Harmonics and Tuning by Ear

Many guitarists use harmonics to tune up by ear, or to check their tuning. For example, they might ping the A string over the 7th fret and then ping the bottom E string over the 5th fret. Or ping over the 7th and 5th frets for any other pair of strings (except B and G which are pinged over the 5th and 4th frets respectively). We might expect these pinged tones to sound exactly the same, ie in unison with no "beating". But, as many guitarists know, tuning by harmonics (pinging) can give errors. A simple test is to tune all the open strings carefully to a good quality tuner. Then "check the tuning" by pinging the aforesaid harmonics. On a well-built guitar, they'll be a bit off. Why??

Guitars are built to play scales based on what is called "equal temperament", which means that an octave is "divided" into 12 equal parts. However, books on music rarely explain what "equal" or "parts" really mean - well, not to any great depth anyway. Instead, they wander off into the world of Greek and Medieval tuning and funny scales derived from harmonics and then adjusted a bit by "tempering" certain notes. Way too much information. It's really quite simple: in equally-tempered tuning, the ratio between the frequencies for any given musical interval (i.e. a pair of notes) is always the same, no matter which note in a scale is the lower of the two notes - and this is why the frets on a guitar have a spacing that decreases by an exact ratio as one goes up the fingerboard. If they did not, they could look like this - a special fingerboard for playing non-Western music.





So, what is this magical "exact ratio" and how is it calculated?

As we know, an octave is made up of 12 semitones (the chromatic scale). It is also well-known that, for the interval of one octave, the upper note is exactly twice the frequency of the lower. Therefore, the trick is to find out what ratio multiplied by itself 12 times comes to a value of 2. Since this is not a maths lesson, I'll tell you that this ratio can be calculated by finding the 12th root of 2, i.e. 12√2. Piece of cake ;-)

According to my Casio Scientific, 12√2 equals 1.059463094 but, even though that's a long number, it's not exact. 1.05946309435 is a little more precise but numbers like 12√2 can not be expressed exactly in decimal form. Be that as it may, now we do know the magic ratio precisely enough for our purpose and we can calculate any interval's frequency ratio, assuming that we know the number of semitones contained therein. But first, what has the foregoing to do with harmonics?

When we ping a string, it will vibrate at a tone higher than the fundamental (open) tone. In the opening paragraph, pinging at the 7th fret was used as an example. Pinging on the A string at the 7th fret causes the string to sound at exactly 3 times it's fundamental frequency of 110Hz, i.e. 330Hz. Pinging the bottom E string at the 5th fret causes that string to sound at exactly 4 times it's fundamental frequency (see next para). Musically, these two tones should be in unison but, on a perfectly-built guitar with perfect strings, they'll be a little off - beating about once every 3 seconds. Again, why?

OK, please recall that equal-temperament is the normal tuning for a guitar and recall also that, to be in tune with rest of the band, we tune our A string to exactly 110Hz (two octaves down from the concert standard A=440). So, what is the tonal "distance" from our A open down to E in semitones? Musically, the interval is a perfect fourth, which is 5 semitones or frets. So, on a guitar with it's equal temperament construction and with the A string tuned to a frequency of exactly 110Hz, the frequency of the low E tone would be: 12√2 raised to the power of 5 and then divided into 110Hz (trust me). Which is a fundamental (open string) frequency of about 82.407Hz. Now we can figure those harmonics: with a correctly tuned guitar and with all else being equal, The third harmonic of A is 330Hz but the 4th harmonic of E (82.407Hz times 4) is 329.628Hz. There we have it - there is a difference of 0.372Hz. Not much you might say, and you're right - it isn't much - only two cents! In fact, if the E string is stiffer in construction than the A, the real difference could easily be negligible.

However, when we tune higher pairs of strings, the difference is more noticeable. The best example is tuning G to B. Pinging B at the 5th fret and G at the 4th until the beats are longer than say two or three seconds will give a G string that sounds horribly sharp when you play an open E chord. Let's do the calculation . . .

Tuning the B string to a tuner should set it to a frequency of 246.942Hz or thereabouts. Then, when we ping the 5th fret, it should sound at 987.767Hz (2 octaves up, i.e. 246.942Hz times 4). Now pinging the G string at the fourth fret sounds the 5th harmonic (open G frequency times 5) and therefore, if we have tuned it perfectly, the open G is now tuned to 987.767Hz divided by 5 = 197.553Hz. However, this frequency is not what your electronic tuner gives out, which is actually 195.998Hz. So, by pinging, we've made the G too sharp. In terms of musical cents, the difference is almost 14 cents - very noticeable to most people.

Now that we know tuning by pinging gives a slightly sharp lower string, what if we started with the top string set to a tuner and then tuned by pinging each pair downward i.e. E/B, B/G, G/D, etc? The errors would add up through these 5 tuning steps. How sharp would the bottom string be? I figure it to be sharp by 21.5 cents. If you then pinged the bottom E at the 5th fret and played the top E open, it wouldn't sound at all good!

Conclusion: Because our scale is based on octaves, it would appear that the only safe harmonics to use while tuning up a guitar are the octaves, (over the 12th or 5th frets). Higher harmonic octaves are less useful because, on real strings, the higher harmonics play sharper than they should due to inharmonicity. Also, I conclude that tuning from an inner string toward the outer strings would give less error that tuning from the top string down to the bottom - for example, see a method used by Kevin Ryan here.

Note: number of cents different = 1200 * log (f2/f1) / log 2

Best regards,

Ted

Strat All Done

Well, here it is . . .

I built this "Schallercaster" guitar as a contrast to my "Yamacaster" (Yamaha Pacifica). I was looking for a jazz/latin/blues sound and had some Schaller Golden 50's pickups lying around with nothing to do. I had bought the neck some time back, was it a Mighty Mite? - I forget. The neck is of the wider variety and pretty flat, thereby suiting my long fingers. The body was bought from the now-defunct Mars Music for 50 bucks. It was a refurbished job and I think they said it was a Cort. However the neck end had extra holes for tilt-neck adjusters, so who knows?

A quick test into the Roland Jazz Chorus 77, showed that the electric sound was as expected - very similar to Seymour Duncan '59's. The pickups have the Alnico V magnets - some find them to be "gutless" but they are clear-sounding with every string audible on chords. Because I had not completed the set-up, the bitch just would not tune up - so it sounded pretty un-musical to the ear. For example, the 3rd string played at the first fret was at least a 1/4 tone sharp. Ouch.

On with the set-up. First, the string grooves in the nut were finalized. A book I have says to have the string just touch the first fret when you push it down between the second and third frets. I stopped a little short of that and also left the wound strings a little higher because I use hybrid gauges 9, 11, 16, 26, 36, 46. Next I set the bridge heights until the strings buzzed slightly at the 12th fret because the neck has a little too much bow and I have to take the neck off to adjust the frikkin inaccessible truss rod. Following that little exercise, I adjusted the rollers on the bridge to set the string widths equal and lined up with the neck.

Next, I plugged the guitar into the trusty ZOOM 707 and adjusted each string's intonation by pinging the 1st harmonic and making the fingered 12th fret show the same. After tuning the guitar using the ZOOM's tuner, I casually struck a bottom E chord and damn near wet my pants, it sounded so good. I fretted an A at the fifth, still good and even E at the 12th - still good! All the chord shapes sounded good enough too, without that flat sound on the 3rd string when playing a bottom C for example.

As you would expect with a hard-tail bridge, string bending is easy and also using the finger to produce vibrato, like classical guitar players, works well. I notice too that this guitar has much louder harmonics on each string compared to the Yamacaster. This is good from the point of having a brightly textured sound but not so good if the strings go out of tune :-(

best regards,

Ted

Strat Step 04

Hello guitar-pickers,

Time for a bit wiring action . . .

Here's the pickguard with the stuff installed but not yet wired. It's a custom leftie from Warmouth. I added two extra holes between the pickups for extra pickup support. Just one volume and one tone control, it's all I need. I'm looking for a Gibson sound - but without having to reach down there to mess with all those 4 knobs and then up there to flip the switch. Too much for my tired old brain ;-)

Two 500K pots (well 470K actually) and a simple 3-way switch. The pots are positioned thus so that I can solder the highs together after a bit of bending (saves a link and is perhaps more reliable). I'll also bend the lows back and solder them to the pot body. This gives a little extra shielding to the pot wiper inside.

You gotta have a plan. Seriously, it goes a lot better if you do a schematic AND a wire routing diagram. Then you can do stuff like hi-lite each connection as it is soldered, etc plus you can scan and save it as a record.

Pickguard wiring completed and checked out OK. With the pots turned up, read about 8K on the vol pot wiper to pot body when the switch is in the bridge position, just under 4K in the middle and about 7K in the neck position. Setting the multimeter to mV and pulling a magnet away from each pickup gave a momentary positive voltage for each one, so they're in phase. Doing that now is so much better than discovering they are out of phase after plugging the guitar into an amp! As you can see, the coil tap wires (brown&white) still need to be insulated.

Later, I'll be connecting the ground wire from the bridge and the shielded cable from the output jack, leaving about 10" length for ease of fitting or removing the pickguard assembly.

Best Regards,

Ted

Strat Step 03

Hello guitar-pickers,

More progress this week. Even had some strings on for a while!

Put the Schaller machines on - silky smooth feel. The holes were 3/8" but, being German, the post sleeves are 10mm - so a bit of filing was needed. I don't like the string tree on the 5th & 6th strings, it will be removed. Too steep an angle over the nut and heavy string wear - even just installing them seriously flattened some windings.

Got the neck bolted on. Fortunately the plate holes lined up with the existing ones in the body. Pity that the truss adjuster is buried in there now. No way I'm going bore an access hole in the body behind the neck pickup.

Here's the pickguard I'll be using. It's a Warmoth custom. They have a a great option selection form and your item is built from your choices. I chose Pearlite, 11-hole, two knobs, leftie, two standard humbuckers. It's great looking.

The Schaller Golden 50's bridge pickup. It'll be mounted with the adjustable poles toward the neck. I may use Schaller 2in1's eventually, because they have two rows of adjustable poles giving a better shot at good string balance.

Next job is to assemble and then wire up the pickguard. Should be fun, but I've done it before many times, so I don't expect problems. I'll be using 500K log. pots a là Gibson and a 0.022 uF capacitor. Just a 3-way Strat style switch, no fancy coil-switching on this job ;-)

Best Regards,

Ted

Strat Step 02

Hello guitar-pickers,

The next step was to fill in the pre-existing tremolo hole in the body. I will be fitting a Schaller hard-tail (non-tremolo) bridge. Full of promise, the saddles are adjustable sideways - in addition to the usual up/down and back/forward. By that means, the string spread will be made perfect and then locked in place with the built-in cam screw.

So, I'll be needing to fill the big tremolo hole where the bridge fixing screws will go.

I had some Yaupon Holly stacked for burning, dried about two years. Yaupon is much like oak - very hard and tough with a fine grain. It grows on my place like weeds. So I cut this piece to go in from underneath (the trem hole has a lip which, in retrospect, I should have removed).

Glued it in with carpenter's white glue.

Then I made another piece to fill in the top.

Now I'm full of admiration for people who can "dutch" holes in furniture.

A bit of wood filler fixed my rough edges!

A lick of black enamel - looks horrible. I'll try some better stuff tomorrow.

P.S. I did try some other enamel but the thinner I used blistered the existing paint. I'm not going to try and fix that - it makes no difference to the sound. So I just sanded it level with 220 grit paper.

Best Regards,

Ted

Strat Step 01

Hello guitar-pickers,

I'm building a Strat-style from bits and pieces. Many of the bits were made by Schaller, in what was West Germany. Since my "Yamacaster" (Pacifica)has been reverted back to single-pole for the neck and middle pickups, I'm going to go Les Paul and use a pair of Schaller Golden 50's humbuckers combined with a Schaller 475 hard-tail bridge. The machines are also by Schaller so, now, I am formally introducing my xpatUSA Schallercaster!
I'll publish progress in a series of posts called "strat step 01" et. subs. I might put a news feed up on my website so that people can put it in their NewsReader if they so desire. So, here we go with the first post . . .

My new neck came without a nut, so it was off to Bellville meat market to get one of these . . .

After leaving it outside for a good while, I cut the middle piece out. Gross, the rotting marrow inside still stunk to high heaven, so my wife kindly boiled it clean.

Next, an 18-point tenon-saw was used to slice off a piece about 3/16". More stench!

After much fret-sawing and filing down to 1/8" thick, here it is

At first, slotting the nut looked like it would be a simple job. Ya know, just figure the required spread of the strings and divide by 5. However, a little Googling revealed that equal spacing between the slots slot is old hat and that the modern way is to have an equal gap between the string edges. Fortunately my math is still up to such complex subjects and the internet has some string-spacing calculators anyway. Also, Stewart-Macdonald has a nifty ruler which increases each spacing by 0.004" as you go from treble to bass. We'll see . . .


The next job is to fill in the pre-existing tremolo hole in the body. I will be fitting a Schaller hard-tail (non-tremolo) bridge for which (of course) the fixing screws go right where the big tremolo hole is.


Best Regards,

Ted

WebSite: http://kronometric.org