String Tension/Gauge/Length affect on tone?

[lloyd_cdb]

I've been thinking about this for a bit. As a math nerd, I was looking at cross strung vs straight strung pianos and the finite element analysis of the harp design within the piano. I was looking to compare and contrast harp design for tensile strength as an afternoon activity. You know... normal afternoon stuff. ANYWAY...

Knowing the math of calculating frequency (I'll put it at the end) I realized I had no knowledge of standard string dimensions to actually apply it. After some digging, I found that string dimensions vary on the make/model for a given note, beyond even the obvious dimensional changes up and down the scale. From what I gather, the plate is designed so that tension on the strings is reasonably equal across the (unwound) strings, while the gauge and speaking length change along the scale. Futily trying to find any information to even work up from, I was wondering if any of you technicians that have changed strings out have any knowledge as to even a rough guide on information like:

- how often does the string gauge step up?
- what is/is there a reasonably typical gauge for A4?

So for the curious, here is some of the math and basics on frequency. The basic variables that factor into frequency produced are length, diameter, density, and tension.

f = (1/Ld)*sqrt[T/(y*pi)]

f = frequency (Hz)
L = Length (cm)
d = diameter (cm)
T = tension (g*cm/s^2)
y = density (g/cm^3)

All else being equal, to lower a string by an octave you would need to do one of the following:

-2x the length
-2x the diameter
-0.25x the tension
-4x density

After starting to look into this, I became curious as to the actual tonal effect these decisions result in. Basically the major curiousity I had in regards to the less mathematical aspect boils down to (as a very extreme case):

Compare two strings. String 1 with diameter = x and tension = z, String 2 with d = 2x, T = 4z. These two strings produce the same frequency. How would the two strings sound compared to one another? Is there anything beyond length of sustain?

EDIT: I'm tired, so if I did anything dumb in here I'll correct it in the morning.

[lloyd_cdb]

And... After reading a bunch more random stuff, I found this fun chart on string tension throughout the ages:



EDIT: If you open this in a new tab, it's much more readable. It won't let me resize it for some reason

[iansinclair]

Neat.  I've never seen that chart; that's kind of fun!  And it does answer for you how often the string gauge changes -- the breaks in tension vs. note give it away.

From a rather general point of view, different gauge strings for the same note (therefore either different tension or length) do make a significant difference in tone, particularly if it is the length that is changed.  The reason is the stiffness of the string itself, which is not negligible.  A stiffer string (larger gauge) theoretically should have less power in the higher harmonics than a floppier one, up to a point.  This is also, however, very much affected by the exact location along the string that it is struck by the hammer.

[chopin2015]

Not a techie, but I am a little nerdy.(Or am I flirting with you? who knows!)
Anyways:
Depends, are there sets of 2/ sets of 3 unisons and how many? each piano is different. I am pretty sure that the gauge is not changed across each string in sets of 3(uncoiled) and sets of 4.  a4 is string no 49...so that would be what I would google. The tension for a4(why don't you just give me a length of the 1st string and the a4 string you need to gauge/add tension?)  I am going to calculate that and respond back to you today(please, do I have to?)

however, as the pitch increases, the gauge and length slightly decreases, so perhaps the tension on each string is actually the same! (it depends on your piano how many rotations of the tuning peg it takes to achieve specific frequency)
 
Also, you can take an impressions of all the pins and bearing points in the harp, and the strings, too! the technician can match strings and gauge to the shape of the harp. Also, there are different points on coiled strings, like the cut-in point. Those vary too. Let me know.

See the chart about piano gauges. This is from a website where piano strings can be purchased for replacement of individuals or complete set.

[lloyd_cdb]

Neat.  I've never seen that chart; that's kind of fun!  And it does answer for you how often the string gauge changes -- the breaks in tension vs. note give it away.

That was my exact reaction!

I figured the steps in that chart were gauge changes. The issue is that the most recent of those is the 1967 Steinway, the next one being the Ibach 1914. I would at least assume some things have changed, mainly due to production quality of strings as well as modern engineering sciences. I know my (~1990) Yamaha C3 has an average tensions of 198.5 lbs (90kg), which is at least near that Steinway.

In addition, regarding wound strings - The tension list ends at the lower end before the scale is done, so I assume that is where the wound strings begin. From looking at the chart there is a varying range of points at which the tension list begins. In regards to the Cristofori/Silbermann, they obviously lacked wound strings as well as had a smaller octave range making sense of the chart. The Erard utilizes wound strings up to C#2 (note #17). The Boesendorfer up to D2 (#18). The Steinway up to C3 (#28). My Y-C3 uses wound strings up to G#/Ab2 (note #24). I don't really have access to any pianos excluding my own at the moment, so is there any modern standardization?

From a rather general point of view, different gauge strings for the same note (therefore either different tension or length) do make a significant difference in tone, particularly if it is the length that is changed.  The reason is the stiffness of the string itself, which is not negligible.  A stiffer string (larger gauge) theoretically should have less power in the higher harmonics than a floppier one, up to a point.  This is also, however, very much affected by the exact location along the string that it is struck by the hammer.

Power as in volume? Length makes sense to me as we see it regularly applied in model sizes. Looking at Steinway models and their tension scales:

S (5'1") - 32k lbs
M (5'7") - 34k
O (5'10")- 42k
A (6'2") - 42k
B (6'11")- 39k
D (8'11")- 45k

The "B" seems odd to me, but excluding it as size increases tension increases. (comparatively across makes, my C3 is 6'1", having ~45k tension)

After more digging, it also seems that tension to compensate for length has a higher impact on approaching the tensile limit of the wire compared to tension compensation for diameter (to the extent that diameter can safely be increased). Which would mean that as a speaking length gets longer for a given note, the wire would need to be lowered in gauge with increased tension to prevent breakage (I think?). Which seems to make sense given the above mentioned tensions.

Not a techie, but I am a little nerdy.(Or am I flirting with you? who knows!)

Hay gurllll!  

Anyways:
Depends, are there sets of 2/ sets of 3 unisons and how many? each piano is different. I am pretty sure that the gauge is not changed across each string in sets of 3(uncoiled) and sets of 4.  a4 is string no 49...so that would be what I would google. The tension for a4(why don't you just give me a length of the 1st string and the a4 string you need to gauge/add tension?)  I am going to calculate that and respond back to you today(please, do I have to?)

Don't all A4 have a set 3? Again, foreign territory for me, all I have is my piano for reference. The gauge doesn't change within each individual note and it makes sense not changing when a wire is looped for 2 speaking lengths. That would mean that it could theoretically change every other note nonetheless (in the 3 voice notes). This is all entirely theoretical. I went on this tangent when I realized there wasn't even a semblence of standardization. So I don't have a speaking length to measure, it's about theoretically creating one and building an optimized scale/harp design.

however, as the pitch increases, the gauge and length slightly decreases, so perhaps the tension on each string is actually the same! (it depends on your piano how many rotations of the tuning peg it takes to achieve specific frequency)

Equal tension across the keyboard would allow a piano to stay in tune longer, so in an ideal world they would all be equal as long as the tone wasn't negatively effected. I believe that's how piano manufacturers aim to design their piano. Given that many of them still use their designs from half a century ago, modern computing can optomize the actual harp design based on the tension they want to achieve across the keyboard, a la:



This would allow them to minimize the weight of the harp and design their tension scales more effectively.

Side Note: While I was searching for the Steinway tension scales, I bumped into this quote on the section of their website explaining the piano industry and got a kick out of it:

"Several adequate pianos are built in Germany and Austria. They are Bechstein, Grotrian, Forster, Bluthner, Fazioli (Italy) and Bosendorfer*. Fazioli and Bosendorfer however, are distinguished only by their high price and perceived obscurity and are not taken seriously among artists, concert halls or music schools."

[lloyd_cdb]

Measuring my own piano, I believe the speaking length is ~15.5" (rough guess without shoving a tape measure between dampers), @440hz and 198.5lbs tension is a #19 gauge piano wire. But if it's widely variable, then I'd be better off just doing it from scratch.

Another thing that popped into my head was spacing between the voicing strings. So within a 3-string note, would a gap of 2mm betwee each of the strings differ from a gap of 4mm assuming they were still struck equally?

[chopin2015]

Measuring my own piano, I believe the speaking length is ~15.5" (rough guess without shoving a tape measure between dampers), @440hz and 198.5lbs tension is a #19 gauge piano wire. But if it's widely variable, then I'd be better off just doing it from scratch.

Another thing that popped into my head was spacing between the voicing strings. So within a 3-string note, would a gap of 2mm betwee each of the strings differ from a gap of 4mm assuming they were still struck equally?

 

I believe that the space between each note probably has an effect on the way you would have to tune and dampen your piano. Even though right of the top, wood and glue are the medium which colors the tone, the strings placed right next to each other would probably be more prone to movement and excitement and I assume work together with relevant strings to create a more rich, LOUDER sound. And texture. However, idk about the highest octaves. Over all, low strings are longest and thickest, spaced apart the most. Space closes in as you shrink length and gauge, and increase pitch(if the string was longer, more tension would be needed to get same high pitch, i think) the space gets smallest as the strings get shorter, because the higner pitch is not as audible now, and will not affect other strings as much if positioned further away. As opposed to lower strings which need to be further apart and away or they would resonate too much. The spacing varies according to the curve shape of the harp?
I assume resonance is important across the whole piano...im not drunk, i swear!

Cant think, gotta go finish practice.