Pure tuning for guitar

How to tune a slide guitar to pure pitch: (you might first like to skip to the paragraph titled 'Important'). QUICK METHOD just tune the top two. 

Tune the low E to a D, but 2 cents sharp. For the time being we will leave the A and D in standard tuning to make this simple. Later you can make them beatless using harmonics if you can be bothered.

Using the G string as our root note (you may therefore want to tune your A string to a G as well) the B string is a major third up, so we will tune this 14 cents FLAT (it's actually closer to 13.7 but 14 will do).

The top E we now tune to a D (the perfect 5th of G) but again, 2 cents sharp so that it is 702 cents (perfect) instead of 700 cents (Equal Temperament).

We now have DGDGBD. The D's and G's should be 2 cents wide. You can check this by playing the harmonics, which should be beatless (not the band). The B is 13.7 cents flat to the G.

Playing the slide across any part of the neck, or indeed fingering a Grand Barre chord across a single fret, will result in a Pure Major Chord, absolutely beat-free and solid. At first this may seem dull and lifeless because all your life you've been listening to beating thirds that are slightly out of tune. You may hear this beatlessness as a lack of 'vibe' but actually the notes are all vibrating together better than ever but our ears aren't used to the different tuning or the lack of beating. Of course you can always later on add vibrato to get it back but that's not the exercise here.

This is what Beethoven, Mozart, Strauss and all the Masters heard when they composed music BECAUSE EQUAL TEMPERAMENT WASN'T UNDERSTOOD PROPERLY until at least the late 19th century. In fact it wasn't until 1917 that W.B.White came up with a formula for tuning a pianoforte accurately to ET, all based on the piano tuner counting the beats between every single string and note, sometimes as much as 7 or 8 beats per second (a real skill). 

That's what piano tuners do - what makes ET what it is and facilitates total freedom in modulation. Every INTERVAL must BEAT on a well tuned ET piano because they are all somewhat out of tune. 

Up until 1917 it was hit and miss, actually constituting a type of quasi-equal temperament. Some of the black keys were still "off" a little! Like harpsichords and clavichords, the first pianos (invented by Bartholemew Christofori) were tuned to meantone or well-temperament as made famous by Johann Strauss and his Das Wohl Temperirte Clavier 24 Preludes.

So when the Grand Masters played music, composed music, or listened to music, equal temperament did not exist (hadn't been invented) so therefore the majority of it was either in pure tuning, meantone, or later, well-temperament. String quartets invariably played using pure tuning, the strings tuned to beatless perfect fifths or fourths (eg viol, viol de gamba) depending on the instrument. Inherently the open high E string is therefore quite out of tune with a fingered low C. Violinsts therefore will always use their fingers to play a high E. Learners use the open strings which is why they sound so bad.

Open strings must be used judiciously, usually only played simultaneously with stopped strings as a fourth or fifth drone, the thirds and sevenths being avoided because of the out-of-tuneness. Interestingly, fiddle players do the opposite and may utilise the open string/s as a major or minor third to add an 'edge' to their playing. That's why fiddles are so often 'off' yet raucous!

Pianos were originally shunned by orchestras because they didn't fit in with the strings and brass which naturally tend towards pure intervals. In those days they weren't yet used to Equal Temperament. In fact a modern violinist has to learn to play their major thirds deliberately sharp by 14 cents to make their instrument be in tune with the other ET instruments. A well-made violin will often "go dead" playing an ET major third because the ribs were filed and sanded by luthiers back in the early days so that the strings resonated with all the main keys eg C, F, G, Bb, Eb etc (or so the myth goes... why the Stradivarius was so special, apparently). The quarter comma meantone that I use on piano was common in the period of the Masters, especially on harpsichord and clavichord, meantone having being developed and formularised by Pietro Aaron since before the 1500s.

Important:

The Human Ear has difficulty discerning difference in pitch of a few cycles per second and only begins to hear separate notes at around half a dozen cents (5-6 % of a semitone). Worth noting here is the difference between cents and cycles per second. Two notes that are X cycles per second apart will beat at X cycles per second. NO-ONE can hear one cent difference as a difference in pitch. But anyone can hear two notes beating, wo-wo-wo-wo-wobble, because of the difference in air pressure as the sound waves hit the eardrum.

Octaves represent a doubling of Hertz eg A as 220, 440, 880 etc. Cents on the other hand divide each semitone into one hundred. At high frequencies a cent may be several Hertzes; in the low register it is the other way around - for instance, the difference between a low E on a bass guitar at 41.2 Hz and the F at the first fret 43.6 Hz is only 2.4 Hertz yet they're still 100 cents apart.

In Equal Temperament the only interval that is perfectly in tune is the octave. The fifth is two cents flat and the fourth is two cents sharp - which explains why it is impossible to tune a guitar using harmonics unless you count the beats and tune adjacent strings so they waver a little - about two beats per second. It is a skill I have yet to master and thus I still rely on a digital tuner. If you tune the A to the E using the harmonics at the 5th and 7th frets, the resultant interval between the two strings will be 2 cents wide - 702 cents instead of 700 in ET. By the time you get to the G (E-A, A-D, D-G) you're 6 cents wide - an out-of-tuneness that the ear CAN discern. That resulting G will sound terrible when fretted to make the major third G# in E major because the ET maj 3rd is already 14 cents sharp and adding 6 cents results in a fifth of a semitone difference. If we can't hear that, we should give up playing music. 20 cents is WAY out of tune in anyone's book.

Roughly speaking pure major intervals (3rd and 7th) are 14 cents flatter than ET, and pure minor intervals are about 17 cents sharp. As mentioned before, the so-called Perfect Fifth and Perfect Fourth are anything BUT perfect, both being out by 2 cents, but we can't hear it other than the beating. The minor and major seconds and sixths are far closer in both temperaments, so much so that substituting them will hardly cause any noticeable difference. But once the ear has learnt pure major and minor thirds and sevenths there is no denying the difference. It is sort of like not having enough sugar in your tea or coffee, or too much, more than the usual. There is a Goldilocks zone where the pure intervals sound 'just right'. It has to be learnt - ear training - if one is accustomed to only hearing music played in Equal Temperament.

Perhaps the most obvious example, where everyone can perceive the difference, is like when listening to a brass band or a string quartet (those musicians if unaccompanied by ET instruments will automatically gravitate to pure intervals because they sound 'thicker', 'stronger', 'better') ... and then the piano or guitar comes in and ruins everything. Whether or not you have realised it, in listening to the brass or strings playing pure intervals for some time, your ears have adjusted and gotten used to it. The introduction of ET piano or guitar will invariably seem an out-of-tune intrusion, an uncomfortable awkward mismatch. In the same way that brass bands sound out of tune (although they're actually playing in tune!), once the ear adjusts to pure intervals suddenly Equal Temperament doesn't sound "quite right". It's a subtle difference but well discernable once learnt.