~ tunings and other musings for creating an equal tempered guitar ~

'a calculated twisting of the tuning pegs opens both old and new worlds of sounds to explore ...'

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In a nutshell. We take the general term 'tuning' a couple of related directions in the following discussions. First there's some ideas about how to tune up our guitars with a tuning fork and a couple of ways to tune. Then follows some of the basics of how our guitars are physically built up so as to play 'in tune.' This concerns the theory of why the frets are located where they are on the fingerboard. The discussion of the physical scale length of the fingerboard and related ideas sets the table for how the frets are measured and placed on the instrument. Two methods are included in this section; the historically earlier 'rule of 18' followed by the mathematics of equal temper. So if the battery goes out on your nearest tuning device ...

tuning fork

Tuning from scratch. This first tuning is probably the most common, arranging the pitches in what is often termed 'standard' or 'concert' tuning. Here are the open string letten name pitches.

string #'s
6
5
4
3
2
1
concert

This twisting of the pegs is most common and works in a wide range of settings. Everything from honky tonks to concert halls. In this method of tuning up we can get our starting pitch / tone by using a tuning fork created to sound 'A' ~ 440.

This pitch corresponds to the A above middle C on the piano. We simply match its pitch with our 5th string, then tune the rest of the strings from this pitch. If you already know the string names this is a breeze. If not, then learn the six of them now and it will be done with once and for all.

1) tune up the 5th string A, open or at the harmonic at the 12th fret to the pitch of the struck fork.

2) find the pitch A at the 5th fret of the low E string and match it up with the open A 5th string.

2a) or tune the low E string to its octave E above, located at the 7th fret of the A string.

3) find the D on the 5th string / 5th fret and tune up the open 4th string D.

4) find the G on the 4th string / 5th fret and tune up the open 3rd string G.

5) find the B on the 3rd string / 4th fret and tune up the open 2nd string B.

6) find the E on the 2nd string / 5th fret and tune up the open 1st string E.

Done correctly, you should be out of strings now :)

Depending on your instrument and the quality of its build, we can check our tuning by repeating this same process up an octave located at the 17th fret.

Also, most of the above method works by matching the pitches of adjacent string harmonics between the 5th and 7th frets. Use the the above open strings for tuning the G and B strings.

string numbers

tuning fork

pitches of the piano keys

string letter names

string harmonics

Open G. The first of our open tunings here is open G. Here we simply tune the open strings to sound a G major triad / chord. As this is also a common tuning for the five string banjo, tis easy to imagine the historical transfer of the tuning from banjo to guitar in our early Americana days. That it makes a super rocking chord machine out of any guitar with a ton of blues in the middle strings as well makes it just a hoot and a holler ... but that's banjo talkin' I'd imagine and we are theorists :)

Regardless, going from standard to concert is a breeze. If your ax is tuned up to standard, un-twist each of the strings to get to the open G magic. Oh, the B string stays the same so ...

string #'s
6
5
4
3
2
1
concert
open G

Banjo's traditionally are four or five strings right? So ... leave off the low 6th string for a brighter sound. On any thing with any sort of a tune-a-matic bridge, easy to loosen the string and just set it out of the way. Historically speaking this open G was a 'gamechanger' for Keith Richards when he first learned it. No surprise then that a dozen or so global hits for band followed.

superhoot lick

wiki ~ Keith Richards

Hawaiian 6 / 9. The next open tuning is a jammer's delight. While taking a lesson with this blues cat Mario, it never went further, much to my later own chagrin, for once we tuned up to play some slide with these pitches, the lesson ended for me. For true be told a whole new universe of coolness immediately opened right up. Hope you have as magical a moment. Here are the pitches evolved from concert.

string #'s
6
5
4
3
2
1
concert
6 / 9

So why 6 / 9? well, those are the two colortones aded to the three note triad. Great slide key, such as Hawaiian slack key styles ( ? ) and there must be some blues in here too. I wish now I had stuck around for more, although did write a suite of tunes based in this tuning.

color tones

wiki ~ slack-key

"From Near Or Far"

Open E. The next open tuning is very much like the open G above but based on E. This is perhaps more of a slide key than the other two as it tunes right into the standard or concert tuning with no transposition thus mixes in better with standard tuned instruments. Examine the pitches.

string #'s
6
5
4
3
2
1
concert
open E

Open E was the favorite slide key of Daune Allman.

wiki ~ Daune Allman

The dots. Multiple queries revealed a continual mystery as to why the dots / fretmarkers are located where they are on the neck. Looking at available pictures of older guitars from various sources, there is some variablilty in placements. So thinking of what was the oldest song we have that is consistently found in modern collections written in the same key centers, and upon examining the instruments I had at the time, the following solution occurred to me.

The English song "Geensleeves" takes us back near the mid 1550's or so before our six string guitars really got their start. Modern scoring of the song finds it mostly centered in the key of E minor with the bridge / refrain going to G major. So in looking at the fret markers on my guitars, the corresponding pitches create E Dorian and G Lydian. In today's theory, these are parented by the D Ionian / D diatonic major scale. While even the lutes of this era are said to be equal temper tuned thanks to their fret placement by the rule of 18', good chance that the modal system of pitch organization still ruled the day. Am I way out of my league here? Absolutely, but who doesn't love a good mystery :)

Here's a picture of the first octave of my old guitar.

wiki ~ "Greensleeves"

open E
3rd / G
5th / A
7th / B
9th / C#
12th / E

So if there was a fret below the nut, to the left in the picture, it would have a fret marker as its octave above does to the far right of the picture. So filling in this marker we then have a sequence. What do it mean? Well in this cluster of a theory book that the dots mark out E Dorian and G Lydian, the pitches of Greensleeves and a portal for a new organization of the pitches when we arpeggiate these pitches and 'correct' the intervals.

sequence

Dorian

Lydian

Building a modern tuned Americana guitar. Our modern builders of today have a fairly solid 200 years or so of instruments to examine and study to bring forth the modern marvels of today. Electronic wizardry aside, we've a rather remarkable instrument today capable of melodies, arpeggios, chords, rhythms, blue notes; a fully chromatic aural spectrum of possibilities.

wiki ~ guitar

Going back even a bit before the guitar to the lute and vihuela, whose 'rule of 18' fret mathematics creates a fret location scheme capable of building an equal temper tuning into any style of fixed fret instruments, we find an evolving artistic that continues today.

wiki ~ lute
wiki ~ vihuela

Our instruments become more accurate over the centuries as the math and tools used to physically locate frets on a piece of fretboard wood improved, thus the intonation of our instruments could also evolve. This gradual physical honing of the instrument's pitches into equal temper fret spacing surely plays a part in that it allows and encourages the evolution of harmony in the American jazz story. This evolution is surely traceable while moving through the decades of our American music history. That pitch doubling in chords is easier to tune than getting the multi-noted Hollywood chords, rich in the color tones, to sound good and in tune all improves on better crafted instruments.

Regardless of how well crafted and in tune an instrument is, generally speaking the fret spacing scheme built into any guitar nowadays is an equal temper layout, which allows us to create all of our intervals, scales, arpeggios, our various chords and pitch clusters, from each of the 12 pitches of the chromatic scale. This marvelous tonal and musical dexterity has the distinctive title here of 'anything from anywhere', which even just as a one page treatise will, in theory, open up a nice new universe of possibilities.

intervals
scales
arpeggios
chords
clusters

Anything from anywhere simply implies that all of our musical sounds and components can be equally created from any of the 12 pitches over the entire range of the instrument. And as we'll see as we progress in this study, our ability to slide our numeric pitch style indicator seamlessly through our popular musical styles is made possible by this pure equality of the 12 pitches.

For today we think little of changing keys, moving ideas chromatically, further extending arpeggios beyond a tonal center, we even can stack complete tonal centers on top of one another and still sound in tune. Advanced musicians often will spontaneously transpose written music on sight to better accommodate a melody to a unique perfomanc setting; a person's vocal range, instrumentation etc. Sky's the limit. Acquiring and strengthening these skills become nice challenges for the career musician.

modulation
chromatic motions
polytonal
transposition

So as we evolve through our musical styles and count up the gradually increasing number of pitches employed, knowing that the structural basis of equal temper tuning provides these artistic resources, could very well provide the intellectual basis for further evolution of our musical resources thus the art it might create. Along this line of thought, we can ride a sequenced arpeggio to get there.

The problem to be solved. Ever wonder why the width between the frets on our guitars, and most other fretted instruments, gradually gets smaller as we ascend in pitch? As pitch is determined by frequency of vibration, measured in cycles per second, it is our fret placement that properly divides up our string length, thus creating the different vibrations whence our pitches.

What is needed for equal tempered fret spacing to work is a way to divide the octave into 12 equal parts or semitones (half steps) while realizing that our numerical cycles per second doubles in number from our root pitch to the octave pitch above it. Somehow we must convert the different pitches into the physically measurable spacing on the guitar fingerboard.

octave

So if the distance from A(4) 220 Hz to A(5) 440 Hz is one octave, we're multiplying 220 Hz times two to get 440 Hz. To divide that octave into twelve equal pitch increments, we need to use that number which, when multiplied by itself twelve times, equals two. In math parlay that number is the 'twelfth root of two.' (1)

The early solutionist. We can trace the solution of this problem by taking the wayback machine some 450 years to the later Renaissance era of Florence, Italy. Here we find Vincenzo Galilei (1529-1591), father of famous astronomer Galileo. Written historical records reveal to us that Papa Galilei's main musical instrument was the lute and that he was an accomplished performer and noted composer. While movable frets were not uncommon in this era, Vincenzio preferred fixed, parallel frets which would produce a consistent intonation over the range of his instrument. Fixed, parallel frets ... seems awfully familiar doesn't it? (3)

wiki ~ Renaissance
wiki ~ V. Galilei

To tune such an instrument, Mr. Galilei is associated with the 'rule of 18', the first written mention of which is in his instructional text for lute. So named for the number used to divide a string length i.e., the distance from nut to bridge, the 'rule of 18' mathematics produces the division of the octave into the 12 half step pitches of equal temper tuning. Interesting to note that keyboard builders, tuners and players would continue to grapple with the same tuning issues for another century and a half or so before accepting and arriving at an equal temper equilibrium in tuning. (4)

While we can trace our guitar ancestors all the way back to the 12 century Moors of the Iberian Peninsula, and our modern version of the classical guitar to the early 19th century Spain, we have in essence used the rule of 18 to create the fret spacing to equal temper tune the pitches of any of the stringed instruments with fixed, parallel frets. (5) And in some parts of the world it is probably still in use today.

wiki ~ Iberian Peninsula

Application of the rule. The rule of 18 places the first fret of a fingerboard 1/18th from the nut, of the total length of the string. The second fret is located 1/18th of the remaining distance plus the distance to the first fret. We consistently measure each fret from the fixed string 'nut' to minimize errors. Do remember that we still have to saw a groove into the fingerboard and hammer in the metal fret. We simply repeat this process for the number of frets we need. Today, with the better calculating, measuring and building resources, the more accurate figure of 17.817 is used for equal temper fret placement and is derived from the math associated with the pure mathematics solution of the 12th root of 2.

This factor of 17.817 is used to calculate the 12 fret locations between the two pitches of the octave. It will work for any scale length and any number of pitches (frets) we need. Here is the formula to find the location of our 1st fret and the math that creates this 17.817 fret spacing constant. We'll examine the numbers associated with a 25.5 inch scale length of string and use the solution of the 12th root of 2 as 1.0594631.

1) Scale length ( 25.5" ) divided by 1.0594631 = 24.0688"
2) Scale length ( 25.5" ) - 24.0688" = 1.4312", the distance from the nut to the first fret.
3) Scale length ( 25.5" ) divided by 1.4312" = 17.817", our fret spacing constant.
4) Finding the placement of the 2nd fret, we now can use the spacing constant divided into our shortened scale length. Thus; 24.0688" divided by 17.817" = 1.351", to this we add our first fret spacing of 1.4312" to get 2.2782", our second fret is located 2.782" from the nut. Cool?
5) Third fret: 25.5" - 2.782" = 22.718" / 17.817 = 1.275" + 2.782" = 4.057"
6) Fourth fret: 25.5" - 4.057" = 21.443" / 17.817 = 1.203" + 4.057" = 5.261"
7) Fifth fret: 25.5" - 5.261" = 20.239" / 17.817 = 1.135" + 5.261" = 6.397"
8) Six fret: 25.5" - 6.397" = 19.103" / 17.817 = 1.072" + 6.397" = 7.469"
9) Seventh fret: 25.5" - 7.469" = 18.031" / 17.817 = 1.012" + 7.469" = 8.481"
10) Eighth fret: 25.5" - 8.481" = 17.019" / 17.817 = 0.995" + 8.481" = 9.436"
11) Ninth fret: 25.5" - 9.436" = 16.064" / 17.817 = 0.901" + 9.436" = 10.338"
12) Tenth fret: 25.5" - 10.338" = 15.162" / 17.817 = 0.8510" + 10.338" = 11.189"
13) Eleventh fret: 25.5" - 11.189" = 14.311" / 17.817 = 0.803" + 11.189" = 11.992"
14) Twelfth fret: 25.5" - 11.992" = 13.508" / 17.817 = 0.758" + 11.992" = 12.750"

Each of the above calculations work with the full scale length to find the distance from the nut to each of the frets. In theory, builders measure each successive fret from the nut, as opposed from the bridge, to minimize goof ups. Having to measure the smaller distance apparently creates a greater accuracy. The octave at the 12 fret, being one half of the string length, provides a reset point of perfect clarity for the frets above #12.

So at some point get your ax and a tape measure and see what you come up with. Having never personally built and fretted a fingerboard, I honestly do not know the difficulty. For purposes here, we simply need to recreate the math for our own edification. This understanding forms the core basis of how our tuning is created and built into our instruments, creating the foundation for our music theory discussions.

Musical styles / tunings. Do we tune our instruments different ways depending on musical style? In some cases we surely do. Early blues players used open G tuning carried over from the banjo. Slide guitar players often lean towards the open tunings. Singers will often open tune their axe's as it can free up their voice to find different nuances with pitch. For example, there's three G's in the open G major chord commonly used by singer/guitarists, so not a lot of wiggle pitchwise, (not being a singer I really shouldn't say.) A best example of voice / open tunings just might be in Joni Mitchell's work. Another might be in Bob Dylan's work, where his natural tone of voice, its expression and inflection, easily overcomes any pitch dominance created by the standard tuning he usually chose for his acoustic git.

 
 

The three G's. Since we sort of dinged the open G chord in our example just above, we surely should note that its close relation, the open G tuning, goes way deep deep deep into our American musical roots. First as a banjo tuning then in its adaption to the guitar as used by our turn of the last century blues players. Recently, the fairly ancient open G tuning went to multi platinum global fame yet again through composer and guitarist Keith Richards with the Rolling Stones, who readily testifies about how his songwriting was epiphanly effected after getting hip to the open G tuning.

open G video

And if you just plain love to rock out loud with your guitar, you owe it to yourself to retune your git and dive into any of riffs and patterns that motors any one of the many wonderful songs that Mr. Richards conjured up for us. And maybe like Keith's epiphany, in minutes you realize a whole new universe of possibilities, one that brings a lot of fun and surely tradition, as its history roots go very deep into the music of Americana. It's amazing how in just rooting around in these open tunings that oldtime cliche licks reveal themselves.

rocking out loud video

This 'open tuning epiphany' personally happened twice for me. First with the open tuning taught to me as the 'Hawaiian 6/9', during a slide guitar lesson. The second was with the open G when playing old time blues. If yet another reason was needed to justify having two or even three servicable guitars, surely having rigs for standard, open G and D tuning could be near the top of the list :)

Hawaiin 6/9 video
open G tuning

For the vocalist and totally just saying ... One of the last things that science has yet to discover is how and if a singing voice is effected when it is accompanied by a guitar tuned to an open tuning versus an standard tuned guitar. For example, in the open G chord included here, there are three G notes that range two octaves. If the melody note being sung is G, how much variability is there in pitch in the voice? Is the voice coerced into this pitch and does that effect its emotional statement?

Does this really matter? Not being a singer I do not know. That said, when I listen to singers who play this chord and sing along, their vocal pitch seems 'determined' by their instrument and sounds 'dull' in regards to artistic coloring. So I muse ... if a voice had to work a bit harder to find their pitch in the mix of an open tuning would there be a more colorful pitch and sound created ...?

wiki ~ open tunings

Three favorite singers; Joni Mitchell, Muddy Waters and Bob Dylan seem to a have a wider variety of pitch as thery go through their verses of their songs. In Ms Mitchell's case, open tunings are a part of her composition. Mr. Waters is a bluesman, so there's all sorts of interpretive nuances. Mr. Dylan just seems to be telling his story and honestly, it's hard to convincingly play his melody lines as single note guitar melodies. Again, just saying here as I be a terrible terrible singer ... but sure do love to hear a singer's stories :) So for the vocalist reading here, try an open tuning sometime, even just to see where it might take your most cherished melody lines and the stories they tell.

wiki ~Joni Mitchell

wiki ~ Muddy Waters

wiki ~ Bob Dylan

That's all for this first chapter folks. Not really sure if we needed to go this far into all this tuning but it is an interesting story spanning mucho millennia. Knowing that to build chords we need a mathematically tighter system than for our original melody pitches in a sense mirrors the theory relationship we often find in our Americana musics.

Ideally, by understanding how our musical system of pitches is created and tuned and how we build that structure into our guitars provides us with; another basis and way into understanding whatever music might come our way, a perspective of the raw source materials and how they are built into art, help us imagine new ways to organically build new ideas into our existing system.

We also can gain a sense of what makes a good instrument and why it is worth having. That as our ears evolve and better understand what we hear, we can hear the virtues of an instrument that just may help better realize our own ideas as well as inspire us to push beyond existing forms and boundaries etc.

Review. Our modern six string guitar has many ancestors from many cultures. Its tuning we can reasonably trace to the later Renaissance of Florence, Italy. Here we find Vincenzio Galilei, an accomplished performer and composer for the lute. We credit Mr. Galilei with reording for posterity what was then called the 'rule of 18', original source unknown (?), which properly laid out the fret spacing on the lute to achieve an equal temperament of tuning of the 12 half steps within the octave. Successive generations of builders refined this rule as newer methods of building and measuring were discovered. Standardizing classical guitar construction began in the early 19th century in Spain, and from that point forward our instruments have been nearly identical in how the frets are positioned and thus, the pitches they create.

vocabulary terms for "building a tuned guitar"

intonation
the quality of being in tune
scales
succession of notes usually by a combination of whole steps and half steps
arpeggio
Italian for "harplike", usually a succession of notes in thirds
chord
simultaneous sounding of multiple pitches
modulation
moving from one key center to another, i.e., changing keys
polytonal
two or more key centers sounding simultaneously
shedding
slang for practicing
cycles
a loop of ordered events
12th root of 2
formula to divide the octave into 12 equal tempered pitches
Vincenzio Galilei
credited as the father of modern tuning on fretted instruments, rule of 18
17.817
factor derived from 12th root of 2 for fret placement

matching quiz

shedding
the quality of being in tune
17.817
succession of notes usually by a combination of whole steps and half steps
arpeggio
Italian for "harplike", usually a succession of notes in thirds
cycles
simultaneous sounding of multiple pitches
Vincenzio Galilei
moving from one key center to another, i.e., changing keys
12th root of 2
two or more key centers sounding simultaneously
intonation
slang for practicing
chord
a loop of ordered events
polytonal
formula to divide the octave into 12 equal tempered pitches
modulation
credited as the father of modern tuning on fretted instruments
scales
factor derived from 12th root of 2 for fret placement

"If I fail I'll face it, but I can't live with not trying."

Jonathan Bender

New York Knicks 2009

Footnotes:
A(5) These designations pitch come from the 88 piano keyboard. The pitch A(4) @220 Hz., is below middle C known as C4, while the pitch A(5) @ 440 Hz., corresponds to the A above middle C on the standard 88 key piano keyboard. Reblitz, Arthur A. Piano Servicing, Tuning and Rebuilding, p. 206. Vestal Press, Maryland. 1993.
(1) Mike Doolin / mike@DoolinGuitars.com / Mr. Doolin took the time to explain this essential component and word the fret placement / tuning problem so eloquently to me by e-mail.
(2) The "12th root of 2." Reblitz, Arthur A. Piano Servicing, Tuning and Rebuilding, p. 206. Vestal Press, Maryland. 1993.
(3) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 162-163. USA Alfred A. Knopf, New York. 2001
(4) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 210-212. USA Alfred A. Knopf, New York. 2001
(5) Denyer, Ralph. The Guitar Handbook, p. 42. Great Britain. Pan Books, London. 1982