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Why you might need to "calculate strings"
If all you will ever do is play a standard-sized guitar in the standard tuning, stringing your guitar is no problem - pre-packaged sets of guitar strings will do just fine. If, you ever need or want to branch out, however, you would need to "calculate" the correct string depending on the size of your instrument, the note you want, and the tension you want. Here are some of the guitar-related areas you could find yourself exploring that would require strings not found in the regular set.
There are the early plucked string instruments: vihuela, Renaissance guitar, Renaissance lute, Baroque guitar, Baroque lute, plus more. Even if you're sure you're never going to own one of those instruments, you could get involved, anyhow. For instance, I converted a modern guitar into a Baroque guitar (see my web page). Some of these instruments have additional bass strings, and I can envision the simple addition of second neck to a modern guitar for these extra basses.
Other eras produced guitar music requiring extra basses. For example, Napolean Coste wrote for a 7-string guitar and Fernando Carulli wrote some music for a 10-string guitar. A 7th string would give us access to the Russian guitar repertoire. (Reading the music would pose a problem, but nothing that couldn't be handled by tablature.)
There is a wealth of chamber works written for the terz guitar, tuned a minor third (3 half-steps) higher than the regular guitar. The capo solution becomes unsatisfactory when the music starts going way up the fingerboard. By the way, a terz guitar with the 3rd string tuned down a half-step could serve as a quasi-Renaissance treble lute.
Alternate tunings that deviate more than slightly from the regular tuning are candidates for custom-selected strings. In my web page on the subject, I point out that you can often shift the whole tuning up or down a half-step or so to keep the over-all tension about right, but there may be cases where some strings are still too far removed from their intended tension. Examples that come to mind are tunings that call for a low C, and my proposed violin tuning with string 6 tuned up to G. Of course you wouldn't change a string to play a single piece in an alternate tuning, but it could be worth your while if you had a collection of such pieces and an extra guitar.
Perhaps your guitar is larger or smaller than the standard guitar. Maybe a child in your family or a student has a half- or three-quarter sized guitar. What are the right strings for it?
The first problem is that string makers who offer a complete range of gauges - usually for "lute strings" - are pretty hard to come by. And when you find a dealer, the prices can be hard to swallow after having gotten used to the prices of pre-packaged sets. But I'll let market forces work that one out. When guitarists get enthused about the areas that require custom-selected strings, all that should change.
How to "calculate strings" - determining MPL
So how do you figure out what string(s) will do the job? What is the relevant string measurement and how do you calculate it?
I've used the word gauge above, and we know that steel-string guitarists and lutenists select strings by gauge. The gauge is the diameter of the cross-section of the string. The problem is: gauge is not a very useful parameter.
That should come as no surprise. We know, for example, that our 3rd string has a larger diameter than the 4th and 5th strings even though the 3rd is higher pitched. The explanation, of course, is that the composition of the treble and bass strings are different.
If we take a look at the physical equation that describes a vibrating string, we see that the frequency depends on length, tension and mass per unit length. If we use the following shorthand:
TENS = tension
FREQ = frequency
LENG = length
MPL = mass per unit length,
and sqrt = square root
the fundamental relationship is:
FREQ = sqrt(TENS/MPL)/LENG/2
Solving for the thing we need to know:
MPL = 9800xTENS/FREQ/FREQ/LENG/LENG/4
where TENS is in kilograms (kg),
FREQ is in cycles per second (cps),
LENG is in meters (m),
and MPL is in grams per meter (g/m).
See footnote 1 if you're curious about where the 9800 came from.
Related formulas have been devised to calculate the string diameter, but you need a different formula for every string composition (see footnote 2.) The beauty of our formula above is that it works for any type of string imaginable - nylon monofilament, wound, gut, steel, graphite-impregnated horse-hair, rubber bands . . . you-name-it. Just plug in your string length, the frequency of the note you want and the desired tension, and out pops the required mass per unit length.
But who the heck knows note frequencies??? You know I always have your best interests at heart, and if you check out Appendix 1, you will find a table of note frequencies. Better yet is Appendix 2, which gives a program in BASIC which does the whole calculation for you. You don't even have to mess with frequencies - just tell it the note you want.
Example: You are experimenting with higher tension on string 3 only. Your set of normal tension strings has the G string sized for a tension of 5.49 kg (see footnote 3) and you wonder what 6.0 kg would sound and feel like. Your guitar has a .650m scale length. The frequency of the G is 196 cps. Plugging everything in yields MPL = 0.91 grams/meter. That will give you a G note at the tension you want - regardless of the string composition: nylon, wound, gut, steel . . .
Example: You want to tune the pair of 5th strings on your Baroque guitar to the upper octave (220 hz) and you only want 3 Kg of tension per string, for a total of 6 Kg for the pair. Suppose the string length is .650m. Plugging everything in yields MPL = 0.36 grams/meter for each string.
Example: For playing John Dowland, you need a low B for your 8-string guitar. You think 6.33 kg will be a good tension. Plugging in LENG=.650, FREQ=61.7 and TENS=6.33 yields MPL=9.63 g/m.
Summary:
Summing up, I implore all string makers to supply the MPL measurement for all of their strings. I propose that grams per meter is a convenient unit for MPL.
It would be nice to see guitarists get enthused about the alternatives to the regular guitar tuning. This would incidentally create an incentive for string makers to offer individual strings spanning the whole gamut of MPL densities.
Footnote 1. Deriving the formula for MPL.
The constant 9800 which seems to pop up out of nowhere comes about in the conversion of natural physical units into units we are more comfortable with. A kilogram is actually a unit of mass, but is used in this context as a unit of force - specifically, the weight of a 1-kg mass at sea level, or, equivalently, the tension in a string with a 1-kg mass hanging from it. (Weight and tension are both forces.)
The conversion of tension from proper units of newtons to units of "kilograms" is gotten from the relationship,
TENS newtons = MASS kg x grav
where grav = 9.8 m/sec/sec, the acceleration due to gravity. The units all cancel out leaving,
TENS = 9.8 MASS
This substitution is made while solving for MPL, BUT . . . after the substitution is made we rename MASS back to TENS. Got it? (Wouldn't it have made more sense to have gotten comfortable with the distinction between mass and force from pre-school on? After all, they're totally different beasts.)
The extra factors of 10 arise from the conversion of MPL in units of kg/m to g/m.
Footnote 2. The old-fashioned string calculation: gauge (= diameter).
Toyohiko Satoh, in his article "A Method For Stringing Lutes", Journal of the Lute Society of America, Vol II, 1969, p44, gives a formula devised by Dr. Helmut Herminghaus:
DIAM = CONST x sqrt(TENS)/FREQ/LENG
where DIAM is the string diameter in mm, and CONST is some constant which depends on the material of the string. The article gives two values: CONST=49 for gut strings, and CONST=54 for nylon strings.
Note that this equation is in agreement with ours. The diameter of the string and the mass per unit length are related as follows:
MPL = DENS x PI x DIAM x DIAM / 4
where DENS is the ordinary density (mass per unit volume) of the string material, and PI = 3.14159 (approx.) Substituting for MPL in our equation yields
CONST = sqrt(9800/PI/DENS)
where DENS is in units of g/m/mm/mm (grams per meter per millimeter squared.) It can be seen that the Herminghaus equation requires a different constant for every density of string material. Worse yet, no CONST will work for a family of wound strings since the density would not be the same for different gauges.
Still, string manufacturers only provide string diameters so that's what you must work with (unless you have a triple beam balance and string samples.) Appendix 3 gives a computer program that calculates desired string diameters.
Footnote 3. A typical set of strings.
A set of values for the tension of normal tension guitar strings is given on D'Addario Pro Arte J45 string envelopes. They used a .648m scale length. The tensions and diameters are:
String Tension Diameter
------ ------- --------
1st, E 6.94 kg 0.71 mm
2nd, B 5.26 kg 0.82 mm
3rd, G 5.49 kg 1.02 mm
4th, D 7.09 kg 0.74 mm
5th, A 6.29 kg 0.86 mm
6th, E 6.33 kg 1.09 mm
Does anybody know the reason for the variation in tension from string to string? It seems likely that, given the chance, many guitarists would want to be in control of the tension of each string.
By the way, the information provided by D'Addario allows us to calculate the constant in the Herminghaus equation (see footnote 2) for the nylon used in D'Addario monofilament strings. It works out to about 57. Herminghaus provided a constant of 54 for nylon strings. I don't know the reason for the discrepancy.
Appendix 1 - Frequencies of the notes
Pitch names are a pain in the neck. The A notes named below (A, A a a') which start the octaves indicate the actual pitch - as opposed to the notated pitch in guitar music. Another wrinkle is that the subscripts and superscripts change at each occurrence of C - not A.
But you can safely ignore all that. To orient yourself, note that the six open guitar string notes are labeled with the string number in parentheses.
To get the frequency of the next lower octave of a note, just divide by 2.
Frequencies (cps)
Octave starts on: A, A a a'
---- ----- ----- -----
Notes: A 55 (5) 110 220 440
Bflat 58.3 116.5 233.1 466.2
B 61.7 123.5 (2) 246.9 493.9
C 65.4 130.8 261.6 523.3
C# 69.3 138.6 277.2 554.4
D 73.4 (4) 146.8 293.7 587.3
D# 77.8 155.6 311.1 622.3
E (6) 82.4 164.8 (1) 329.6 659.3
F 87.3 174.6 349.2 698.5
F# 92.5 185 370 740
G 98 (3) 196 392 784
G# 103.8 207.7 415.3 830.6
Appendix 2 - Program to calculate MPL
This BASIC program calculates the mass per unit length, MPL, for a string that will have the desired tension for a desired note.
If you want to take it off this web page and use it, make sure you download the unrendered HTML. BASIC uses "<" and ">" signs in the conventional "less than" and "greater than" sense. But HTML uses those characters to bracket instructions for your browser. Thus they might be misinterpreted, or not even shown, in the program code below. In the Explorer browser, click View; click Source; scroll down to this point; and save the lines of program code in a BASIC file called STRINGM.BAS .
10 'STRINGM - calculate mass per unit length, MPL, for a string. 20 'Programmed by Donald Sauter. 30 ROOT#=2#^(1!/12!) 50 PRINT 60 PRINT "This program calculates the Mass Per Unit Length, MPL, of the 70 PRINT "string that will give you the desired tension for a given note. 80 PRINT "This value for MPL is appropriate for any string composition, 90 PRINT "monofilament or wound. 100 PRINT 110 PRINT "The program goes the other way, too. You feed it the MPL and it 120 PRINT "will calculate the tension. 130 PRINT 140 PRINT "The relationship is MPL = 9800 x TENS/FREQ/FREQ/LENG/LENG/4 150 PRINT 158 STRLEN=.65 160 INPUT "String length (meters) (=.650): ",ANS$ 170 IF ANS$<>"" THEN STRLEN=VAL(ANS$) 180 IF STRLEN>1 THEN BEEP : PRINT "WARNING: That's a mighty long string!" 190 'DO-UNTIL operator is done 200 PRINT 210 PRINT " 4 220 PRINT " 4 -O- 230 PRINT " 3 _________ Choose the number corresponding to the 240 PRINT " 3 _________ octave within which your desired note 250 PRINT " 3 _________ will fall. Octaves begin on A notes. 260 PRINT " 3 ____O____ (This is a *guitar music* staff - notes 270 PRINT " 2 _________ sound an octave lower than written. The 280 PRINT " 2 ___ highest A pictured here is 440 cps.) 290 PRINT " 2 -O- 300 PRINT " 1 310 PRINT " 1 320 PRINT 330 INPUT "Desired octave number (0-4, probably): ",IOCT 340 IF IOCT>4 THEN BEEP : PRINT "WARNING: That's a mighty high note!" 350 IF IOCT<0 THEN BEEP : PRINT "WARNING: That's a mighty low note!" 360 OCTOFF=-IOCT+3 370 PRINT 380 PRINT " A A# B C C# D D# E F F# G G#" 390 INPUT "Note number (0 1 2 3 4 5 6 7 8 9 10 11): ",NOTNUM 400 IF NOTNUM>11 THEN BEEP :PRINT "Try again." :GOTO 390 410 T2MFL=-1 :'Calculate MPL from tension flag. 420 PRINT 430 INPUT "String tension (kg) *OR* hit to input MPL instead: ",TENS 440 IF TENS=0 THEN T2MFL=0 : INPUT "Mass Per unit Length (g/m): ",MPL 450 FREQ=220*ROOT#^NOTNUM 460 FREQ=FREQ/2^OCTOFF 470 ' 480 'IF calculating MPL from tension 490 IF NOT(T2MFL) THEN 620 500 MPL=9800*TENS/4/FREQ/FREQ/STRLEN/STRLEN 510 PRINT 520 PRINT "For: Length (m) = ";STRLEN 530 PRINT " Frequency (cps) ="; 540 PRINT USING "####.##";FREQ 550 PRINT " Tension (kg) ="; 560 PRINT USING "###.###";TENS 570 PRINT "MASS PER UNIT LENGTH ="; 580 PRINT USING "###.####";MPL; 590 PRINT " grams/meter. (Any string composition.) 600 GOTO 720 610 ' 620 'ELSE calculating tension from MPL 630 TENS=MPL*4/9800*FREQ*FREQ*STRLEN*STRLEN 640 PRINT 650 PRINT "For: Length (m) = ";STRLEN 660 PRINT " Frequency (cps) ="; 670 PRINT USING "####.##";FREQ 680 PRINT " Mass per length (g/m) ="; 690 PRINT USING "###.####";MPL 700 PRINT "TENSION (kg) ="; 710 PRINT USING "###.###";TENS 720 'ENDIF 730 PRINT "===============================================================" 740 INPUT "Another calculation? (Y/N =yes): ",ANS$ 750 IF ANS$="" OR ANS$="y" OR ANS$="Y" THEN GOTO 190 760 'END-DO operator is done 770 END
Appendix 3 - Program to calculate string diameter
This BASIC program calculates the Diameter (gauge) for a string that will have the desired tension for a desired note. As discussed above, this only makes sense for monofilament strings, even though string manufacturers talk in terms of gauge for wound strings as well. Again, you must extract the unrendered HTML. In the Explorer browser, click View; click Source; scroll down to this point; and save the lines of program code as STRINGD.BAS .
10 'STRINGD - calculate string diameter. 20 'Programmed by Donald Sauter. 30 ROOT#=2#^(1!/12!) 40 PRINT 50 PRINT "This program calculates the Diameter, or gauge, of the string 60 PRINT "that will give you the desired tension for a given note. 70 PRINT "Unfortunately, this only makes sense for monofilaments, and 80 PRINT "you need to supply a constant which depends on the material. 90 PRINT "(Values for gut and nylon are provided here.) 100 PRINT 110 PRINT "The program goes the other way, too. You feed it the diameter 120 PRINT "and it will calculate the tension. 130 PRINT 140 PRINT "The relationship is FREQ x LENG x DIAM = const x sqrt(TENS) 150 PRINT "See Satoh, Journal of the Lute Society of America, Vol II 1969. 160 PRINT 170 STRLEN=.65 180 INPUT "String length (m) (=.650): ",ANS$ 190 IF ANS$<>"" THEN STRLEN=VAL(ANS$) 200 IF STRLEN>1 THEN BEEP : PRINT "WARNING: That's a mighty *long* string!" 210 'DO-UNTIL operator is done 220 PRINT 230 PRINT " 4 240 PRINT " 4 -O- 250 PRINT " 3 _________ Choose the number corresponding to the 260 PRINT " 3 _________ octave within which your desired note 270 PRINT " 3 _________ will fall. Octaves begin on A notes. 280 PRINT " 3 ____O____ (This is a *guitar music* staff - notes 290 PRINT " 2 _________ sound an octave lower than written. The 300 PRINT " 2 ___ highest A pictured here is 440 cps.) 310 PRINT " 2 -O- 320 PRINT " 1 330 PRINT " 1 340 PRINT 350 INPUT "Desired octave number (0-4, probably): ",IOCT 360 IF IOCT>4 THEN BEEP : PRINT "WARNING: That's a mighty high note!" 370 IF IOCT<0 THEN BEEP : PRINT "WARNING: That's a mighty low note!" 380 OCTOFF=-IOCT+3 390 PRINT 400 PRINT " A A# B C C# D D# E F F# G G#" 410 INPUT "Note number (0 1 2 3 4 5 6 7 8 9 10 11): ",NOTNUM 420 PRINT 430 DFLG=-1 :'Calculate diam flag. 440 INPUT "Tension (kg) *OR* hit to input Diameter instead: ",TENS 450 IF TENS=0 THEN INPUT "String diameter (mm): ",DMM : DFLG=0 460 FUDGE=57 :'d'Addario nylon. 470 INPUT "Constant (49=gut =57=DAddario nylon): ",ANS$ 480 IF ANS$<>"" THEN FUDGE=VAL(ANS$) 490 FREQ=220*ROOT#^NOTNUM 500 FREQ=FREQ/2^OCTOFF 510 ' 520 'IF CALCULATING DIAMETER 530 IF NOT(DFLG) THEN 710 540 DMM=FUDGE*SQR(TENS)/STRLEN/FREQ 550 DIN=DMM/25.4 560 PRINT 570 PRINT "For: Length (m) = ";STRLEN 580 PRINT " Frequency (cps) ="; 590 PRINT USING "####.##";FREQ 600 PRINT " Tension (kg) ="; 610 PRINT USING "###.###";TENS 620 PRINT " Material constant = ";FUDGE 630 PRINT "DIAMETER ="; 640 PRINT USING "##.###";DMM; 650 PRINT "mm or "; 660 PRINT USING ".####";DIN; 670 PRINT "in. "; 680 PRINT "(For monofilaments only!) 690 GOTO 810 700 ' 710 'ELSE calculating tension 720 TENS=(FREQ*STRLEN*DMM/FUDGE)^2 730 PRINT 740 PRINT "For: Length (m) = ";STRLEN 750 PRINT " Frequency (cps) ="; 760 PRINT USING "####.##";FREQ 770 PRINT " Diameter (mm) = ";DMM;" (Monofilament only!) 780 PRINT " Material constant = ";FUDGE 790 PRINT "TENSION (kg) ="; 800 PRINT USING "###.###";TENS 810 'ENDIF 820 PRINT "===============================================================" 830 INPUT "Another calculation? (Y/N =yes): ",ANS$ 840 IF ANS$="" OR ANS$="y" OR ANS$="Y" THEN GOTO 210 850 'END-DO operator is done 860 END
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