished 7/26-comma meantone.

I was going to say: it's still 'no cigar', but I'd say it's
closer than an eyelash away. 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:

ratio Semitones

tempered tone = (10/ 9)*((81/80)^(6/13)) = ~1.92
tempered semitone = (16/15)*((81/80)^(9/26)) = ~1.19

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:

3/1 commas
= 78/26
= 60/26 + 18/26
= 30/13 + 18/26
= (5*6/13) + (2*9/26) commas

which (I don't know if there's any significance) can be
further reduced to:

= 5*(3*4) + 2*(3*3) / 26
= ((2^2)*3*5) + (2*(3^2)) / (2*13) commas

(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:

50-tET 6/13-&-9/26-comma
AND 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

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!

> [back to Paul]
>
> and decided that 19-tone equal tempermant was a close
> enough approximation,

Absolutely.

> 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.

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

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

> 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.

This is all absolutely true, except that I would say his
optimal tuning was the 6/13- & 9/26-comma meantone, and
even advocating 50-tET 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 give
a bit of weight to Dave Hill's hypothesis that Woolhouse
was more willing to accept the tuning status-quo than he
books makes him seem.

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-tET approximates both 3- and
5-limit JI and relatively little about 50-tET, how about
showing us exactly why 50-tET is better. Or if it's not
better, then tell us what descriptions best characterize
the comparison of these two temperaments.

REFERENCES
----------

Smith, Robert. 1749.
_Harmonics, or the Philosophy of Musical Sounds_.
Cambridge.
[2nd edition: 1759, London.]

Herschel, J. W. F. ?date?
'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.]

Woolhouse, W. S. B. 1835.
_Essay on Musical Intervals, Harmonics,
and the Temperament of the Musical Scale, &c._.
J. Souter, London. xii + 84 p.

Fischer, J. Cree. 1907.
_Piano Tuning_.
reprint: Dover 1975.

Mandelbaum, M. Joel. 1961.
_Multiple Division of the Octave
and the Tonal Resources of the 19-Tone Equal Temperament_.
PhD dissertation, University of Indiana. (unpublished)

Bosanquet, R. H. M. 1876.
_An Elementary Treatise on Musical Intervals and Temperament_.
Macmillan & Co., London.
[Reprint: 1987. Ed. Rudolf Rasch. ]
[With introduction, bibliography and index.]
[Diapason Press, Utrecht. ]

-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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