Why fretted instruments can never be in perfect tune in all keys

[Brutus1999]

Maybe someone already has posted on this. Someone recently commented that the second fretted B note on the A string often sounds sharp.

The reason for this is because in a "PERFECT CHORD" for example, in order to have the overtones (secondary sound waves--imagine a wave hitting the side of a tub and making smaller waves along with the big wave although that's not really an accurate metaphor) -- for the overtones to sound "PERFECT" on a G scale, the C note should be vibrating at 1.33333333.... times the frequency of the lower G note, and the D note should be vibrating at 1.50000... times the frequency of that G note.

You could set up a piano that way, and you could set up the frets on a stringed instrument that way. However, it would only be in tune IN THAT KEY. In order for the A note to be in tune with that D note for the key of D, that A note would have to be vibrating at 1.5 times the frequency of the D note. The same would go for the E note in relation to the A, the B in relation to the E, and all the way around seven frets at a time until we get to a G note, which would NOT be in tune with the original G note. (I'm leaving out all the other notes of the scale which would similarly be "off" all along the way.) And then putting it in tune for the key of C, well, the G note in the key of C would have a different frequency than the G note in the key of D.

To compromise, musician/mathematicians figured out that the two notes that sounded best together have double the frequency and so decided to give them the same letter. C and C' (or an octave higher). What kind of scale can go up an equal number of "vibrations" to get to the doubled frequency? Well, they took the TWELFTH ROOT of two -- in other words, a number, multiplied by the ROOT "C" (for example) which would create the next note C# and then multiplying by the same number, the next note "D", etc. until, after twelve times, it becomes the doubled "higher octave" C.

They picked that because five frets (half steps) up, the vibration is 1.3348 times the root note, which is pretty close to 1.333, and seven frets up, the vibration is 1.4983 times the root note, which is very close to 1.500.

However, the other notes on that scale do not EXACTLY, PERFECTLY create the mathematically perfect, "sweet" overtones. In particular, the second half step is 1.122 instead of 1.125 (a little flat, often not a big deal) but the fourth half-step is about 1.256 instead of 1.250 (a bit sharp enough to hear). On a mandolin, to sharpen up the A string so that it doesn't sound that teeny bit flat means that when fretted at the second fret, (to be the fourth half-step in the G scale) -- well, that B note is going to be even a little sharper. And if you change keys, the whole thing becomes a big mess! So "perfect tempered scale" in one key would be a big mess if you change keys.

(This problem is also compounded by the thickness of the strings, but that is another issue dealing with "compensation" etc.)

The "EQUAL" tempered scale is the best "compromise" to be able to play in different keys even though each note will not create the perfect overtone, the main notes are pretty close. On a fretless instrument, one can ever so slightly adjust the string length. On a fretted instrument, a combination of set up (because poor set up can exaggerate this mathematical problem) and also technique (slight bending to sharpen a flat note, etc.) can make it sound pretty good.
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Below is a chart I copied from (gasp!) Wikipedia -- or you can just go there and read about "EQUAL Tempered" scales on stringed instruments.

Okay, that's enough. There will be a quiz next Monday!

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Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation Error
Unison (C) 1.000000 0 = 1.000000 0.00 0
Minor second (C♯/D♭) 1.059463 100 = 1.066667 111.73 −11.73
Major second (D) 1.122462 200 = 1.125000 203.91 −3.91
Minor third (D♯/E♭) 1.189207 300 = 1.200000 315.64 −15.64
Major third (E) 1.259921 400 = 1.250000 386.31 +13.69
Perfect fourth (F) 1.334840 500 = 1.333333 498.04 +1.96
Augmented fourth (F♯/G♭) 1.414214 600 = 1.400000 582.51 +17.49
Perfect fifth (G) 1.498307 700 = 1.500000 701.96 −1.96
Minor sixth (G♯/A♭) 1.587401 800 = 1.600000 813.69 −13.69
Major sixth (A) 1.681793 900 = 1.666667 884.36 +15.64
Minor seventh (A♯/B♭) 1.781797 1000 = 1.750000 968.83 +31.17
Major seventh (B) 1.887749 1100 = 1.875000 1088.27 +11.73
Octave (C) 2.000000 1200 = 2.000000 1200.00 0

[mrmando]

Well, what we need is an Laser Fret System(TM), or LFS. Your instrument is fretless, but equipped with a multibeam laser projector. You select which key you'll be playing in, and presto! The LFS projects a series of frets onto your fingerboard in the optimum positions for that key. Assuming that all the beams are projected from a single point, the frets would even be fanned! Is the next song in a different key? No problemo, just select it and watch the frets shift ever so slightly to a new position! Or change keys on the fly with the optional foot pedal!

I've never made a fanned fretboard, but they look to me as if they're designed to have a "vanishing point" at which all the frets would converge if extended far enough off the fretboard. Trouble is, the fretboards usually look as though the vanishing point is at least a couple of feet off the treble side of the instrument. The LFS projector would have to be located at the vanishing point, which might make holding the instrument a little awkward.

[bribhoy]

But I like the sound of the mandolin - so you're telling me I prefer music to maths?

Actually, I'm OK with that.

Brian

[resophil]

When my daughter was in Grade 9, she won the city science fair with this exact concept, except she started the other way round, by looking for the ratio between adjacent steps of the tempered scale. Interestingly enough, her display board was entitled "Music and Math" as well...

What she attempted to do was to find the decimal equivalent of the coefficient in the Minor Second interval. In other words, If you take a scale length measurement number, divide if by this coefficient once, then divide the result by the same number, and so on for 12 times, you will end up with half the original number. That was in the days before computers were everywhere, no laptops to be had, but we took in a small PC that I had in the day where she had used a spreadsheet to compute these figures. Very impressive display from a 14 year old girl in the "pre-computer age..."

Then I helped her build a small banjo which had the frets cut according to a printout that the spreadsheet provided. She cut fret slots and I hammered in frets for her and dressed them to make it playable. It worked like a charm. There were a couple of string instrument players in the judge's panel who were really taken with the little banjo, and it helped her to the prize!

[Brutus1999]

Just so folks don't go nuts trying to tune these things mathematically perfectly -- just play with a banjo player and whatever you do will sound sweet (I can say that...I play banjo too...)

[Dale Ludewig]

mrmando- on a fanned fretboard, each string (or course, on a mandolin) is a different scale length and every one of them is tempered and hence, out of tune. It's a beautiful thing. Imagine what a fanned fretless fingerboard would be like to play!

[re simmers]

Not to further complicate things, but when math gets to a certain level you will find that it is not a perfect system either.

Bob

[foldedpath]

It's also not a perfect system when you're doubling melody with a fiddler who has their G, D, and E strings tuned in perfect 5ths from their A string referenced at 440 Hz.

Or a piper, playing tunes with a "C Supernatural" somewhere between C and C# (which is a really cool interval in Scottish and Irish music).

Somehow it all works, sort of. We do the best we can.

[Willie Poole]

People have always said that I had a good ear but when I am sitting around playing any of my mandolins I am constantly tuning but when playing with all of the other band members mine sound like they are always in tune....I can`t explain it but I guess if it is close that is good enough....

All of this info is way over my head....

Willie

[OldSausage]

Well, and I always thought it was the banjo picker's fault.

[Ivan Kelsall]

Read here :- en.wikipedia.org/wiki/Pythagorean_comma,
Ivan

[Lee]

The word "perfect" shouldn't be used.
It's interesting that you specified the strings' harmonicity from 1.00000 up 2.000000. When we tune, there's also the string's internal harmonicity. Also, the piano will react differently when it's a small vertical instrument compared to a full size Steinway or Mason. "Perfect" tuning doesn't need so much attention, unless you like to. I certainly do, because it's like tuning from 4186.01 Hz down to 27.500 Hz and everything in between, plus their harmonicities.

[mrmando]

mrmando- on a fanned fretboard, each string (or course, on a mandolin) is a different scale length and every one of them is tempered and hence, out of tune. It's a beautiful thing. Imagine what a fanned fretless fingerboard would be like to play!

Well, then instead of projecting the frets with a laser, you could use LEDs embedded in the fingerboard.

[Capt. E]

I also play a Cajun style one row button accordion, which is diatonic (tuned to one key), and they are tuned in what is called "just" tuning where selected notes are a bit "off" perfect 440. See this little video to get an idea of the sound you get: http://www.youtube.com/watch?v=1TtwSPU5pCk
In fact this is common in the accordion world where tunings can be as much as fifteen "cents" off which produces a characteristic wavering sound you hear especially in French musette tunings. Cajun instruments are usually about 5 "cents" off. This improves the sound of the chords, especially where thirds are involved. You can't do this with a fixed note chromatic instrument as it will sound "bad" (to use a very technical term).

[John Ritchhart]

It's OK for me because one ear is lower than the other.

[Stephen Perry]

Takes me 10 minutes to get used to guitar. About 20 for a piano. Assuming they're in tune. Comes from playing bowed strings. Gets one away from that fret crutch! But the fingertip damps the sound, so continuous excitation is in order, so the pick gets replaced by a bow. Otherwise identical to a mandolin!!!