Russell Collection of Early
Keyboard Instruments

The graph below shows the tensions resulting from these calculations and gives smooth fitted curves showing the 'ideal' tensions for both the 8' and 4' registers as well.

The string diameters for the Russell Collection 1769 Taskin harpsichord which is un-marked, can be calculated using these tensions, the string lengths, the density of the stringing materials used in each part of the compass and the pitch. This can be done using Taylor's formula re-written in terms of the diameter:

d = string diameter in metres	f = frequency in Hz
l = length of string in metres	T = string tension in kg wt
= string material density	g = acceleration of gravity = 9.8 m/sec²

However, before these calculations can be carried out it is necessary to determine the notes at which the material transitions from iron to yellow brass and from yellow brass to red brass occur. This is easily done graphically, and the graph of the scalings shown below illustrates the method.

This graph shows the scaling curves of both the 8' and 4' strings drawn in heavy curves. Light straight lines are shown for 8' pythagorean iron scalings (13¹/₆ pouce = 356.5mm), 4' pythagorean iron scalings (12¹/₃ pouce = 333.9mm), pythagorean yellow brass scalings (11 pouce = 297.8mm) and for pythagorean red brass scalings (9 pouce = 243.6mm). The latter two are those already determined for the 1764/83 Goermans/Taskin harpsichord [link]. The transitions from one material to another occur where the straight Pythagorean lines cross the actual scaling curves.

Hence from the graph it is clear that for this instrument the notes to be strung in each material are:

String Material	8'	4'
red brass:	F1 to F	F1 to D
yellow brass:	F to c	E to B
iron:	c to f³	c to f³

All of the information that is needed to calculate the string diameters for the 1769 Taskin double-manual harpsichord is now to hand. This calculation can be done using the string tensions which have been calculated for the 1764/83 Goermans/Taskin harpsichord [link] and shown above in Figure 1, the string lengths, the material densities and the pitch. These factors were entered as appropriate into Taylor's formula as given above.

The 'ideal' string tension are those given by the fitted curve from the graph shown in Figure 1 for both the 4' and 8' stringing. This was used to calculate the 'ideal' string diameter, different for e very note, that would give these 'ideal' tensions. The eighteenth-century French string diameters chosen for the 1769 Taskin harpsichord approximate these 'ideal' diameters as closely as possible and the transition from one diameter to the next was selected to be between the notes for which the 'ideal' diameters were on either side of the average of the French gauge system string diameters. In this way the tension in each string was as near as possible to that which may have been used by Taskin² .

These calculations are shown in Table 1 and Table 2 below.

Note	Frequency	Length		'Ideal' Tension	'Ideal' Diameter	Actual Diameter	French Gauge
f³	1318.5	148.5	2463	2.53	0.164	0.211	9
e³	1244.5	155	2463	2.59	0.168	0.211	9
e³	1174.7	162.5	2463	2.65	0.172	0.211	9
d³	1108.7	169.5	2463	2.71	0.177	0.211	9
c³	1046.5	177.5	2463	2.77	0.181	0.211	9
c³	987.8	187	2463	2.84	0.184	0.211	9
b²	932.3	196.5	2463	2.89	0.187	0.211	9
b²	880	205	2463	2.96	0.192	0.211	9
a²	830.6	217	2463	3.02	0.194	0.211	9
g²	784	228	2463	3.08	0.198	0.211	9
g²	740	240.5	2463	3.14	0.201	0.211	9
f²	698.5	253.5	2463	3.21	0.204	0.211	9
f²	659.3	268	2463	3.27	0.206	0.211	9
e²	622.3	283	2463	3.34	0.209	0.211	9
e²	587.3	299	2463	3.4	0.212	0.211	9
d²	554.4	317.5	2463	3.46	0.213	0.211	9
c²	523.3	336.5	2463	3.54	0.215	0.211	9
c²	493.9	356.5	2463	3.6	0.217	0.211	9
b¹	466.2	378.5	2463	3.66	0.218	0.211	9
b¹	440	402	2463	3.74	0.22	0.211	9
a¹	415.3	427	2463	3.8	0.221	0.211	9
g¹	392	453.5	2463	3.87	0.223	0.211	9
g¹	370	480.5	2463	3.95	0.225	0.237	8
f¹	349.2	508.5	2463	4.02	0.227	0.237	8
f¹	329.6	537.5	2463	4.09	0.23	0.237	8
e¹	311.1	569	2463	4.17	0.232	0.237	8
e¹	293.7	602	2463	4.25	0.235	0.237	8
d¹	277.2	637.5	2463	4.32	0.237	0.237	8
c¹	261.6	674	2463	4.4	0.24	0.237	8
c¹	246.9	711	2463	4.48	0.243	0.237	8
b	233.1	749	2463	4.57	0.247	0.237	8
b	220	788	2463	4.66	0.251	0.237	8
a	207.7	825.5	2463	4.75	0.256	0.266	7
g	196	865	2463	4.85	0.262	0.266	7
g	185	905	2463	4.96	0.268	0.266	7
f	174.6	944	2463	5.06	0.275	0.266	7
f	164.8	982.5	2463	5.18	0.283	0.299	6
e	155.6	1021	2463	5.31	0.292	0.299	6
e	146.8	1061	2463	5.45	0.302	0.299	6
d	138.6	1100.5	2463	5.61	0.313	0.337	5
c	130.8	1138	2463	5.78	0.325	0.337	5
c	123.5	1177.5	2639	5.97	0.327	0.337	5
B	116.5	1217	2639	6.19	0.341	0.337	5
B	110	1256	2639	6.42	0.357	0.337	5
A	103.8	1293	2639	6.66	0.374	0.378	4
G	98	1332	2639	6.9	0.392	0.378	4
G	92.5	1369	2639	7.11	0.41	0.426	3
F	87.3	1407.5	2639	7.3	0.428	0.426	3
F	82.4	1445	2780	7.45	0.435	0.426	3
E	77.8	1479	2780	7.57	0.454	0.426	3
E	73.4	1515	2780	7.66	0.472	0.479	2
D	69.3	1548	2780	7.71	0.491	0.479	2
C	65.4	1584	2780	7.73	0.509	0.479	2
C	61.7	1617	2780	7.73	0.528	0.538	1
B1	58.3	1647	2780	7.71	0.549	0.538	1
B1	55	1674	2780	7.67	0.571	0.538	1
A1	51.9	1704.5	2780	7.61	0.591	0.605	0
G1	49	1721	2780	7.53	0.617	0.605	0
G1	46.2	1739	2780	7.44	0.643	0.605	0
F1	43.7	1752.5	2780	7.32	0.671	0.68	0
F1	41.2	1764	2780	7.18	0.699	0.68	0
		Total tension:		304.79

Note	Frequency	Length		'Ideal' Tension	'Ideal' Diameter	Actual Diameter	French Gauge
f³	2637.02	69	2463	2.12	0.161	0.187	10
e³	2489.02	72.5	2463	2.18	0.165	0.187	10
e³	2349.32	75	2463	2.24	0.171	0.187	10
d³	2217.46	79	2463	2.3	0.174	0.187	10
c³	2093	81.5	2463	2.36	0.181	0.187	10
c³	1975.53	85	2463	2.43	0.187	0.187	10
b²	1864.66	89	2463	2.49	0.192	0.187	10
b²	1760	93.5	2463	2.55	0.196	0.187	10
a²	1661.22	97.5	2463	2.62	0.201	0.211	9
g²	1567.98	103	2463	2.68	0.204	0.211	9
g²	1479.98	107.5	2463	2.75	0.21	0.211	9
f²	1396.91	114	2463	2.82	0.212	0.211	9
f²	1318.51	121	2463	2.88	0.214	0.211	9
e²	1244.51	127.5	2463	2.96	0.218	0.211	9
e²	1174.66	135	2463	3.02	0.221	0.211	9
d²	1108.73	143.5	2463	3.08	0.222	0.211	9
c²	1046.5	152.5	2463	3.15	0.224	0.211	9
c²	987.77	162	2463	3.22	0.226	0.237	8
b¹	932.33	172.5	2463	3.28	0.227	0.237	8
b¹	880	183	2463	3.36	0.229	0.237	8
a¹	830.61	195	2463	3.43	0.23	0.237	8
g¹	783.99	207.5	2463	3.49	0.231	0.237	8
g¹	739.99	221	2463	3.57	0.233	0.237	8
f¹	698.46	236	2463	3.64	0.233	0.237	8
f¹	659.26	250	2463	3.71	0.235	0.237	8
e¹	622.25	265.5	2463	3.78	0.237	0.237	8
e¹	587.33	282	2463	3.85	0.239	0.237	8
d¹	554.37	300	2463	3.93	0.24	0.237	8
c¹	523.25	318	2463	4	0.242	0.237	8
c¹	493.88	338	2463	4.08	0.244	0.237	8
b	466.16	358	2463	4.16	0.246	0.237	8
b	440	377.5	2463	4.24	0.25	0.237	8
a	415.3	398	2463	4.32	0.253	0.266	7
g	392	419.5	2463	4.39	0.257	0.266	7
g	369.99	441.5	2463	4.47	0.261	0.266	7
f	349.23	465	2463	4.55	0.265	0.266	7
f	329.63	488	2463	4.63	0.27	0.266	7
e	311.13	511	2463	4.71	0.275	0.266	7
eb	293.66	534.5	2463	4.79	0.281	0.266	7
d	277.18	557.5	2463	4.87	0.288	0.299	6
c	261.63	581	2463	4.95	0.295	0.299	6
c	246.94	605.5	2463	5.03	0.302	0.299	6
B	233.08	629.5	2639	5.1	0.3	0.299	6
B	220	652.5	2639	5.18	0.309	0.299	6
A	207.65	676	2639	5.27	0.318	0.299	6
G	196	700.5	2639	5.35	0.328	0.337	5
G	185	725.5	2639	5.42	0.338	0.337	5
F	174.61	749.5	2639	5.5	0.349	0.337	5
F	164.81	774.5	2639	5.58	0.36	0.337	5
E	155.56	799	2639	5.67	0.373	0.378	4
E	146.83	823	2639	5.74	0.386	0.378	4
D	138.59	848.5	2780	5.82	0.389	0.378	4
C	130.81	873	2780	5.88	0.403	0.378	4
C	123.47	899.5	2780	4.95	0.38	0.426	3
B1	116.54	924	2780	5.99	0.431	0.426	3
B1	110	949	2780	6.01	0.445	0.426	3
A1	103.83	976.5	2780	6.03	0.459	0.479	2
G1	98	1001	2780	6.03	0.475	0.479	2
G1	92.5	1024.5	2780	6.01	0.491	0.479	2
F1	87.31	1048.5	2780	5.98	0.507	0.538	1
F1	82.41	1073	2780	5.92	0.522	0.538	1
		Total tension:		254.51

calculating the diameters which come closest to giving these tensions for each note in the compass. First of all however, it is necessary to determine the notes at which the material transitions from iron to yellow brass and from yellow brass to applying these to the, enables both the string transition notes to be calculated using the same string transition scalings, as well as the string gauges and diameters which result in the same tensions These calculations give the following results:

	8'	4'
Iron:	Gauge 9 = 0.211mm - g¹ to f³	Gauge 10 = 0.187mm - b² to f³
	Gauge 8 = 0.237mm - b to g¹	Gauge 9 = 0.211mm - c² to a²
	Gauge 7 = 0.266mm - f to a	Gauge 8 = 0.237mm - b to c²
	Gauge 6 = 0.299mm - e to f	Gauge 7 = 0.299mm - e to a
	Gauge 5 = 0.337mm - c & d	Gauge 6 = 0.299mm - c to d
Yellow Brass:	Gauge 5 = 0.337mm - B to c	Gauge 6 = 0.299mm - A to B
	Gauge 4 = 0.378mm - G & A	Gauge 5 = 0.337mm - F to G
	Gauge 3 = 0.426mm - F & G	Gauge 4 = 0.378mm - E & E
Red Brass:	Gauge 3 = 0.426mm - E & F	Gauge 4 = 0.378mm - C& D
	Gauge 2 = 0.479mm - Cto E	Gauge 3 = 0.426mm - B1 to C
	Gauge 1 = 0.538mm - B1 to C	Gauge 2 = 0.479mm - G1 to A1
	Gauge 0 = 0.605mm - G1 to A1	Gauge 1 = 0.538mm - F1 & F1
	Gauge 00 = 0.680mm - F1 & F1

If the available modern strings do not have exactly the same diameters as those given above then the transition notes from one string size to the next should be modified slightly one way or the other to compensate for the difference in the string diameter. This could be done by matching the available diameters to the 'ideal' diameters in Tables 1 and 2. The notes at which the string materials change should, however, follow the scheme above.

Taskin's pitch was probably about a¹ = 409Hz, which is slightly more than a semitone below modern pitch (a¹ = 440Hz). No difficulty should be encountered using the above stringing materials and gauge numbers at a pitch of a¹ = 415Hz (ie one semitone below modern pitch). The gauge diameters and material transition notes given above would, however, not be valid if the instrument were tuned to modern pitch (a¹ = 440Hz), in addition to which most modern soft-wire equivalents of the eighteenth-century strings would break at this pitch.

The red brass, yellow brass and iron used for the stringing of plucked keyboard instruments in eighteenth-century France were all relatively soft ductile materials. Phosphor-bronze, spring brass and patent steel strings do not make good modern substitutes for the historical materials. Rather, one should try to find red brass, yellow brass and iron which are as soft and weak as possible but which will still maintain their pitch without breaking. Normally suitable stringing materials stretch and drop in pitch as soon as they are placed on the instrument, and they usually take several weeks to several months before s tabilising and maintaining a constant pitch. Normal experience is that the musical quality of the strings also improves during the period that they stretch and stabilize in their physical properties.

FOOTNOTES:

^1.Jean-Louis Val, 'Une d'termination de la taille des cordes de clavecin employ'es en France au XVIIIiSme siScle', Revue de musicologie, 56, N" 2 (1970) 208-14.

^2.Grant O'Brien, 'Some principles of eighteenth-century harpsichord stringing and their application', The Organ Yearbook, 12 (1981) 160-76.

References:

• Jean-Louis Val, 'Une d'termination de la taille des cordes de clavecin employ'es en France au XVIIIiSme siScle', Revue de musicologie, 56, N" 2 (1970) 208-14.
• Grant O'Brien, 'Some principles of eighteenth-century harpsichord stringing and their application', The Organ Yearbook, 12 (1981) 160-76.
• Martha Goodway and J. Scott Odell, 'The metallurgy of 17th- and 18th-century music wire', The Historical Harpsichord Volume 2, Howard Schott editor, (Pendragon Press, Stuyvesant, NY, 1987).

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