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. 

108 cycles per prana = 432Hz

You may quote me: "I have gravitated towards a digital keyboard because I find it easier to push buttons to switch temperaments than use a saw and hammer on my other instruments".

How many 432ers know that before 1834 there was no way to measure cycles per second? The whole concept of A = 432 Hz as some sort of musical magic (a right-brained fantasy) is based on a supposed connection between pitch, Vedic numbers, Pythagoras's Music of the Spheres, and the Western system of measuring pitch in Hertz values which are cycles per second. Hertz or Hz references were completely unknown to the Grand Masters of Classical music who predated 1834, often by centuries - Vivaldi, Handel, Bach, Hayden, Mozart, Beethoven, Schubert, Schumann, Mendelsohn, Chopin. None of them used the same pitch for tuning or if they did, it was by sheer accident. 'A' could have been anywhere between 425 and 450 and they wouldn't have even known its reference value. Claims that Vivaldi used 432 Hz are just nonsense! 

At first I, too, thought that the "second" was arbitrary, nothing but yet another human construct. It turns out that it may not be after all. But it still doesn't make a clear-cut difference without exceptions.

Upon inquiry I was surprised to find a correlating time unit in... wait for it... Vedic cosmology, also connected with 108, 432, etc.  Without going too far into the philosophy, the Vedas are held to be composed - not by humans - but by God or empowered persons (specifically Vyasadeva aka Vedavyasa - I guess for those of Abrahamic faiths, somewhat like Moses and the Ten Commandments). So, in Hindu, Jain and Buddhists' etc minds, the time unit used wouldn't be considered arbitrary at all, but on the contrary, "God-given".

There are several Kala (Vedic time) systems depending on what time periods you are measuring - the blink of an eye, a breath, a day; a month, a year, the manifestation of the material world. Their system of diurnal timekeeping is different yet similar enough to end on a proportionate unit. To start with they divide the day not into 24:60:60 hours:minutes:seconds but with 30 and 2 (to begin with) to get a muhurta then a ghati or danda of 24 minutes.  From here it varies from system to system. One I used divides a muhurta of 48 minutes into 30 laghu of 96 secs, then by 15, 5, and 3 down to the respective units of a kashtha of 6.4 seconds, a kshana of 1.28 sec and a nimesha of 426 milliseconds.

Straight away I'm thinking it may not take much juggling to convert 1.28 into harmonic equivalents of the frequencies we are dealing with (read: "want"). Instead of cycles per second we now have cycles per kshana. Surely if we are going to use the "sacred geometry of the Vedas" as a basis for our magical thinking then our pitch will have to be based on the orbital frequency of the Earth and a pure tuning system like Just Intonation instead of Equal Temperament, using cycles per kshana instead of cycles per second (cps or Hz)!

Turns out not much of it makes sense. Drat!

432Hz = 552.96 cycles per kshana; and 256Hz = 327.68 c.p.k.

Not bad, close to a C# and an E respectively. Sort of.

So, using 432 cycles per kshana what do we get? (Comparisons are to note values from 432Hz ET)

432 cpk = 337.5 Hz (halfway b/n a G and an F#); and

256 cpk = 400 Hz (somewhere b/n G and G#).

Forget it! However, if we use another system (I have no idea what they're called) based on 30:2:30:2:6:10 we get a better result. Incidentally that ratio is 1/3600th, showing promise for whole integers, common demoninators, etc that 432ers fancy. The previous system is 1/3375ths, probably why it went nowhere as far as the results we are looking for... 426 milliseconds ain't gonna cut it.

For want of a better name, the muhurta-ghati-kala-pala-prana-vipala (30:2:30:2:6:10) method gives either a prana of 4 seconds or a vipala of 0.4 sec. It should be obvious that by using the larger prana unit of 4 seconds, all we are in fact doing is changing the octave. This, and only this, is where the connection with the Western second to the Vedic prana kicks in - 432 cycles per prana becomes A6 @ 1728Hz, the A two octaves higher. 

It also becomes apparent that to use the smaller vipala for calculations as it means you are converting by 2/5ths which will lead nowhere, eg:

432Hz x 0.4 sec (1 vipala) = 172.8 cpv (cycles per vipala), so for the same pitch referenced in Vedic terms instead of the Western 'arbitrary' second:

432Hz = 172.8 cpv (F? 171.4)

512Hz = 204.8 cpv (G#? 203.9)

Again referenced to A=432Hz ET. But if we reference to Just Intonation with C = 256:

432Hz = 172.8 cpv (F? 170.6)

512Hz = 204.8 cpv (G# exactly - finally, a direct hit!)

Close, but... nah, nothing like the results we get converting the frequency of the Earth's orbit around the Sun into cycles per seconds: 1 ÷ 31556925.250732 = 0.0000000317; which raised to the 32nd octave = 136.1Hz, EXACTLY C# in the A=432Hz Equal Temperament scale - the so-called Om note (ॐ)!

By the way, do you know how traditionally they used to tune up (Indian musicians) before a concert? They set pitch to the singer – what ever it is, it then becomes the shadja. Fair dinkum. Now, THAT is truly ARBITRARY! Legend also has it that the shadja (the root note, usually around C#) is found by the "orchestra guru" meditating on the Heart chakra which supposedly vibrates to the Earth's orbital frequency. He then hums it and the ensemble tunes to that. Hard to believe I know but in days of yore they lit fire sacrifices with mantras, not matches.

Conclusion:

108 cycles per prana = 432Hz; but it doesn't make a lot of sense with the Indian scale which is based on the Shadja!

This is hard to put into words succinctly. When we use the prana unit, a conflict arises when trying to establish a correlation between natural harmony and the vibrational frequency of 432Hz tuning using Vedic units of time measurement and 'Sacred Geometry' i.e. the 'magical properties' of whole integers such as 256 and 432. These represent C and A in the Pythagorean Temperament but are derived from the arbitrary Western second. Trying to use Vedic units of vipala, kshana, etc. however, it completely breaks down.

Yet another conflict arises where although Pythagorean (and derivatives thereof - the ONLY temperament with A & C @432 & 256) uses whole integer ratios, it actually sounds pretty terrible imho (unlike JI which also uses whole integer ratios but has pure thirds and is very harmonious despite its wide 4ths and narrow 5ths). Equal Temperament based on A=432Hz sounds a lot better than PythT but then every note in the scale is technically out of tune with the A - vibrating on a frequency that does NOT harmonically concur, causing beating.

Even if we do use the prana unit, which corresponds with A=432Hz frequencies (being a few octaves higher) and the traditional Indian method of tuning (of which Just Intonation and Ptolemaic Intense Diatonic Scale are mere simplifications) it breaks down again -depending on which 'magic number' 256 or 432 we chose as C or A - because a C of 256 gives an A of 426.6; and an A of 432 creates C = 256.9 (close, but again, no cigar! Remember we're talking about harmonics converging via whole integer ratios etc).

It is at about this point that I leave the desk and wander over to my keyboard and play some soothing chords in meantone. Ahhh, I just love those pure thirds!

Part 3: The 432Hz tuning theory

The 432Hz Tuning Theory Part 3.

[# it's worth noting from the outset: the only scale you'll find that has C=256Hz and A=432Hz is PYTHAGOREAN - and virtually NO-ONE uses that temperament now, not even New Age 432ers!]

This is hopefully my last post on the controversial "432Hz tuning system"...  or maybe there should be a fifth, and a flat seventh. I'm no bliss ninny, and I feel I've wasted more time and energy than this fallacy warranted. The pics in Part 2 show non-ET fretting for scales (shall we say 'rational'?) based on ratios, in an attempt to get closer to true notes, to avoid the compromises in equitempered scales with their harmonically sharp and flat 2nds, 3rds, 6ths and 7ths. I kinda feel this may be missing the point - if you want to play pure ratio music, why not use simpler design for an eloquently simple system - but I understand some musicians have the talent and the brainpower to revel in this sort of thing!

 (and so, thus far..  ..mathematically how it just doesn't add up; how merely shifting the pitch from A=440 to 432 achieves little as it retains the inherent out-of-tuneness of ET; how C=256Hz gives an A of 430.5 or 426.6 depending on which temperament is used; etc... I've covered all that, about how it partially works with Pythagorean Temperament, keeping 256 and 432 but losing the 'Om' C# 136.1; and anyway which can't be played on modern instruments fixed to ET without a massive rebuild and/or retune, such as flute, clarinet, saxophone, xylophone, keyboards and fretted string istruments - but what gets me peeved most of all is that the Charlatan who claims to have "invented 432" not only shamelessly uses an undisclosed version of ET (where the numbers mysteriously add up without explaining how) under the guise of the otherwise non-existent q.v. "harmonic equal tempered tuning" which, if it exists at all, is probably just modified to make the numbers work and has no provable basis in musicality. It is one of the many "Golden Elements" they claim to be in tune with, empowered by, and able to harness. One last thing about this Shamanic Trance Music Con-Person Amanda Bossman or whatever... among their many "attributes" listed on their website is Semantic Artist - in other words, a word juggler (read: bullshit artist).... Aha! Something without inconsistencies! You could visit any of their many sites for a good chuckle - some of the creative jargon (read: invented terminology) for New Age techniques and their 'mastery' of various forms of "scientific mysticism" are extraordinary fabrications. Pages and pages of esoteric potpourri but unlike any other website about temperament which always presents detailed information, these sites waffle on without one single chart or descriptive break-down of the scale, showing formulae of how it's calculated)

But to go out on a positive - from what I have learnt about ratios, shrutis and pure tuning from my lifelong interest in classical Indian music, also bhajans and kirtans, and most recently at the many sensible, rational, non-New Age musician's sites while doing this research, I now feel even stronger about avoiding when I can, the impure notes of equal temperament in melody, whether singing or playing an instrument capable of microtones - to find more expression through beautifully harmonious note couplings - especially the 3rd/5th and 5th/b7th with 15 cents increase from the normal 300 in ET. So pure, so powerful! If you don't play guitar yourself, see if you can get a musician friend to try this out and show you.

I first learnt this decades ago, but only now the exact values. It's not an invention- the classical 'Indian' tuning based on harmonic ratios that are real was historically used and still is to this day since before modern tuning was invented and widely adopted. You can do this on any guitar with open pure tuning and barre the chords, but I mainly use it with slide guitar, playing just the top two strings, with a partial open tuning that's a quick and easy variation from regular EADGBE. Tune the B 13.7 cents flat (the Waves app has optional markers on the tuning guage for it), making it a perfect major third (I'll explain later) and the E next to it down to a D 2 cents sharp, a perfect fifth. This D in the partial G open tuning is now a Perfect Fifth because the fifth in standard Equal Temperament is actually 2 cents flat of the natural harmonic note. Ignoring the lower E and A strings (you can always tune them to D and G if you want to barre 6-note chords ie. DGDGBD) you can now easily play a Perfect Major chord across the top four strings.

So why the B @ 13.7 cents flat? Because it's making a major third interval that is Perfect, unlike the ET third which "beats" because it is 13.7% out of tune (sharp). If you don't understand that last part of the sentence, just go with this for the time being or go to wikipedia. You now have, in order across the strings, D-G-B-D making a Perfect Major chord starting from the extra bass note (a fifth). Barre it across the twelfth fret with the slide, gracing up from just below the fret. It will sound terrible at first until you get it right over the fret and BLAM! Isn't that just the most beautiful chord you've ever heard?

Part of this is from the extra harmonics let through by the use of slide, especially at the octave where they are the strongest. Normally, fingering a barre chord there will bar the harmonics (pun intended) from resonating, cutting them off. Here, they are not only let through under the slide because of the light pressure, but with the perfect tonic, third and fifth all resonating purely together, there is nil beating. None, zero, zilch, diddely-squat, sweet FA. Perhaps the only time most people have ever heard anything like it would be hymns sung by monastic choirs (without the pipe organ) in an abbey; any good Barbershop Quartet; or at least modern a cappella groups. Singers in such scenarios naturally gravitate towards notes that fully resonate with each other when free from the dictated pitch of an instrument.

Now the best bit.. just using the top unwound strings at the 12th fret, what you've got right there is a B4 and a D4, the third and fifth of Gmajor, 3 semitones apart, but unlike there being the usual 300 cents between them as in ET, with this Pure Tuning it's more like 315, what it should be for a Perfect Major Chord. And when you slide it up 3 frets, you've got a fifth and a flat seventh, also 3 semitones but 315 cents apart instead of 300 ET. You need to play this slightly sharper (above the fret) to get it right because of the flatter B string. But listen to that power! It oozes out naturally because to get that sort of interval, first of all, you can't use EADGBE (there's no minor third between the strings needed for both previous intervals) and even if you did, you'd have to play the slide diagonally to change 300 to 315. I repeat, If you don't play guitar yourself, see if you can get a musician friend to try this out and show you. It has to be heard to be believed!

What's this got to do with A=432Hz? Sweet FA, zilch, zip, nil, NOTHING! Because it sounds so good regardless of what pitch it's played in. To say 440Hz makes you ill and 432Hz is better is about as silly as saying music sounds better or worse in certain keys, like D# is really "gay", or Gb is '"stupid".

The vibe of music is all about the performers' consciousness - their expression, frame of mind, heart, mood, and hey, even their state of health. To say that A=432Hz promotes Universal Harmony because it contains ॐ=C#=136.1Hz is just fodder for gullible bliss ninnies who also believe that unicorns fart rainbow glitter. No-one can put a label on the Absolute Truth by designating a specific frequency. That is absurd and totally contradictory to the transcendence of the Supreme Whole that Om represents. The sacred syllable ॐ not only encompasses all vibrational frequencies, it simultaneously represents them and eludes confinement to any one level or note. 

All types of people enjoy music - whether they be religious, agnostic, or atheists. Everyone is free to listen to whatever suits their tastes, and I really don't have a problem with that although at times I sound like a zealot. What irks me though is when self-glorifying New Age Charlatans pimp the music scene with their branded 'spirituality', preying on the gullible bliss ninnies that maintain this pollyanna industry.

For those seriously interested in becoming 'spiritually charged', ie. closer to the Supreme Lord, there are authorised processes which involve spiritual activities and producing vibrations, such as chanting the Lord's Holy Name , prayer and meditation, reading and reciting scripture, singing songs of praise, etc. One cannot obtain the Lord's mercy simply by dialling up some frequency on a machine. It takes work and a loving attitude. God is not a menial order-carrier who can be controlled by a proclaimed mastery of any 'spiritual powers' or mystic opulence. Those rascals who promote themselves as Seers and Masters whilst charging a price are simply engaged in manipulating material energy for some personal benefit under the guise of 'spirituality'.   

Musical temperaments

The Tromlitz flute, invented in 1785 preceded the saxophone by 55 years and was the first flute to be able to play all major and minor semitones of 5 and 4 commas, meaning it had no limitations playing all the different flats and sharps in the 24-note octave of, say, Just Intonation. For instance, Mozart wrote his flute pieces using different values for sharps and flats (as everyone did, that's all that was known at the time). A good flautist was adept at playing them fluently, and an average player failed to discriminate, thus rendering the compositions weak and insipid, devoid of the true tonal colourings the Master intended. Not that I like much classical music. But they're now saying it sounds crap unless it is performed using the temperament of the period it was composed in. Makes sense.

In the 19th century most contemporary musicians and composers were moving from various meantone eg ¼comma to well-temperaments like Werckmeister and Kirnberger, but still the different size semitones were still indispensable for playing pure thirds etc. Many people don't realise or are mistaken in thinking that Equal Temperament was widespread at the time. As far as pianos go, it was in fact Quasi-equal temperament because until 1917 there hadn't been a foolproof method for tuning ET - other than fretted instruments, which to this day still approximate the ancient "Rule of 18" to position the frets. (27" scale, ÷ 18 = first fret 1½" from the nut, therefore 2nd fret is 25.5"÷18 = 1.41666666" etc. The Indians and the Chinese have been doing it this way since the year dot)

So my question is, like the Tromlitz flute, do saxophones have special fingerings for small and large semitones to accommodate sharps (which are slightly flatter) and flats (which are slightly sharper), as in G# Ab, or is all this adjusted randomly by embouchure? (before I lost my right index finger, when I played the flute I used the simpler, slightly off-key incorrect fingering to play E, the same as Ian Anderson. According to his website, he was corrected later in his career by his daughter!)

So I'm assuming that the Saxophone, like most instruments made before the 20th Century, were not crafted to play in Equal Temperament. Even early keyboards had split black keys for playing either sharps or flats, and were tuned with temperaments that didn't facilitate modulation to all keys.

Man, I dig those pure thirds! If I put my keyboard in Equal Temperament it lasts about 15 seconds before I switch it back. My default is to Kirnberger III for modern music requiring the usual amount of modulation but for anything that only uses the 8 pure majors I absolutely love ¼comma meantone (Eb, Bb, F, C, G, D, A, and E, all major chords). Only rarely do I notice the narrow fifths of about 4 or 5 cents. It doesn't seem to bug me, but I can no longer stand the impure major third of Equal Temperament which is a whopping 13.7 cents sharp. It's about as pretty as a hat full of arseholes.

http://www.flutehistory.com/Resources/Documents/MozartTromlitzFlute.php3