Optimization of tone and dynamical touch weight of upright and grandpianos.

H.J. Velo

http://home.kpn.nl/velo68

1.0. Introduction.

If we want to optimize the sound and touch during developing or restoration of an uprightor grand piano we have to ask ourselves: what is meant by optimum sound and optimumtouch. The sound of a modern instrument differs of course from the sound of those from anearlier period. The optimization method described in this article is limited to moderninstruments, that is to say, not for historical instruments.In case we are restoring an instrument, we must realize that the optimization is limited bythe properties of the instrument we are restoring, which can not easily be modified, such asthe elasticity of the hammer shanks and keys, the characteristics of the soundboard etc. etc.What firstly can be optimized is the string scale. One of the most important factors here isto try to ensure that the inharmonicity constant matches as good as possible that of a largeconcert grand, such as for example a Fazioli 278 or a Steinway-D. This will not be 100 %achievable in all cases, especially in the bass section of smaller instruments. However, it hasbeen found that, compared to the original scale, an improvement is often possible.String optimization can be carried out with the program “Easy String Calc.” Thisprogram offers the possibility quickly to optimize the string scale in two steps, namely firstsimultaneously all plain strings and after that simultaneously all bass strings. By means ofa special procedure the inharmonicity in the bass can be made almost equal to theinharmonicity of a large concert grand.See reference 5.1 and reference 5.2 for what can be achieved by using Easy String Calc.The second which can be optimized is the dynamical touch weight by choosing the righthammer weight with respect to the relation hammer movement/key front-end movement,the so called Strike Ratio, in order to obtain a dynamical behaviour of the action which iscomparable to the dynamical behaviour of a large concert grand and also a gradually slopeof the dynamical touch weight over the whole scale of the instrument, that is to say withoutlarge differences of the dynamical touch weight between adjacent keys.It also became evident that tone quality is also determined by hammer weight, but we haveto keep in mind the weight of the hammers can not be chosen without also considering theirdetermining factor in the dynamical behaviour of the action.

2.0 Definitions

The Strike Ratio R is the ratio between hammer-movement/key movement at the front ofthe key.

The Strike Weight SW is the mass of the hammer/shank assembly with the assemblysupported at the hammer-shank pivot point, the hammer shank horizontal, the hammerpointing upward, with the hammer resting on a digital weight –measuring scale with aresolution of 0,1 gram.

The Hammer Shank Strike Weight HSW is the mass of the hammer/shank with thehammer/shank supported at the hammer-shank pivot point and the hammer shank restingon a on a digital weight –measuring scale, with a resolution of 0,1 gram, placed support inorder to keep the shank horizontal.

The weight of the hammer has to be measured, if necessary with a digital scale with aresolution of 0,1 gram.

3.0 What is the relation between sound and touch.

When we are talking about the touch, a correct downweight is not sufficient to describethe behaviour of a key in the action, because this describes only the static behaviour ofthe action. For an excellent touch, the dynamic behaviour of the action is of greatimportance.

Fig. 1.

Figure 1 shows a drawing of a grand action. Figure 2 shows the result of an analysis of anaction of one key of a grand. This analysis is carried out with the program ACTCAL(This is short for action calculation). The data of the action of one key is entered in thelight green cells in column J. In row 40 through 46 of figure 2 a number of calculateddata are shown.In row 47 through 57 a table shows the calculated dynamical behaviour for several parts of theaction of one note and the dynamical behaviour of the complete action of one note. The completedynamical behaviour is also shown in a diagram. In column E we see, the main contribution to thedynamical behaviour is provided by the hammer, respectively the Strike Weight. One way toadjust the dynamical behaviour is to change the weighting of the key, that is to say re-arrangingthe place and the number of the key leads and/or applying a whippen spring. It turns out however,that this hardly has any effect on the total dynamical behaviour. A change in the dynamicalbehaviour has to be carried out mainly by changing the weight of the hammer and/or the StrikeRatio of the action.Suppose we want to modify the Strike Weight from 13 gram to 14 gram, for instance toachieve a better sound quality. We have to take care that, by an equal depressing time ofthe key, the sound volume and the dynamical behaviour will not change.It can be proved, when the Strike Weight SW1 is modified into SW2, by an equaldepressing time of the key, one can keep the kinetic energy, by which the hammer hits thestring and consequently the sound volume, unchanged. One can use the following formulato calculate the Strike Ratio, which formula is sufficient accurate. See appendix.

)/(. 2112 SWSWRR (1)

Suppose the Strike Ratio R1 = 5,47, SW1 =13 gram en SW2= 14gram.

The new Ratio becomes: 27,5)14/13(.47,52 R (2)The forgoing calculation has been experimentally checked as follows:First the down weight, the Strike Ratio R1 and the Strike Weight SW1 of a to beinvestigated key was measured.

After that this key was depressed by the following method: a weight was placed at thefront of the key and the key was kept by hand in its rest position. Subsequently the keywas allowed abruptly to go down and the sound volume was measured.Then the Strike Weight was raised by 1 gram by placing a small weight near the hammer.Using formula 1 the new Strike Ratio R2 was calculated. By changing the position of thecapstan, the Strike Ratio was made equal to the calculated Strike Ratio R2 .Then the downweight was corrected to its original value.Again the sound volume was measured in the same manner as described before.The result was that the sound volume was the same as the previously measured value.So by using formula (1), we have achieved the sound volume in both cases is equal. Nowwe have to check what the influence of this modification will be on the dynamicalbehaviour. Let’s have a look to the following calculation carried out with the programACTCAL. See figure 2 and 3.The Strike Weight is changed from 13 to 14 gram and the Strike Ratio is by moving thecapstan 1,65 mm. (See cell J20) in the direction of the fulcrum changed from 5,47 to 5,27as was calculated in formula (2). The down weight is, by changing the weighting of thekey, made again 51,87 gram.In the situation with a Strike Weight of 13 gram is the required force at a depressing timeof 10 ms. 8468,02 gf. In the situation with a Strike Weight of 14 gram is the requiredforce at a depressing time of 10 ms. 8451,45 gf. The difference amounts 8468,02/8451,45=1,00078or 0,08 %.It appears that by modifying of the Strike Ratio, when the Strike Weight has beenmodified, the sound volume and the dynamical behaviour can be kept equal.

1314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162

A B C D E F G H I J K LMT Weight key with,if present, keyleds 161,1 gramF Distance between front key and balancepoint key 249 mm.E Distance between balancepoint key-center top of capstan 140 mm.D Distance between center top of capstan and rotating point whippen 62,9 mm.C Distance between rotating point whippen and center bottom knuckle 96,4 mm.B Distance between center bottom knuckle - rotating-point hammer shank 22,36 mm.A Distance between rotating-point hammer shank and top of hammer 142 mm.V1 Displacement capstan (+ is increasing distance capstan -balance point key) mm.V2 Displacement knuckle (+ is increasing distance. knuckle - rot. point hammer shank) mm.V3 Displacement balance point (+ is increasing distance capstan -balance point key) mm.VONDH Displacement whippen (+ is displacement in direct. of rear-end key) mm. 0SW Strike weight 13 gramMO Weight whippen, measured on the spot where whippen contacts capstan. 21,65 gram 21,65WR Fr iction coefficient knuckle-repetitionlever" 0,104 0,104WTO Friction key plus friction capstan - whippen. 3,5 gram 3,5LF Length of key from its front edge to its pivot point 261 mm.LB Length of key from its back edge to its pivot point 245,1 mm.LC Total compensation lead weights 39,44 gramOVC Compensation whippen spring gramTC Total compensation keylead weights plus compensation whippen spring 39,44 gram1e loodje Weight 19,50 gram distance to bal. point 128 mm. Comp. 10,02 gram2e loodje Weight 19,50 gram distance to bal. point 110 mm. Comp. 8,61 gram3e loodje Weight 19,50 gram distance to bal. point 150 mm. Comp. 11,75 gram4e loodje Weight 12,60 gram distance to bal. point 179 mm. Comp. 9,06 gram5e loodje Weight gram distance to bal. point mm. Comp. 0,00 gram6e loodje Weight gram distance to bal. point mm. Comp. 0,00 gramTGL Total weight keyleads 71,10 gramTouch weight: 51,87 gramfriction between knuckle and repetition lever: 8,59 gramTotal friction: 10,9 gramBalance weight: 40,97 gramRelation of hammer-movement/movement of the front of the key: 5,47Pressure of knuckle on repetition lever: 82,56 gramUplift weight: 30,07 gramInfluence of several parts on the dynamical touch weightPress-time key lead whippen hammer Total

msec gf gf gf gf gf1280 0,037 0,028 0,007 0,44 52,38640 0,149 0,112 0,027 1,77 53,92320 0,597 0,449 0,107 7,07 60,09160 2,389 1,796 0,427 28,26 84,7580 9,557 7,182 1,707 113,06 183,3740 38,226 28,729 6,828 452,23 577,8820 152,905 114,916 27,312 1808,90 2155,9110 611,621 459,663 109,247 7235,62 8468,02

Merk Serienr. 0Type Toetsnr 0NootCalculation Lead diam. mm lenght mm Weight

15 10 19,5 gram

Maximum 1,5 mm.!!

Demomech. Teflongrand 10

100

1000

10000

100000

1280 640 320 160 80 40 20 10

Depress-time (msec)

Forc

e (g

f)

Fig. 2.

1314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162

A B C D E F G H I J K LMT Weight key with,if present, keyleds 161,1 gramF Distance between front key and balancepoint key 249 mm.E Distance between balancepoint key-center top of capstan 140 mm.D Distance between center top of capstan and rotating point whippen 62,9 mm.C Distance between rotating point whippen and center bottom knuckle 96,4 mm.B Distance between center bottom knuckle - rotating-point hammer shank 22,36 mm.A Distance between rotating-point hammer shank and top of hammer 142 mm.V1 Displacement capstan (+ is increasing distance capstan -balance point key) -1,65 mm.V2 Displacement knuckle (+ is increasing distance. knuckle - rot. point hammer shank) mm.V3 Displacement balance point (+ is increasing distance capstan -balance point key) mm.VONDH Displacement whippen (+ is displacement in direct. of rear-end key) mm. 0SW Strike weight 14 gramMO Weight whippen, measured on the spot where whippen contacts capstan. 21,65 gram 21,65WR Fr iction coefficient knuckle-repetitionlever" 0,104 0,104WTO Friction key plus friction capstan - whippen. 3,5 gram 3,5LF Length of key from its front edge to its pivot point 261 mm.LB Length of key from its back edge to its pivot point 245,1 mm.LC Total compensation lead weights 42,13 gramOVC Compensation whippen spring gramTC Total compensation keylead weights plus compensation whippen spring 42,13 gram1e loodje Weight 19,50 gram distance to bal. point 90 mm. Comp. 7,05 gram2e loodje Weight 19,50 gram distance to bal. point 128 mm. Comp. 10,02 gram3e loodje Weight 19,50 gram distance to bal. point 150 mm. Comp. 11,75 gram4e loodje Weight 19,50 gram distance to bal. point 170 mm. Comp. 13,31 gram5e loodje Weight gram distance to bal. point mm. Comp. 0,00 gram6e loodje Weight gram distance to bal. point mm. Comp. 0,00 gramTGL Total weight keyleads 78,00 gramTouch weight: 51,89 gramfriction between knuckle and repetition lever: 9,25 gramTotal friction: 11,18 gramBalance weight: 40,71 gramRelation of hammer-movement/movement of the front of the key: 5,27Pressure of knuckle on repetition lever: 88,91 gramUplift weight: 29,53 gramInfluence of several parts on the dynamical touch weightPress-time key lead whippen hammer Total

msec gf gf gf gf gf1280 0,034 0,030 0,006 0,44 52,40640 0,138 0,119 0,024 1,77 53,94320 0,551 0,475 0,096 7,08 60,09160 2,206 1,901 0,386 28,32 84,7080 8,824 7,603 1,542 113,27 183,1340 35,296 30,411 6,170 453,10 576,8620 141,182 121,643 24,678 1812,39 2151,7810 564,730 486,572 98,713 7249,54 8451,45

Merk Serienr. 0Type Toetsnr 0NootCalculation Lead diam. mm lenght mm Weight

15 10 19,5 gram

Maximum 1,5 mm.!!

Demomech. Teflongrand 10

100

1000

10000

100000

1280 640 320 160 80 40 20 10

Depress-time (msec)

Forc

e (g

f)

Fig. 3.

A same analyses can be done for an upright action, which leads to similar results. Theprogram Sound-Touch has been developed based on the above. Figure 4 shows a part ofthe program. (up to key 18).By measuring the Strike Weight of a specific key, the key number and the Strike Weightcan be entered in cells D10 and D11. After that, the program will calculate the optimumStrike Ratio and the relevant Strike Weight for all keys, providing an even Strike Weightcurve, resulting in an even dynamical behaviour for all keys. Altering the Strike Weight incell D11 results in another optimal Strike Ratio and Strike Weight for all keys. In columnD en E the down weight and up weight can be entered. The program calculates the frictionand if the friction is higher than required, an error message will show up in column H. Theuncallibrated Strike Weight must be entered in column L. The program then calculates thecompensation in grams. If for a specific key the Strike Weight has to be lowered, then thecolour of corresponding cell in column M becomes brown. If for a specific key the StrikeWeight has to be increased a small piece of lead wire has to be mounted in the hammer. Incolumn N the diameter of the lead wire can be entered. If for a specific key the Strike

Weight has to be increased, the length of the lead wire is calculated and visible in columnO. See also figure 5.

678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152

A B C D E F G H I J K L M N OBrandTypeSerial number Ratio is is the ratio hammer movement/front key movement

16 Optimum Ratio 5,5512

If the ratio of the black keys differs from the white keys, enter this ratio in cell J12.

Enter inKey dip Let off Key dip Letoff sub-

mm. mm. mm. mm. jointed11 9,6 11,0 9,6 column

1,7 the un calli-brated

DW strike-Freq. Downwght Up weight Error Srike- Key- weight Weight Diameter Length

Key Note DW UW Measured Required message weight weighting compensation Lead wire Lead wireNr. (Hz). Gram Gram Gram = Gram Friction Gram Gram (gram) (gram) mm. (mm)-8 2C 16,4 16 52-7 2C# 17,3 16 52-6 2D 18,4 16 52-5 2D# 19,4 16 52-4 2E 20,6 16 52-3 2F 21,8 16 52-2 2F# 23,1 16 52-1 2G 24,5 16 520 2G# 26 16 521 2A 27,5 52 28 12 16 12,5 52 12,2 0,3 4 2,22 2A# 29,1 52 26 13 16 12,5 52 12,1 0,4 4 2,73 2B 30,9 52 15 18,5 16 Error 12,5 52 12,7 -0,2 44 1C 32,7 52 28 12 16 12,4 52 12,2 0,2 4 1,65 1C# 34,6 52 24 14 16 12,4 52 12,2 0,2 4 1,46 1D 36,7 52 27 12,5 16 12,4 52 12,1 0,3 4 1,97 1D# 38,9 52 29 11,5 16 12,3 52 12,5 -0,2 48 1E 41,2 52 28 12 16 12,3 52 12,0 0,3 4 2,29 1F 43,7 52 26 13 16 12,3 52 12,5 -0,2 410 1F# 46,2 52 24 14 16 12,2 52 12,0 0,2 4 1,711 1G 49 52 27 12,5 16 12,2 52 12,1 0,1 4 0,712 1G# 51,9 52 19 16,5 16 Error 12,2 52 12,5 -0,3 413 1A 55 52 29 11,5 16 12,1 52 12,1 0,0 4 0,214 1A# 58,3 51 27 12 15 12,1 51 12,0 0,1 4 0,615 1B 61,7 51 28 11,5 15 12,0 51 12,3 -0,3 416 C 65,4 51 29 11 15 12,0 51 11,8 0,2 4 1,417 C# 69,3 51 26 12,5 15 12,0 51 11,7 0,3 4 1,818 D 73,4 51 31 10 15 11,9 51 11,7 0,2 4 1,5

Black keys

On this way the capstans are on one line!

White keysto callibrate the hammers before they are glued to the

hammer shank, ewnter in cell K17 a #

If a new set of hammers will be applied and one wishes

Friction

Number measured keyStrikewieght measured key

Fig. 4.

In the graph below is shown the uncalibrated- and calibrated Strike Weight

STRIKEWEIGHT

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

-8 -5 -2 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91

Key number

Gra

m

Callibrated Strikeweight Uncallibrated strikeweight or hammer shank weight plus weight uncallibrated hammers.

Fig. 5.

In the above example the Strike Ratio of the white and black keys is equal. If the StrikeRatio of the white and black keys is unequal, one can conclude to place the capstans notin line. If we want to have the capstans in line, we can make the Strike Ratio equal by, forinstance, moving the balance point. A third possibility to keep the capstans in line whenthe Strike Ratio of the white and black keys are unequal, is to change the Strike Weightof the black keys. To avoid problems when adjusting the action, the difference in StrikeRatio may not be more than 7 %. In the following example is for the Strike Ratio of theblack keys 5,7 entered in cell J12. The Strike Weight of the black keys will be lower thanin the case the Strike Ratio was 5,55. The program will calculate the appropriate StrikeWeight for all keys.

6789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354

A B C D E F G H I J K L M N OBrandTypeSerial number Ratio is is the ratio hammer movement/front key movement

16 Optimum Ratio 5,5512

If the ratio of the black keys differs from the white keys, enter this ratio in cell J12. 5,7

Enter inKey dip Let off Key dip Letoff sub-

mm. mm. mm. mm. jointed11 9,6 10,7 9,3 column

1,7 the un calli-brated

DW strike-Freq. Downwght Up weight Error Srike- Key- weight Weight Diameter Length

Key Note DW UW Measured Required message weight weighting compensation Lead wire Lead wireNr. (Hz). Gram Gram Gram = Gram Friction Gram Gram (gram) (gram) mm. (mm)-8 2C 16,4 16 52-7 2C# 17,3 16 52-6 2D 18,4 16 52-5 2D# 19,4 16 52-4 2E 20,6 16 52-3 2F 21,8 16 52-2 2F# 23,1 16 52-1 2G 24,5 16 520 2G# 26 16 521 2A 27,5 52 28 12 16 12,5 52 12,2 0,3 4 2,22 2A# 29,1 52 26 13 16 11,8 52 12,1 -0,3 43 2B 30,9 52 15 18,5 16 Error 12,5 52 12,7 -0,2 44 1C 32,7 52 28 12 16 12,4 52 12,2 0,2 4 1,65 1C# 34,6 52 24 14 16 11,8 52 12,2 -0,4 46 1D 36,7 52 27 12,5 16 12,4 52 12,1 0,3 4 1,97 1D# 38,9 52 29 11,5 16 11,7 52 12,5 -0,8 48 1E 41,2 52 28 12 16 12,3 52 12,0 0,3 4 2,29 1F 43,7 52 26 13 16 12,3 52 12,5 -0,2 410 1F# 46,2 52 24 14 16 11,6 52 12,0 -0,4 411 1G 49 52 27 12,5 16 12,2 52 12,1 0,1 4 0,712 1G# 51,9 52 19 16,5 16 Error 11,5 52 12,5 -1,0 413 1A 55 52 29 11,5 16 12,1 52 12,1 0,0 4 0,214 1A# 58,3 51 27 12 15 11,5 51 12,0 -0,5 415 1B 61,7 51 28 11,5 15 12,0 51 12,3 -0,3 416 C 65,4 51 29 11 15 12,0 51 11,8 0,2 4 1,417 C# 69,3 51 26 12,5 15 11,3 51 11,7 -0,4 418 D 73,4 51 31 10 15 11,9 51 11,7 0,2 4 1,519 D# 77,8 51 30 10,5 15 11,2 51 11,5 -0,3 420 E 82,4 51 30 10,5 15 11,8 51 11,6 0,2 4 1,5

Friction

Number measured keyStrikewieght measured key

Black keys

On this way the capstans are on one line!

White keysto callibrate the hammers before they are glued to the

hammer shank, ewnter in cell K17 a #

If a new set of hammers will be applied and one wishes

Fig. 6.

In figure 7 it is clearly visible that the Strike Weight of the black keys is less than the StrikeWeight of the white keys.

STRIKEWEIGHT

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

-8 -5 -2 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91

Key number

Gra

m

Callibrated Strikeweight Uncallibrated strikeweight or hammer shank weight plus weight uncallibrated hammers.

Fig. 7.

4.0 Summary.

In the foregoing points is described, how the sound and the touch of a grand- or upright pianocan be optimised. The relevant programs are user friendly.In cell D11 of the program Sound-Touch also a value can be entered with one digit behind thedecimal sign, for example 12,2 instead of 12.

5.0 References.

5.1 H.J. Velo: The importance of a low inharmonicity in the bass: http:// home.kpn.nl/~velo68/

5.2 H.J. Velo: The importance of a low inharmonicity in the bass. EuroPiano …..

5.3 H.J. Velo: A measuring device for measuring the uplift and down weight of a piano- orgrand action.: http:// home.kpn.nl/~velo68/

5.4 H.J. Velo: Easy String Calc. A program to analyse and optimize the scale of pianos andgrand-pianos.: http:// home.kpn.nl/~velo68/

6.0 Finally:

Those interested can receive the user manual. In this manual is among others described the tobe applying measuring devices and tools. Send an e-mail to vpeilaoo@hetnet.nl .

APPENDIX

Derivation of formula (1)

I = Moment of inertia (kgm2)ω = Angular velocity (rad/sec)d = key dip front-end key (m)Lhs = Length hammer shank (m)Lkf = Length front–end key (m)mh = Mass hammer (kg)mhs = Mass hammer shank (kg)t = depress time front-end key (sec)SW= Strike Weight (kg)R = Strike Ratio

The kinetic energy of a rotating object can be described by:2

21 IEkin

Suppose we have two situations:2111 2

1 IEkin (3)

and2222 2

1 IEkin (4)

and we want to make both kinetic energies equal by changing the Strike Ratio R, when the StrikeWeights SW1 and SW2 are unequal.

ω1 is determined bytR

Ldkf

11 * (5)

ω2 is determined bytR

Ldkf

22 * (6)

Substituting (5) into (3) gives:2

111 )(21 R

LdIEkf

kin (7)

Substituting (6) into (4) gives:2

222 )(21 R

LdIEkf

kin (8)

Set Ekin1 = Ekin2: 222

211 )(2

1)(21 R

LdIR

LdI

kfkf

(9)

Simplified: 222

211 RIRI (10)

The moment of inertia of the hammer/shank assembly of hammer 1 is the sum of the moment ofinertia of hammer 1 and the moment of inertia of the hammer shank and is determined by:

3

22

11hs

hshshLmLmI (11)

The moment of inertia of the hammer/shank assembly of hammer 2 is the sum of the moment ofinertia of hammer 2 and the moment of inertia of the hammer shank and is determined by:

3

22

21hs

hshshLmLmI (12)

Substituting (11) and (12) into (10) gives:22

2

221

22

1 )3

()3

( RLmmRLmLm hshsh

hshshsh (13)

Dividing formula (13) by Lhs2 gives:

222

211 )

31()

31( RmmRmm hshhsh (14)

From (14) it follows:

)3131

(2

121

22

hsh

hsh

mm

mmRR

(15)

From (15) it follows:

hsh

hsh

mm

mmRR

3131

2

112

(16)

Do we consider the hammer/shank assembly as a point mass with a mass equal to the Strike WeightSW rotating around an axis at a distance of this axis equal to Lhs , the moment of inertia is forhammer/shank assembly 1:

211 hsLSWI (17)

and for hammer/shank assembly 2:2

22 hsLSWI (18)Do we substitute (17) and (18) into (10) we find:

22

22

21

21 RLSWRLSW hshs (19)

Dividing by Lhs2 gives:

222

211 RSWRSW (20)

From this it follows:

2

112 SWSWRR (21)

Formula (21) is much simpler than formula (16)

We must now investigate if formula (21) is accurate enough. The Strike Weight is the sum of thehammer weight and half the hammer shank weight, so

hsh mmSW 21

11 (22) and hsh mmSW 21

22 (23)

Substituting (22) and (23) into formula (21) gives:

hsh

hsh

mm

mmRR

2121

2

112

(24)

We compare formula (24) with formula (16) by entering for mh1 3.75 gram, for mh2 4.75 gram andfor mhs 2.5 gram in both formulas.The result of formula (24) is 0.91287*R1 and the result of formula (16) is 0.906033*R1.The deviation is:

%75.0100*)90603.091287.01( Deviation (25)

This deviation is small enough to use formula (21) which is equal to formula (1)