Just Intonation and Locating Pitches

[Budda]

Here's someone that inspired me to study this further:

http://www.youtube.com/watch?v=V2c9J...eature=related

[warmingtone]

Beautiful...

Can you see why I pursue with the tremolo arm these days? With slide of course you can scoop into notes and play between the notes, so too with a good trem...

I use Kahlers, LSR's and locking tuners...



The idea of trems seem to be much maligned for some reason but done tastefully can give you access to so much.

On my new guitar I am thinking of working them still further. I'm even considering adding a b-bender type of thing into the fine tuners as I use a fair bit of behind the nut bends, especially with harmonics and I'm thinking this might be a further addition to this kind of effect...

[w3stie]

Originally Posted by Budda

Ever notice how Barber Shop Quartets and really great Choirs sound so "Harmonious"?

Is that why the BeachBoys sounded so good? This has been one of the most interesting and informative threads I've read. It seems the guitar is behind the eight ball from the start; what with equal temperament plus the vagaries of the guitar fretboard. I'm still struggling to understand the feitlen and other tuning systems, and wondering what will make my guitar sound in tune for the E major C major and A major chords. I'm beginning to believe it may not be my innate lack of musical skill, but something to do with the instrument itself.

Thanks to all the contributers here!

[klasaine]

Because we all (all insts in the west anyway) use the 'equal temperament' system C, A and E triads will always be slightly a miss from one another. That's the compromise for being able to play 'basically' in tune in all keys. Horn and string players learn to subtly adjust as they play depending on the key. Guitar players need to learn to do what they can given the limitations of set frets. We can tweak here and there. Most probably do it w/o even knowing it.

If you want to really get into the science this book is FANTASTIC! It's also one the standard classics of audio science.
http://books.google.com/books?id=Hgc...page&q&f=false
As a 'teaser' ... it will explain to you why, in order for a piano to play in tune in all keys in 'just' intonation (using the natural overtone series as it's basis), it would need to be about 100 feet wide and about as long.
The chapter on the discovery and science behind a wound string is worth the price of admission alone.

[warmingtone]

Originally Posted by w3stie

Is that why the BeachBoys sounded so good? This has been one of the most interesting and informative threads I've read. It seems the guitar is behind the eight ball from the start; what with equal temperament plus the vagaries of the guitar fretboard. I'm still struggling to understand the feitlen and other tuning systems, and wondering what will make my guitar sound in tune for the E major C major and A major chords. I'm beginning to believe it may not be my innate lack of musical skill, but something to do with the instrument itself.

Thanks to all the contributers here!

Well...we discussed the barbershop quartet 'close harmony thing' but there is a bunch of stuff at work here.

Part of it is perhaps the tuning aspects, but frequently they are playing with equal temperament instruments anyway.

A lot more is that these parts are often extremely well thought out and follow good voice leading rules for every part...we are not just talking harmony here, but harmony in motion and every voice having a distinct function and following rules that help considerably and is something of a lost art in most circles.

...

The guitar has it's problems, as do most instruments...but then, we accept these things in most of the music we hear and enjoy, so there is no need to panic.

On guitar, it is worth noting some of the problem spots and avoiding them.

Other tuning systems like the feitlen system generally moves the nut a little closer. An earvana or compensated nut will even move each string a different amount. feitlins system is to move the nut closer in and to offset the tuning a little bit all over...but they don't move the frets and there is still a compromise there. Permanent modifications and alterations to the way you tune are the cost of these things, but they can work at least to some 'degree'...with does of hype added...

...

Technique plays some part in things though....consider this...

Every time you fret a note on the guitar, you are bending the string a little down to the fret. The action in part controls how much, but many will push quite hard on fretted notes righ down to the board or even tight over the fret itself and with big frets and light strings, this can pull things way out.

As mentioned int the parallel "tuning thread" the idea of tuning by harmonics on the 5th and 7th frets was 'boo hooed' a little, but at least in this method you are eliminating the fretting of notes even if that 7th fret harmonic is from the overtone series and not the equal temperament fifth. However, it is so close that it can be a useful additional tuning method I believe because fretted notes to open strings are potentially far worse!

...

And of course, as we have discussed, and it you look at the way those frets are 'adjusted' on those fretboards that were posted, the problems are far more towards the nut end of the fretboard than say in the middle area. So, you have several potential problems, you have possibly a hard grip with a higher than fretted note nut, and more intonation problems (particularly with thirds) and the requirement for more pressure (pulling some strings out of tune) and a combination of slightly bent fretted notes with open strings...all are likely to cause problems.

So, somethings are technique, more advanced players will often even correct for tuneness of chords by controlling the pressure or even slightly bending notes within a chord. Others will avoid some voicing that are particularly 'problematic' such as an open A major chord.

One thing to do is to tune a guitar as best as you can, play say an open A chord or some progression low down near the nut, then try the exact same notes on a lower string set in the middle of the neck...generally you are going to find things are more 'in tune".

You can also avoid a lot of duplication of notes, especially things like thirds where it is going to be glaringly obvious if they are 'out of tune' perhaps....

A lot of the real problem notes are 'sharp' so unless you have a way of bending down from a note (one of the reasons I use trems on my guitars nowadays and thats limited to of course as all the strings are going to waver) there are limits to what you can do.

But in the end, before thinking that they guitar is just inherently flawed, consider all the great music and player you have seen who can play in tune, even without freitlin like adjustments or radical frets or whatever. Obviously it can be done plenty well enough.

...

I asked the question on the other thread, and I suppose it's worth asking here even more so. How many people really get and know how to adjust for intonation and have had their guitar properly set up from head to tail with attention to the instrument having half a chance of playing in tune? How many have had a really good look at their technique to see if their grip or whatever is causing things t be a little out of tune? How many have considered that not only might a slightly thicker string not only 'sound good' (because the pickups have more vibrating metal to magnetically sense say) but may also aid in some of the technical tuning things involved with technique?

[w3stie]

Hi warmingtone, thanks for the input. I've had a classical guitar for many years, and although not an expensive one, it's pretty good, and I haven't noticed any problems with intonation. I just re-tuned it and the Emaj sounds rich and full compared to the electric. So I've had another go at setting the intonation on the electric, and it's much improved. I set the intonation by trying to match the 12th fret harmonic with the corresponding fretted note. I can still hear some problems with the c# in Amaj, but maybe I'm getting closer.

Anyway, not wanting to hijack the thread, but this has been very instructive.

[Budda]

No problem. It's not a HiJack!

I enjoy reading everybody's thoughts on topic, and also the offshoots. ;)

[Larry F]

It's been a while since I've thought of this stuff, but I have always thought of the just scale as a modification of the Pythagorean. To generate the C Pythagorean, I start on the note F and successively add pure perfect 5ths. A pure 5th is appoximately 702 cents, while the 12-note equal 5th is 700 cents. If you take the series of 5ths and put them in a scale form, you get C D E F G A B. In order to see the need for just intonation, I'll put them into triads:

C E G
D F A
E G B
F A C
G B D
A C E
B D F

The problem is that, while the roots and 5ths are pure, the thirds are not. Consider the note E. In the 12-note equal scale, it is 400 cents. In the Pythagorean scale, it is 408 cents. The pure E (based on a 5:4 ratio) is 386 cents. The different between 408 and 386 is 22 cents. This is called the Syntonic comma. This value can be added or subtracted from specific notes from the Pythagorean scale. Looking at the triads below, the some notes are written normally and some are written with a -k next to them. The normal-looking notes are Pythagorean and the -k notes are those that are subtracted from the Pythagorean notes by a Syntonic comma, making them pure. Here they are:

C E-k G
(D F A)
E-k G B-k
F A-k C
G B-k D
A-k C E-k
B-k D F

As I mentioned above, I studied tuning theory with Easley Blackwood. You could tell he really wanted to get behind just intonation. The problem was what he regarded as a fatal ambiguity that is in the D minor chord. Here is the usual tuning of the D minor chord in just:

D- F A-

Here, both the notes D and A are a syntonic comma lower than when they are used in F, G, and Am triads (I am leaving out B dim.). When music cadences on ii V, the usual practice is to keep D in the same voice. But because D is a Syntonic comma lower on Dm, followed by a Pythagorean D in the G chord, something is wrong. Are you supposed to bend the D-k up to G when keeping D in the same voice? Blackwood conjectured that as this ambiguity started to become more pronounced in music, a need was created for an alternative to just. That's where we get into meantone, I think. I am not a historical person, so I am not sure about meantone, where there is one type (based on k/4) or what. I think there was a flowering of tuning systems following the "demise" of the just scale.

This brings us back to the Barbershop Quartet guys. I am sure their music is filled with ii V passages. That would be where you get the D-k, followed by D (sorry for switching so much between Roman numerals and C major). As I said, it has been a while since I thought about this, but I think this is how just works.

[warmingtone]

I read this recently, though had heard of the idea before, and thought this might be of interest. It was found on the site that promoted those true temperament boards. Before one think of associations with equal temperament, it does not apply, but there is no doubt that temperament and tuning continued to move for quite some time...

About Key Colour
"Key colour" is absent in equal temperament . As all its intervals are equal, the blend of intervals is the same in every key, so all keys sound alike. "Key colour" is a feature of temperaments with unequal intervals, like Meantone and Well Temper. Before equal temperament came along in the mid 19th century, "Key Colour" - also known as "Key Character", or, in German, "Affekt" - was a familiar and generally accepted part of musical expression.

"By Beethoven's day, the concept of "Key Character" (in which different keys conveyed specific emotional meanings), was much refined. A widely read and influential list of keys and their affective qualities, written by Christian Friedrich Daniel Schubart and published posthumously in 1806, contained the fashionable descriptions for all major and minor keys." from "Six Degrees Of Tonality - The Well Tempered Piano" by Edward Foote.

AFFECTIVE KEY CHARACTERISTICS
from Christian Schubart's Ideen zu einer Aesthetik der Tonkunst (1806)

C Major
Completely Pure. Its character is: innocence, simplicity, naïvety, children's talk.

C Minor
Declaration of love and at the same time the lament of unhappy love. All languishing, longing, sighing of the love-sick soul lies in this key.

Db Major
A leering key, degenerating into grief and rapture. It cannot laugh, but it can smile; it cannot howl, but it can at least grimace its crying.--Consequently only unusual characters and feelings can be brought out in this key.

C# Minor
Penitential lamentation, intimate conversation with God, the friend and help-meet of life; sighs of disappointed friendship and love lie in its radius.

D Major
The key of triumph, of Hallejuahs, of war-cries, of victory-rejoicing. Thus, the inviting symphonies, the marches, holiday songs and heaven-rejoicing choruses are set in this key.

D Minor
Melancholy womanliness, the spleen and humours brood.

Eb Major
The key of love, of devotion, of intimate conversation with God.

D# Minor
Feelings of the anxiety of the soul's deepest distress, of brooding despair, of blackest depresssion, of the most gloomy condition of the soul. Every fear, every hesitation of the shuddering heart, breathes out of horrible D# minor. If ghosts could speak, their speech would approximate this key.

E Major
Noisy shouts of joy, laughing pleasure and not yet complete, full delight lies in E Major.

E minor
Naïve, womanly innocent declaration of love, lament without grumbling; sighs accompanied by few tears; this key speaks of the imminent hope of resolving in the pure happiness of C major.

F Major
Complaisance & Calm.

F Minor
Deep depression, funereal lament, groans of misery and longing for the grave.

F# Major
Triumph over difficulty, free sigh of relief utered when hurdles are surmounted; echo of a soul which has fiercely struggled and finally conquered lies in all uses of this key.

F# Minor
A gloomy key: it tugs at passion as a dog biting a dress. Resentment and discontent are its language.

G Major
Everything rustic, idyllic and lyrical, every calm and satisfied passion, every tender gratitude for true friendship and faithful love,--in a word every gentle and peaceful emotion of the heart is correctly expressed by this key.

G Minor
Discontent, uneasiness, worry about a failed scheme; bad-tempered gnashing of teeth; in a word: resentment and dislike.

Ab Major
Key of the grave. Death, grave, putrefaction, judgment, eternity lie in its radius.

Ab Minor
Grumbler, heart squeezed until it suffocates; wailing lament, difficult struggle; in a word, the color of this key is everything struggling with difficulty.

A Major
This key includes declarations of innocent love, satisfaction with one's state of affairs; hope of seeing one's beloved again when parting; youthful cheerfulness and trust in God.

A minor
Pious womanliness and tenderness of character.

Bb Major
Cheerful love, clear conscience, hope aspiration for a better world.

Bb minor
A quaint creature, often dressed in the garment of night. It is somewhat surly and very seldom takes on a pleasant countenance. Mocking God and the world; discontented with itself and with everything; preparation for suicide sounds in this key.

B Major
Strongly coloured, announcing wild passions, composed from the most glaring colours. Anger, rage, jealousy, fury, despair and every burden of the heart lies in its sphere.

B Minor
This is as it were the key of patience, of calm awaiting ones's fate and of submission to divine dispensation.

Translated by Rita Steblin in A History of Key Characteristics in the 18th and Early 19th Centuries. UMI Research Press (1983).

Plagiarised from various (acknowledged) sources by Paul Guy

One does get some feeling with the guitar of some of this though with different keys and voicings of chords however, that they are not all "equal" to each other in character...but it could be just associations with songs and sounds we are familiar with I guess

[klasaine]

"D Major
The key of triumph, of Hallelujahs, of war-cries, of victory-rejoicing. Thus, the inviting symphonies, the marches, holiday songs and heaven-rejoicing choruses are set in this key."

I guess Beethoven didn't read this paper.

[warmingtone]

Well...the belief in such things was probably always a bit subjective...or perhaps it was more from a statistical analysis...it's a curious kind of phrase there "by Beethoven's day the character was much refined..." I would have thought that as equal temperament marched on the effects of different keys would have become less...but they were strange times I guess.

Still it is an interesting idea that I've heard some still give some credence to. I do wonder though if to some degree, whether its the voicings or by association or whatever, the guitar does have some idiosyncrasies in it. You kind of notice it with some song writing perhaps...certain keys have a kind of 'sound' on the guitar at times...not sure I'd go as far as this list...still, interesting all the same.

"Six Degrees Of Tonality"...is this like 'six degrees of separation'?

[klasaine]

I absolutely think that tonal centers, not necessarily keys, but the overall tonality of a piece or section has a color and emotional feel regardless of tuning system. The only thing is is that I firmly believe it's personal and it's learned.

*I play with plenty of musicians who don't use the equally tempered system (various ethnic gigs). Their sense of and descriptions of keys in relations to emotions and feelings are as diverse as their various cultural backgrounds. They also play fine with us westerners - as long as we're playing in their keys(s).

[Budda]

Steve Kimock uses a technique similar to Cello Vibrato Technique, to actually push Notes Flat a little.
Pretty cool!

[warmingtone]

Originally Posted by Budda

Steve Kimock uses a technique similar to Cello Vibrato Technique, to actually push Notes Flat a little.
Pretty cool!

I'll have to look at his playing some more, he's new to me though I have kind of heard the name...that technique is hard to do...

I was influenced by a book I found somewhere that listed all these 'jazz articulations' and notations when I was studying, I noticed how many can be used by horn players but so few by guitar players, after that I always wanted to bend down and scoop into notes and have a variety of vibrato and play with the 'intonation' of notes.

Miles Davis was a big influence particularly, how excruciatingly beautiful his notes can be. His 'phrasing' is of course unique and beautiful and obvious, but his articulations of both the attack and the decay of notes is equally impressive and with this kind of control all he need do is play a single "note" to be great.

It's only really been in the last two years that I have put the effort into putting together guitars that are really capable of such things. My strat copies I was playing were ok for a bit, but you really need to know that the tuning is stable and will return to pitch reliably to have the control with trems.

I had a jam today so back to the pick for a bit but the fingers also give a bit more of a range of attack I find as well. Also, new set of strings and switched back to the tele for the day, felt good and great for that music...but I'm spending the evening re-intonating again the new guitar and it still needs more 'tinkering' time.

[Budda]

Yeah, Fingers for me also.

I can't stand Picks anymore.

[ryokan]

Originally Posted by warmingtone

Miles Davis was a big influence particularly, how excruciatingly beautiful his notes can be. His 'phrasing' is of course unique and beautiful and obvious, but his articulations of both the attack and the decay of notes is equally impressive and with this kind of control all he need do is play a single "note" to be great.

So true! There is no one else who can play more beautifully and interestingly. I LOVE how he plays so hesitantly, quietly and often hits notes that are slightly out of tune. This is particularly evident on the 'Porgy and Bess' recordings with Gil Evans. I'm still trying to figure out how to incorporate that in my guitar playing...

[Ed Foote]

Greetings,
I have, for many years, watched the pros tune their guitars by ear,(I began tuning pianos in recording studios in 1977, well before most of these advanced tuning machines). It was amazing how often a guitarist would get lost trying to place the B string in tune. There is a reason for this, and it has to do with how wide the G-E sixth is!
For the six string guitar to play evenly in all keys, it must be tempered according to a plan. It will sound just a little off everywhere, but every key will be equally off. Keep in mind that all fourths are tuned just a little wide from perfect, also the major third, but more so. You have four fourths and one maj 3rd on a guitar. *Cumulative error is what you must guard against.* (Wide means that either the upper note is sharpened from Just, or the lower note lowered).

In order to tune a slightly wide fourth, you must use a test note to play each of the notes of the fourth with, and compare the speed of the beating that results.
With a fourth, the beating occurs at a pitch two octaves above the lower note. When your guitar is tuned you should hear virtually no beating in the fourths, maybe on the B-E, but even then once a second is about the max you need allow. This is what makes tuning by fourths so hard, there is very little information to go by when the interval approaches pure.
The test for a tempered fourth is called a maj3rd-maj6th test. By use of a test note below the interval, you will form a 3rd and a 6th using the lower and upper(respectively) notes of the fourth. The speed of the beating in the 6ths is very slightly faster than the speed of the third; both are rapid enough to use for comparison.

1) To tune the D-G fourth, play the open D string and the Bb below it on the A string. Listen to the beats, if you cannot hear any, lower the Bb a little, this "test" third has to be wide, so lower the Bb until the beating is about 3 times a second. After you have identified the speed of the Bb-D maj.3rd , play the Bb and the open G string together.
This 6th should beat barely faster than the Bb-D major third, It must not beat slower!!. This is the test for a tempered fourth.
2) Following that, tune the A string to the D (use the F on the lowest E string, same test with the beats ).
3) Then tune your low E string to the A. since you don't have a test note below, just bring the E up until you hear no beating at all. Then compare the low E string to the 2nd fret B on the A string above, this should be very close to a perfect fifth, and compare it to the second fret on the D string, if this is not a clean octave, try to slightly sharpen the fretted string and see if it makes a cleaner octave, then move the E string up or down as necessary.
4) After the four lower strings are tuned, you should be able to play the G-B third on the open strings and tune the B string as purely as possible, then sharpen it until you hear approx 3 beats per second, ( you may use a metronome at 90 for a half speed count). this widens the interval, which you must, in order to make the final string fall into place).
5) Then tune the E string to the B by use of the open G string as a test note. This fourth is high enough that many guitars require a slight beating, you must decide how fast the G-B third is to be allowed to get.
6) tune the high E string to the middle E, and then finally check that the B string is a clear octave with the second fret B on the A string.

The real trick in getting the guitar to sound evenly is to get a solid base on the four large strings, and then determine how the B string is to divide the G-E sixth . Martins and Gibson both need different amounts of tempering on the B, so there is no set number to aim for.
When the high E string is a perfect octave to the 2nd fret on the D string, moving the B string down will slow the beating on its G-B third, (When you narrow a widened interval, you approach purity), but increase the beating with the fourth,( lowering the B further widens the interval). This is where your ears get to make a musical decision, and you will soon find
that you are tempering, instead of just hopefully hunting harmonics and hearing hell's howl. {Perhaps this process will engender a little sympathy for the piano tuner faced with placing 230 strings together, nobody knows how hard we.......................(:)}}})

It does take several times tuning step by step like this to get the hang of it, but once practiced, the B string will no longer be in mysterious territory.

good luck,
Ed Foote
Precision Piano Works
Nashville, Tn.

[klasaine]

Wow! Very enlightening.
Thank you ED.
Come back some more.

[Thorpey]

This thread is truly a gold mine of information

I think that maybe it should become a sticky

All those in favour... Say I

[slowpinky]

Fascinating stuff

- that book looks great Klasaine - certainly an easier read than Helmholtz....

http://books.google.com/books?id=x_A...Music%22&hl=en