Long scale length, low tuning, string tension, flexibility, construction, and inharmonicity

Discussion in 'Extended Range Guitars' started by Hollowway, Nov 1, 2017.

  1. Hollowway

    Hollowway Extended Ranger

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    Ok, guys, I’ve been googling up a storm tonight, and I want to try to get to the bottom of this issue about why we need such higher tension on longer scale instruments, and what to do about it.

    So, consider that bass strings have tensions in the 30-40 lbs range. And they STILL flop around against the frets more than a 11 lbs high E string on a strat.

    Now, I understand that the sheer length requires substantially higher tension. But can we not figure out an equation to help determine what that tension should be? We now stiffness goes down with the cube of the length, so that’s a HUGE drop in stiffness for a small increase in length. But we don’t have an equation to help pick a string thickness.

    Then, there’s the core thickness (hex or heavy core) that makes the string less flexible, allowing for a smaller diameter string. But wouldn’t that inherently increase inharmonicity?

    As a practice issue, I have a 30” guitar tuned to G#0, with a .142” Kalium. And it STILL feels too floppy. And I have an instrument at 32” that I want to tune to G#0, but I am not sure if the increase in scale length will make that .142” feel tighter, due to the stretching the extra 2”, or if it will feel looser, for that increased length.

    I’m looking at piano stuff online, because clearly that has been figured out (based on scale length issues and string diameter). But it’s heady stuff, and it’s not easy to slog through.

    Anyway, give me your thoughts. And @ixlramp @bostjan @Winspear get in here!
     
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  2. Hollowway

    Hollowway Extended Ranger

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  3. Hollowway

    Hollowway Extended Ranger

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    A little more reading.... So, I’m quite sure I’m way over simplifying this, but if stiffness is related to the length of the string cubed, then we could say that 32” scale length string needs to be roughly twice as stiff as a 25.5” scale length string. Because 32” is about 1.25 times as long as 25.5”. So, all other variables being constant, if we increase he length 1.25 times, then we need to cube that, which is 1.95. So the 32” string needs to be 1.95 times as stiff as the 25.5” string to compensate for the new scale length. Which means that if a strat guitar string is, say, 17 lbs, we’d need a 32” string to be about 34 lbs. And that seems right in line with what bass string tensions are.
     
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  4. Winspear

    Winspear Tom Winspear

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    Interesting thoughts! Seems like a good theory. Right now I don't know much more about it than my preferences, and I believe I posted before about weighing up the pros and cons of a longer scale (for example would there perhaps be a case where a shorter scale will actually allow a thinner string and sound brighter than the longer scale which may need a thicker string - depending what gauges are available). As you said, needing a thicker string would increase the inharmonicity. But perhaps the scale length more than offsets it. I wonder if that inharmonicity calculator may show something useful...

    Here's some rough numbers going off what you've said above:
    len 32
    e1 .100 ckwng == 34.39#
    len 30
    e1 .095 ckwng == 27.17#
    len 25.5
    e1 .090 ckwng == 17.66#

    Whilst I know I'd enjoy those exact steps in feel, I do think it's a bit excessive. The 32" would be a lot more like a bass than the guitar, and I feel like you could easily get away with 90 still on the 30" and perhaps 95 on the 32" from my previous experiences. But I do agree a longer scale hasn't resulted in much of an ability to use a thinner string in my experience - aside from subtle changes like 25.5 to 27 drop A 70 vs 68 for example. The more extreme ERG tunings I've definitely found a need to retain great tension and benefit only from the brighter tone of the scale but not a gauge decrease.

    I actually messaged Durero about this because he has the 34 (or was it 36?) tuned only to 7 string standard B. I was wondering what his thoughts on needing to increase the tension were - and he actually said he hasn't found a need to providing that the picking position is compensated for (i.e. your hand remains very close to the bridge). This is one thing I am considering about long scale design - moving the bridge up a bit rather than having it right back like a bass, even though I prefer that aesthetic I do find it a little awkward for muting and do feel like I'd be able to get a better playing tone with a more conventional bridge position. Shame about the reach to the nut...
     
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  5. InfinityCollision

    InfinityCollision SS.org Regular

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    Bear in mind that string geometry is a factor here (round vs hex core, number of windings, diameter of core and windings), along with material differences between core and windings.

    I believe this (inverse) relation should be squared, rather than cubic?

    The general concept still checks out though. A 20lb E2 on a LP is roughly equivalent to a 38lb string on a 34" bass. Even a 19lb string on 25.5" needs about 34lbs when scaled out proportionally. Given that electric bass tensions are probably more aligned with more traditional electric guitar tensions (.010-.012 E4 string) than the ultralight sets many metal players favor, this fits perfectly: a 23lb .052 string would match to about 43.5lbs tension at 34".

    Of course, there are additional considerations in each instrument's typical playstyle and desired timbre. Still, it's worth considering... Perhaps it's still too aggressive? Perceived tension doesn't necessarily follow such simple numbers.

    Yes. Inharmonicity is strongly related to core diameter, at a rate similar to core diameter's effect upon string stiffness.
     
    Last edited: Nov 1, 2017
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  6. Beheroth

    Beheroth SS.org Regular

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    I think there is confusion between stiffness (wich depends of the strings material mechanical properties and the construction of the string) and tension (wich is related to the scale length, the linear weight of the string and the angle after nut and bridge and the tuning of course).
    I think ideally you need a LESS stiff string and a bigger gauge
     
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  7. bostjan

    bostjan MicroMetal Contributor

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    The stiffness of a string is the ratio between the force on the string and the deflection.

    k = F / d

    The stiffness at slack is also given by the material properties, theoretically as

    k = A E / L

    where A is the cross-sectional area of the string (pi times radius squared or pi times diameter squared divided by four), E is the modulus of elasticity of the material or composite, and L is the length of the string.

    So, it's tricky to say this increases that squared or whatever, because, it depends on what you hold constant. If we vary length but use exactly the same string at exactly the same tension, then stiffness is inversely proportional to length. But on a guitar, you are going to adjust the tension or the string gauge or whatever, so the relationship is not really quite as meaningful.

    Inharmonicity is also directly proportional to the modulus of elasticity, so using a softer material for a string yields lower inharmonicity, but gives less stiffness. Say you decrease the elastic modulus by 10%, you decrease the inharmonicity by 10%, but you also decrease the stiffness by 10%.

    First harmonic inharmonicity due to stiffness alone is given by

    G1 = pi² d² Y / ( 128 L^4 f0² rho )

    Where d is diameter of the string, Y is the composite Young's modulus (about 2 x 10^11 for unwound strings, but significantly lower for wound strings), L is the length of the vibrating part of the string (scale length, if the note in question is the open string), f0 is the fundamental frequency of the note, and rho is the density of the composite (about 8 x 10³ kg/m³ for unwound strings, but significantly lower for wound strings).

    It's a good guess that scale length pretty handily dominates that formula, though. G1 is like 1/L^4. Twice the scale length is 1/16th of the inharmonicity, all else equal (which it's not, because you'd be using a smaller diameter string to obtain the same note, which decreases inharmonicity even more).

    As far as stiffness goes, there's the situation of the tension, and the intrinsic material stiffness at slack, which should sum up linearly to give total apparent stiffness. With the extrinsic component increasing stiffness like L² and the other decreasing it like 1/L, the L² component (external tension) should dominate the behaviour at "longer lengths."

    So what's a "longer length?"

    pi d² E / 4L >> pi rho f0² L² d²

    or more simply

    E/4 >> rho f0² L

    Since E/4 is a big number (10^10 Pa), and rho (10³ kg) f0² (100 1/s²) L (1 m) multiplies out for low range guitars to 10^4 Pa, this is just never ever the case. At higher pitch, this is simply not an issue.

    So how do we get length to dominate stiffness? Well, I think we can't, unless you can make the pitch higher, the density much much greater, the length much much greater, or some combination of a little of each.

    Thus is the struggle of tuning an ERG.
     
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  8. Dayn

    Dayn silly person

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    Is your concern about perceived tension, rather than actual tension? From what I understand, the longer the scale length, the less perceived tension you'll have. I don't think there's a way around that whatsoever. There's more string, so there's more give. There's more to stretch, so it'll stretch more. Only way around it is to use a shorter scale. ...Or find a string that won't flex, which would sound dull if it didn't break from tuning it.

    I'm not sure how stiffness works in string construction. All I know is that it certainly exists.

    Density, however? Of course. Construction being the same, the denser string has more mass which has higher tension for a given frequency. Hence why stainless steel strings usually have slightly higher tension than the equivalent nickel strings.

    But again, density doesn't equate to stiffness. I've used an .080 Dunlop bass string for E1 at 27". Stiff, dull, couldn't intonate. Using a bigger .090 Kalium guitar string? Thicker, higher tension, but it's far more flexible, sounds brighter, and intonates well.

    I think at a certain point we just need to accept the guitar is an imperfect instrument, as annoying as that is.
     
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  9. Hollowway

    Hollowway Extended Ranger

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    @Dayn Yeah, so I’m not really concerned about tension per se. But, I’d like the string to play and feel the same way as the others. But since it doesn’t play or feel the same way at he same tension (because it’s too floppy) I know that the equation for choosing an appropriate string for low tuning at long lengths either has more variables, or is not linear.

    The primary reason for this thread is that I cannot put a string with 23 lbs of tension on a high E at 25.5” and a low G#0 at 32” and expect any sort of consistency. The Hugh string will break, and the low string is too floppy. Now, we know how to work with regular guitar strings and lengths, but the idea of tuning at guitar to G#0 at 32” is new, and we have no way of knowing what would be the best plan there. Heavy core, thicker string, higher tension, etc are all variables we can tweak, but we’re really just doing trial and error. I’d like a little more objective way if figuring it out.
     
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  10. Hollowway

    Hollowway Extended Ranger

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    @bostjan So, given my situation, and your math, what string size would feel like a regular guitar string tension for a 32” G#0? I just estimating (aka pulling a number out of my ass with just enough math to be dangerous) in the mid 30 lbs for tension, and then cross reference that on the Kalium spreadsheet. In other words, I think that the extra length means I should double the “regular” tension numbers for it to perform the same way.
     
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  11. Zhysick

    Zhysick Sick of all

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    And this is the reason why I stopped playing 8 strings... because if you even get an equation for this you will still need to find where to buy that string... you have Kallium there but here in Europe it's a pain in the ass and Kallium is very expensive considering the shipping and customs...

    Really hope you can find an equation for this so the manufacturers can start doing something useful in the strings industry...
     
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  12. ixlramp

    ixlramp SS.org Regular

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    Seems inevitable considering the amount of mass and momentum they have grows by the square of gauge and it is unrealistic to match that with the necessary increase in tension. My impression is that you cannot practically keep the vibration of large strings under control to the extent thin strings are, it would require too much tension to be playable (or would break). You just have to play larger strings more carefully and/or use a higher action off the fretboard.
    Low pitches lose meaning if played too fast so it's perhaps best to not tremolo-pick a low G#0.
    I feel it would be very complex to model, and if done i expect the results will be unplayable and impractical. If you raise the tension the frequency increases so you have to raise the gauge, which then makes it floppier still and needing even more tension. It's a nightmare.

    It seems to me that it is an unreasonable expectaton that we can have big heavy things with a low vibration frequency vibrating like tiny light things, it's fighting against the laws of nature. Big heavy things are floppy with large excursion, the only way to reduce excursion is to raise stiffness or tension very high, but that would then also raise their frequency of vibration.

    Personally i would look at the gradual per-string increase in tension from high to low across the other strings and just extrapolate that, such that the hands get used to a uniform progression of tension. Then adjust technique on the lower floppier strings.
     
    Last edited: Nov 2, 2017
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  13. bostjan

    bostjan MicroMetal Contributor

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    For me, personally, I always bump up the string gauges on the wound strings over what I use for plain strings. A wound string will never really feel like a plain string, but I think the higher tension just feels better and that too much tension on plain steels feels like I'm trying to saw my fingertips open when I play.

    I think that following the same principle as you get into separating the ERG strings from the "normal strings." And if anything is double wrapped of whatever other significant change in construction, it should be considered as something different from the other strings, in my mind. If you use a tapered string, you introduce even more subtle challenges in achieving a crisp tone.

    So, sticking with Kalium, your options are 0.136", 0.142", 0.150", 0.158", 0.166", 0.174", or 0.182"

    At 32", this should give you tension options of 24.6N, 26.7N, 29.7N, 32.8N, 36.3N, 39.1N, or 43.4N.

    Stiffness is difficult to calculate without the elastic modulus of the string.

    Let's take a .046" E string on a strat. L = 647.7 mm, d = 1.17 mm, f0 = 82.41 Hz, E = ~60 GPa, k ~ 100 kN/m

    E/4 is about 15 GPa.
    rho f0² L is about 1300 Pa

    For your .142" G# string, I simply can't even guess what E is. It's safe to say less than 60 GPa, but how much...IDK.

    rho f0² L is about 1550 Pa, so it's actually higher, but I have a feeling that E is significantly lower than for the E string. It ought to be, because the core takes up a lot less of the string's volume.

    I guess "I don't know" is a pretty horrible answer. But the shear stiffness of the string would be easy enough to measure. If there's time tomorrow, I'll grab the thickest Kalium I have and see what sort of stiffness I get.
     
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  14. Beheroth

    Beheroth SS.org Regular

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    nope, it's the other way around : nickel is heavier than steel

    I think you mean lbs insted of N, otherwise it's like granpa's dick : floppy

    Back on the subject : i think all those maths are pretty pointless, you just need to figure out what FEELS good to you and is adapted to your style of playing.
    Also, at those scale lengths you're pretty much in bass territory and bass players struggle with tuning this low
     
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  15. bostjan

    bostjan MicroMetal Contributor

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    Did I get sloppy with my units of measure? Sorry.

    Bulk nickel is heavier than bulk steel. Strings are weird, though, because rarely are nickel strings made of pure nickel...usually they are made from an alloy of iron and nickel. In fact, usually that alloy contains way more iron than steel. Stainless strings might actually contain as much nickel as regular nickel steel strings. :lol: None of the manufacturers really ever explain which grades of materials they use. D'Addario has extensive unit weight tables published, which tell us that "Nickel Plated Steel" guitar strings are significantly less dense, in terms of mass per unit length, for the same commercial thickness gauge, than "Stainless Steel" guitar strings, but that the "Phosphor Bronze" strings are significantly heavier than either.

    Whether this is true for other manufacturers, in general, or not is a bit of a guess, but it seems to be the case for most brands of stainless strings I've tried.
     
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  16. Beheroth

    Beheroth SS.org Regular

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    i'm looking at the d'addario charts right now : nickel strings have sligthly higher tension than the stainless equivalent (up to a pound for a bass string).

    not trying to mess with you, just saying
     
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  17. GunpointMetal

    GunpointMetal SS.org Regular

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    I'm doing A0 on a 30" and from my experience, once you get that low, your options are get higher string tension closer to bass and have your guitar sound like a bass or go with "guitar" tension and adjust your playing style. I'm running a 118 from Kalium for that A and it moves A LOT when I hit it hard, and I can fret any note sharp if I dig in, but with a little moderation it plays well and balances with the other strings. I can't say much to inharmonicity, because I'm not expecting anything that low with some gain on it to have an audible fundamental anyways, but it does carry enough pitch information to sound good with the right notes and bad with the wrong ones instead of just being a noise or anything. I'm talking with a local guy about building a 32"-29" FF headless 10-string so I can try for E0, and I'm guessing I'll be pushing the limits of what Kalium can put on a guitar, lol.
     
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  18. Winspear

    Winspear Tom Winspear

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    Looks like the bass steels are lighter whilst the guitar steels are heavier. Weird!
     
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  19. Winspear

    Winspear Tom Winspear

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    I'm really curious to test what can be done with an Evertune to at least take pitch stability out of the equation. Of course, fretbuzz is not solved, but if pitch stability is the main issue then an Evertune would allow increased performance and/or a lower gauge too.
    I'm also putting a high priority on getting the pickup for the lowest string only separated and moved a cm or so closer to the bridge. Multichannel setups like that are definitely going to be the way forward to me, even if not taking advantage of individual EQ and instead being summed to mono - custom positioning could solve a lot I'm sure.
     
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  20. GunpointMetal

    GunpointMetal SS.org Regular

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    I've been curious about the Evertune for pitch stability as well, but I read somewhere that they add several pounds to a guitar's weight....8/9 string guitars already weigh enough.
     
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