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In the second edition of his book Harmonics or the Philosophy of Musical Sounds (1) Robert Smith applies his 'Equal Harmony' temperament to the harpsichord as well as to the organ. Indeed he claims that he commissioned a certain Mr. Kirkman, doubtless the well-known London maker Jacob Kirckman, to make a harpsichord according to his ingenious design and, presumably, to tune it to his temperament. (2)
Smith's defines his 'Equal Harmony' temperament in Section 8 of the second edition of his book. In this temperament the octave is divided into 50 equal parts, of which the tone is 8 parts, the major limma or [diatonic] semitone is 5 parts, the minor limma or [chromatic] semitone is 3 parts and the diesis, or the difference between the major and minor limma or between the diatonic and chromatic semitones is 2 parts. I find it useful to think of this in terms of the modern definition of the octave equal to 1200 cents. If an octave of 1200 cents is divided into 50 equal parts then each of these small parts of the scale is just
Thus Smith's scale is made up by piling
his small intervals, each of 24 cents, one on top of the other and observing
the above rules for defining each of the different sizes of musical interval.
This is shown in Table 1 below where the various
intervals produced by this procedure are named, taking each of the seven
notes of the diatonic scale as far as double sharps and double flats. Table
1 also compares the pitches of the notes of Smith's scale to those
of normal ¼-comma meantone temperament.
| Note | Equal
Harmony |
¼-comma
Meantone |
Note | Equal
Harmony |
¼-comma
Meantone |
| c | 0 | 0 | g |
552 | |
b |
24 | f |
576 | 579.47 | |
| d |
48 | 600 | |||
| c |
72 | 76.05 | g |
624 | 620.53 |
| 96 | f |
648 | |||
| d |
120 | 117.11 | 672 | ||
| c |
144 | g | 696 | 696.58 | |
| 168 | 720 | ||||
| d | 192 | 193.16 | a |
744 | |
| 216 | g |
768 | 772.63 | ||
| e |
240 | 792 | |||
| d |
264 | 269.21 | a |
816 | 813.69 |
| 288 | g |
840 | |||
| e |
312 | 310.27 | 864 | ||
| d |
336 | a | 888 | 889.74 | |
| f |
360 | 912 | |||
| e | 384 | 386.31 | b |
936 | |
| 404 | a |
960 | 965.748 | ||
| f |
432 | 984 | |||
| e |
456 | b |
1008 | 1006.84 | |
| 480 | a |
1032 | |||
| f | 504 | 503.42 | c |
1056 | |
| e |
528 | b | 1080 | 1082.89 | |
| g |
552 | 1140 | |||
| f |
576 | 579.47 | c |
1128 | |
| 600 | b |
1152 | |||
| g |
624 | 620.53 | 1176 | ||
| f |
648 | c | 1200 | 1200 |
Table 1 - Smith's 'Equal Harmony' temperament expressing the pitches of the notes in cents and comparing these pitches with those of normal ¼-comma meantone temperament
Neglecting the redundant octave-c there are 35 named notes in this scale, these being the 7 notes of the diatonic scale, with these sharpened and flattened (14 notes) and these all double-sharpened and double-flattened (14 notes). Smith called the un-named notes of the scale 'superfluous' consonances.
Clearly Smith's temperament is a kind of exaggerated ¼-comma meantone system. The major thirds are 4 cents narrower than pure, the minor (chromatic) semitones are even smaller than in meantone making the enharmonically-related notes even further from one another than in meantone (48 cents apart instead of 41 cents in meantone temperament). The fourths and fifths are only half a cent further from being pure than in meantone, and the minor thirds are about 2½ cents closer to being pure than in ¼-comma meantone. Nonetheless from it is clear that there is really very little difference between Smith's 'Equal Harmony' and normal ¼-comma meantone temperament.
We have already seen from the previous
papers that the application of Smith's 'Equal Harmony' to the Parker organs
resulted in a scale not with 35 named notes, but rather with only 16 -
the 'normal' 12 notes of the chromatic scale plus the enharmonically-related
notes of c, e,
g and b
- the additional 4 enharmonic notes enabling a musician to play consonances
in 4 additional major tonalities. Thus the range of major tonalities playable
on the Thomas Parker Foundling Hospital or the Laigh Room organs increases
from 6 (two flats to three sharps in the key signature plus C major - these
were called the notes of the 'vulgar' scale by Smith) to 8 (three flats
to four sharps in the key signature plus C major).
Smith's First System for applying his 'Equal Harmony' to the harpsichord
Smith's First System for applying his 'Equal
Harmony' to the harpsichord makes use of an existing, normal harpsichord
with at least two sets of 8' registers and strings. Effectively the instrument
becomes a single 8' instrument since only one string is used at a time,
although both registers are held on permanently (Smith suggests tying the
two 8' register knobs at the right-hand end of the keywell together with
a piece of string). He adds a small channel and capping piece of brass
supported by small pillars placed in the wrestplank, and tunes each of
the two strings played by the 12 keys of the normal chromatic scale to
the two enharmonically-related notes corresponding to that keylever. The
following is a quotation (3) from Smith relating
to Plate 1 below:
Plate 1 - Figure 66 from the second edition of Smith's Harmonics or the Philosophy of Musical Sounds showing his First System.
Pl. xxvi. The 66th figure represents the
heads of two jacks standing as usual upon one key, with their pens pointing
opposite ways under the strings on each side of them, as G
and A, the back unison being
raised to A. And abcd
represents a small brass square of the size in the figure, whose shorter
leg ab is made very thin and placed between the jacks with its flat
sides facing them; and the longer leg bced, being placed directly
over, and parallel to the next couple of strings that are closest together,
is filed four square, and slides lengthways in two square notches at c
and d made in the parallel sides fcg, hdi of a long
brass plate turned up like the sides of a long shallow trough, which is
supported a little above the strings by a row of small brass pillars placed
between the larger intervals of the strings, as at r, s,
&c, (but farther asunder) and screwed fast into the pinboard
of the harpsichord.
These pillars have long necks passing through the holes r, s, & c in the bottom of the trough, and the nuts r, s, &c are screwed upon the necks down to the bottom, to hold it fast upon the shoulders of the pillars. And a brass lid FGHI with oblong holes R, S, &c corresponding to r, s, &c, being laid upon the trough fg hi, the upper nuts R, S, &c must be screwed upon the same necks, to keep the lid tightish upon the longer leg of the square abcd and others of the same size. A slit mn is made in the lid for a short round pin e in the longer leg cd to come thro' it, and to move in it to and fro by a touch of the finger laid upon the pin. There must be as many such squares as keys or couples of jacks, and the trough and lid may be each of one piece or consist of two or three pieces joined together at the necks of the pillars or any where else.
While the jacks t, u are kept at their full height by holding down their key, with your finger laid upon the pin e push the leg ab against the far jack and mark the edge, or inner side of it with a line drawn close by the upper edge of the leg ab; and after the square is drawn back, make such another mark upon the edge of the near jack. Then from a small slender pin cut off a piece of a proper length measured from the point, and taking hold of its thicker end with a pair of pliers, press the point into the inner edge of the jack, a little above the mark and far enough to stick fast in it, and do the like to the opposite jack. Let each pin project from its jack about a quarter of the space between the two jacks, leaving about half of it void in the middle between the opposite ends of the pins, as represented in the figure.
Now when the two jacks are again raised
by their key and kept at their full height, by drawing the square backwards
with your finger laid upon the pin e in the longer leg, the shorter
leg
ab will come under the pin in the near jack, and keep it suspended
with its pen above the string G
which therefore will be silent while the far jack plays alone upon the
string A; or, by pushing the
square forward with your finger at e the leg ab will go under the pin in
the far jack, and suspend its pen above the string A,
while the near jack plays alone upon the string G.
When all the strings of the back unison are tuned to the notes in the upper line in art 11th all their jacks must be suspended on the shorter leg of the squares; and then all the fore jacks will strike the sounds of the vulgar scale; and when other flat or sharp sounds are required in any piece of music, they must first be introduced by holding down the keys of their usual substitutes, one by one, and by drawing back the corresponding squares with a finger laid upon their pins at e. So long as you choose to play upon this changeable scale, keep the knobs of the right-hand stops of a double harpsichord tyed together by a string.
When the strings are tuned unisons again, you may play upon them without removing this mechanism, provided you first draw every pin e towards the middle of the slit mn, in the lid FI, till it be opposite to the angular notch o, and then draw the lid lengthways by the button p, till the notch o embraces the pin e and keeps the shorter leg ab in the middle of the void space between the ends of the pins in the opposite jacks: otherwise these pins may sometimes strike against the shorter legs of the squares. If that middle space be too narrow, try whether it may not be widened a little by separating the sliders with some very thin wedges put between them: perhaps a little may be planed off from the back edges of the sliders without hurting them.
I have described this mechanism to fully, I think, that any man who works true in brass may easily apply it at a small expence to any harpsichord ready made, and take it quite away without the least damage to the instrument. I have used it some years in my own harpsichord with great pleasure and no other inconvenience than that of removing the music book in order to touch the pins in the brass squares behind it. But the following mechanism for the reception of which little preparation must be made in the fabric of a new harpsichord, is quite free from that inconvenience, and changes any sound together with all its octaves in an instant, without putting down their keys."
Smith's Second System for applying his 'Equal Harmony' to the harpsichord
The Second System proposed by Robert Smith
(4), and the one which he claims was used to make
a harpsichord for him by Kirckman, would involve making an instrument especially
for the 'Equal Harmony' temperament rather than adapting an already-existing
harpsichord as in the First System just described. This system, like the
first, uses an instrument with two choirs of 8' strings played as a single
8' and not in unison. Smith says that the compass of the instrument which
he commissioned was the somewhat-conservative G1 to e3,
rather than the, by this time (1759), more common 5-octave compass of F1
to f3 (normally without F1).
Probably as a result of his earlier experience
with the harpsichord which he modified using his First System, Smith realised
that because the enharmonically related notes for d (c
and e),
of a (g
and b),
e
(d
and f, and b (a
and c) (5)
were rarely used musically there was little point in providing an instrument
with these enharmonic notes. He therefore begins his discussion by pointing
out clearly that the a, b, d and e keylevers
were to be equipped each with only one string and jack (6),
whereas all of the others were to be equipped with two strings and jacks
in order to provide the enharmonically related notes for each of the other
keylevers, including the g keylever which has f
as its enharmonically related equivalent. The top part of Plate
2 (Smith's Fig.67) shows a schematic representation of the jacks, strings
and tuning pins which he proposes for this system. Here it is clear that
the notes a, b, d and e all have only one string
whereas all of the others have two strings providing the enharmonically
related notes for each keylever. The figure shows the jacks plucking the
strings sounding the strings needed in the flat keys.
The two lower diagrams in show the upper
two registers which Smith proposes to use in this harpsichord. His system
involves altogether 6 moveable upper registers some of which have slots
large enough that they do not displace any of the jacks at the upper level,
but with some jackslots which are close-fitting and move the jacks laterally
back and forth in the usual way. Register No 1 operates only the jacks
for the enharmonically related notes A
and B, but does not affect any
of the other jacks. The second register, No 2, operates the jacks of B
and C as well as F
and G. The third register, not shown, operates the jacks for C
and D as well as those of F
and G, and the bottom three registers
then operate only one pair of jacks each for each of the remaining three
pairs of enharmonically related notes.
Plate 2 - Fig. 67 from the second edition
of Smith's Harmonics or the Philosophy of Musical Sounds showing
a plan view of the jacks/registers/wrestplank (top) and, at the bottom,
the top register used to control the a/b
jacks and the second register used to control the jacks for the enharmonically-related
notes b/c and
f/g.
Plate 3 shows a cross-sectional
view of the jacks, registers and the ends of the keylevers on which the
jacks rest. This time the jacks are shown positioned so as to engage the
strings needed to play in sharp keys. The seventh and lowest register rs
is cut so as to hold the jacks for the notes a, b, d
and e in a permanently engaged position. The other six registers
can be engaged and disengaged using stop levers similar to those shown
in the top part of Plate 4. Here only the three treble
register levers are shown. The right-most register lever y has a
pin at its top end which passes through the uppermost 1st register
through a slotted hole in the top register like that shown at l
and fits into a tight hole in the 2nd register so moving it
back and forth and engaging and disengaging the jacks operated by it (f/g
and b/c). The
next register lever is pivoted at o and has a hole at its top right-hand
end which is large enough that it doesn't foul on the pivot of the right-most
lever y. The pin at the top end of this lever passes down through
slotted holes l in the top four registers into a tight hole in the
5th register which engages and disengages the jacks operating
on the f and e
enharmonically-related strings. Similarly the other levers operate the
registers for the notes shown in Smith's Fig. 70 at the bottom of Plate
4. Note that the pins in the top ends of the of the register levers
are placed conveniently in the gaps left in the registers by the lack of
doubled jacks for the b2, d3 and e3 notes.
On the bass side these would presumably have been placed at the bass end
of the registers and in the gaps left at the position of A1
and B1 neither of which requires doubled
jacks.
Plate 3 - Fig. 68 from the second edition of Smith's Harmonics or the Philosophy of Musical Sounds showing a sectional view of the jacks, registers and the ends of the keylevers in the harpsichord made by Jacob Kirckman for Smith.
Plate 4 - Figs. 69 (above) and 70 (below) from the second edition of Smith's Harmonics or the Philosophy of Musical Sounds showing the register-lever arrangement and the front stop-lever arrangement in the harpsichord made for Smith by Kirckman.
Moving the first two stop levers at the
bass end of the keywell to the right, and positioning all of the other
keylevers in the left-hand position enables the jacks to be positioned
in such a way as to play all of the notes of the 'vulgar' scale. As sharps
are added to the key signature beyond the F
and G provided by the 'vulgar'
scale, the levers are to be moved into the right-hand position by selecting
the register knobs progressively further to the treble side of the instrument,
thus making the logic of Smith's placement of the levers clear. Similarly,
starting from the position of the levers in their position for the 'vulgar'
scale, extra flats can be added by moving first the second-left and then
the left-most keylever into their left-hand positions.
Table 2 shows a schematic
representation of the available major tonalities using Smith's Second System.
Below these major key signatures are shown the relevant note which this
temperament provides in both of its enharmonic forms. The major tonalities
C, G
and D are not playable because
of the lack of availability of f,
c
and c
respectively. This also shows the logic of providing just one keylever
to double up the function of the registers involving d
and g, and b
and f
at the extremes of the available major tonalities which can be used in
the way described by Smith. The number of major tonalities playable on
a harpsichord made according to Smith's Second System is 14 encompassing
D major to G
major, or one fewer than using his First System. This is many more than
the 8 major tonalities playable on the Foundling Hospital or the Laigh
Room Thomas Parker organs. Even so a harpsichord made according to Smith's
Second System would not be able to play the continuo for a performance
of the whole of Messiah, which needs a c
for example.
In conclusion
Musically, in those tonalities which are
playable in Smith's 'Equal Harmony' temperament and using his Second System
, the tuning quality of a major triad is almost identical with 
-comma
meantone temperament, except that normal 
-comma
meantone has major thirds that are 4 cents wider instead of 4 cents narrower
than pure. Chromaticism is even more poignant in Smith's temperament than
in any of the normal seventeenth- and eighteenth-century cyclical or regular
temperaments which use major thirds which are either pure or slightly wide.
This poignancy is achieved because the narrow thirds of Smith's temperament
effectively result in the characteristic small chromatic semitones. This
effect becomes especially apparent and musically effective in chromatic
passages and in enharmonic modulations because of the wide spacing of the
enharmonically-related notes.
Although Smith's 'Equal Harmony' temperament has gained little popularity and is considered an obscure and little-known temperament, it is clearly not without its musical advantages. Nonetheless it is true that the difference between Smith's temperament and the easy-to-tune ¼-comma meantone temperament is only marginal in its musical effectiveness or in its use on an instrument equipped with the enharmonic lever arrangements suggested by Smith. It is of course possible to use ¼-comma meantone temperament equally well on either an organ or a harpsichord equipped for Smith's 'Equal Harmony' without modification to the physical operating mechanism. Perhaps the musical qualities and advantages of the temperament do not quite live up to the mathematical elegance of Smith's ingenious tuning system and the mechanical artifice and sophistication with which he puts it into practice, whether on the organ or on the harpsichord.
Footnotes:
(1) Robert Smith, Harmonics or the Philosophy of Musical Sounds, (Cambridge, 1749; second edition 1759; postscript, 1762; facsimile of the 1749 edition, Da Capo Press, New York, 1966)
(2) Robert Smith, Harmonics or the Philosophy of Musical Sounds, (Cambridge, 2/1759) 186.
(3) Page 173, Robert Smith, Harmonics or the Philosophy of Musical Sounds, (Cambridge, 2/1759).
(4) Page 177, Robert Smith, Harmonics or the Philosophy of Musical Sounds, (Cambridge, 2/1759).
(5) Effectively
this means that he limits the range of major tonalities to go from D
on the flat side (5 flats) to G
on the sharp side (8 sharps). Therefore as applied to the harpsichord Smith's
second System strongly favours pieces written in sharp keys, rather than
those with flats.
(6) The weight of the missing jack for each of these notes was to be compensated for by adding a lead weight equivalent to that of the jack to the end of each of the corresponding keylevers.
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