Just Intonation and Locating Pitches

[Budda]

Something I have been fooling with for awhile now, that I thought I'd share here.
I find it very interesting.

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

This is because the Naturally Occuring Pitches of the Overtone Series ring so well together. They Blend so smoothly.

If there is no Piano or other Equal Temperment Instrument around to screw them up, these Vocalists will usually go straight to the more Harmonious "Justly Intonated" Pitches.

Not that Equal Temperment is bad. Far from it. But, the Resonance created by Justly Intonated Pitches is just so compelling and full sounding, that they are hard to escape.

Anyway, if you want to hear these for yourself, they are easily accessed right on your Guitar's Strings!

Here's a good way to do just that:

On an A String, the 12th. Fret Harmonic will give you an Octave of the Open A Note. This could be expressed as a 2:1 Ratio.

What about splitting the String into 1/3s?

This is basically playing the Harmonic at the 7th. Fret.

And this Pitch (3:1) is the Perfect 5th. of A. An E Note.

See how this Pitch appears so strongly? This helps to explain WHY a Perfect 5th. is so fundamentally important.

Power Chords anyone?

Split the String into 1/4s and you get another Octave of the Fundamental (4:1). Or, 2 Octaves above the Fundamental.

This is right over the 5th. Fret.

Split the String by 1/5s, (5:1 Ratio) and you'll arrive at a new Pitch. This is the Major 3rd., or C#.

You can find this Pitch right behind the 4th. Fret. Not directly over the 4th. Fret, but slightly behind it. A little closer to the Nut.

Now you have the Building Blocks of the Major Triad.
In this Example, it was the A Major Triad, spelled A E C#

When you use Octave Reduction to bring all of these Pitches into Intervallic Space, you end up with what looks like Stacked 3rds.:

A C# E

And this is as far as it goes with most Western Music Theory.
It's called "5 Limit" Tonality, as it goes as far as the 5th. Partial (5:1 Ratio).

*Now, listen to that JI (Justly Intonated) Major 3rd.
Compare it to the Fretted Major 3rd. on the G String, 6th. Fret.

Hear how Sharp the Fretted (ET, or Equal Temperment) 3rd. sounds in comparison?

Ever wonder why there is such a strong desire to "slightly Bend/Tweak" that "Minor 3rd" C Note, whenever playing those "5th. Fret Pentatonic Minor Blues Box Licks"?



WADDAYATHINK?

[68thinline]

Originally Posted by Budda

WADDAYATHINK?

I think the Ancient Greeks were waaaay ahead of their time.

[ddewerd]

Originally Posted by 68thinline

I think the Ancient Greeks were waaaay ahead of their time.

yup, I believe it was Pythagoras

Pythagorean Tuning

Music and Math

Greek Music History

Why this scale won't work

Cheers,
Doug

[Budda]

So, does any of this information filter down into your playing?

[ddewerd]

Originally Posted by Budda

So, does any of this information filter down into your playing?

Absolutely........


NOT


I find it interesting, but there's really not much you can do about it in terms of tuning your guitar or thinking about the relationships - at least for me.

I have enough struggles with expanding my music by applying "regular" theory, much less trying to incorporate some of these quite esoteric (at least to me) concepts.

It's kind of funny that this came up - I am reading a book called "The Physics of Music" (yeah I know, I'm a geek) which went into quite a bit of the history of how Pythagorsa did all this. He started with a single string stretched across a board, and then figured out the 3:2 ratio by inserting a "fret" at the point where he got the harmonics and it sounded good. He never actually did anything more than that, but he then took that ratio and applied it mathematically to come up with all the other ratios. He left it to others to figure out how to apply it.

I find it quite interesting, but not something I would use. Sort of like all the rest of the useless trivia that's stored in my head.

Cheers,
Doug

[warmingtone]

Hey...something that I treat seriously and play around in my, errr, playing...

Firstly, the guitar is a particularly compromised instrument in relation to tuning, this needs to be accepted, avoid the worst effects, or find alternatives...

...

So...notes towards the end of the string, so the nut end where we can fret, are typically more compromised than others, third intervals are going to be far worse than 4ths and 5ths, octaves should be in tune sevenths are usually dissonant anyway, but are close to the octaves...so basically, a 3rd or 6th interval should be a "red flag"...minor 3rds less so as the third is already 'flattened' which tends to be a nice sound and can stand to be a little 'sharp', but a Major 3rd a little sharp can sound particularly bad.

So...that C# on the second fret of an A major chord...always problematic and most people will have heard it...I tend to play that triad up the neck wherever possible, or avoid the third, perhaps by using a power cord or a sus 2 sound. Many of the root position and chords will suffer from this problem, a G chord with the open B strings for instance, I tend to favor fretting the D note at the 3rd fret to avoid it.

...

The harmonics things can be really useful, both as contrast and variety and for range, sometimes they can even make things easier, and sometimes in part for their 'in tune-ness'...

In a recent thread I wrote an arrangement of 'Good Bye pork pie hat for solo guitar, at bars 5 and 6 I came up with this eventually...

......G-7......................Ab7........................ ........... .Em7.................................A7…………….

---|---------------------------------------------------------|------*12~~~~~-------------------*5~~~~~~~~~----|
---|-------10~~~~~~~~~--8~~~~~~~-----------|------*12~~~~~-------------------*5~~~~~~~~~----|
---|-------10~~~~~~~~~--9~~~~~~~-----------|------*12~~~~~--------*12-------*5~~~~~~~~~----|
0-|-------10~~~~~~~~~-----------------------------|--*12~~~~---------*12------------------*7-----11s10-|
---|--*12~~~~~---8--------10--8---------------*12-|~~~~----------------------------*4----------*7~~~~-----|
---|--------------10-------------------10-8---------------|--------------------------------------------------------------|

It's an unusual melody, but the Em7-A7 change is an excellent opportunity to use some harmonics. Note that I used 12th fret,5th fret and 4th fret harmonics to play the 'melody and that the last bar slides from C#-C on the D string, quite high where it sounds more in tune and allows the harmonics to ring thru...

So, yep, it influences my use of harmonics like this.

There are a lot more harmonics than those given of course, they are increasingly "off the fret" as you can imagine, there are a few around the 2nd fret, and of course they get increasingly closer together...it's kind of the opposite of how a guitar is fretted.

...

I've been influenced a lot by players like Jeff Beck who listened intently to Bulgarian choirs whose intonation and sense of harmony are even more attuned than the old barber shop quartet thing. He uses both harmonics and the tremolo arm to constantly adjust the intonation of notes...

...

Some of the Barbershop quartet sound though is actually produced by the type and closeness of their harmony. On the guitar spread wide in fourths, this is not the more common chord formations.

So, I do tend to favour close voicings, even 2nd intervals if possible, in smaller chord formations that have less if any doubling.

The other aspect to such sound is that this kind of thing works with counterpoint as much as it does with "harmony"...closely related, but treated differently...it's all about 'voiceleading' and the way simultaneous melodies or voices in the harmony move and there are a heap of 'rules' regarding this that the barbershop and good close harmony writers work with...so, I do think about that a bit....

...

The other thing that is very influential to me is a part of putting together my guitars and part time 'inventing' that I do....

So, the guitar is compromised, more so than the piano in many regards...the biggest problem is really major3rd intervals, either played together, in skips, exposed in a melody, or outlined in a line...

Now, bending is an obvious bonus to the guitar, players will often bend slightly flat' and vibrato perhaps just below pitch as it sounds 'sweeter' due to the intonation issues.

...

As noted, instead of playing a sharpened edgy major third, many people will bend the minor third slightly sharp towards the more 'just' pitches...

This is a large part of the 'blues third' but not completely, it is from the old 'field holler' days and the vocal tradition, the enforced learning of hymns that influenced harmony and the banning and punishment early on for retaining their original language and song, etc. However, if you study west African speech and scale pitches in those traditions, you will hear their own flat third intonation.

On that, it is interesting to think about accents and the way people talk or emote different things in the way they talk. Many accents and speech contours use flattened intervals and interval outlines...many, many things to think about in this aspect of musicology. For instance, ever noticed the striking differences between accents and contours in speech from people who live in wide flat plains to those with a lot of peaks and valleys, in general?

...

But, a digression...

As I say, I work extensively on putting my own guitars together and a lot to allow me to work with these ideas...

This is my tele...



It incorporates a Kahler trem that keeps the bridge rock solid (doesn't move like a fender or fulcrum trem does, more like a bigsby in a way) and is adjustable for intonation, height and string spread and has other unique features and potential...

All the intervals are potentially a little out, sharp always sounds 'edgy' or even 'off', you can only bend up...except with a tremolo system. So, I use the tremolo subtly to make chords and notes 'sweeter' or more in tune...

As well as the bending thing, consider also vibrato. This too will sharpen notes, slightly bending them at an above the fretted pitch, so even more sharp. vibrato on a blues third is going to sound sweet, vibrato on an octave may sound 'out of tune'. Again the vibrato, a little like hank marvin's use of it, can vibrato around and below a fretted pitch. Even on a chord where the third may sound a little sharp, using the vibrato to flatten everything a little with vibrato can make it sound sweeter and more 'in tune'...

I also use it much like a slide player would, again a technique that allows to bend and vibrato below a pitch and contributes to that particular sound...

...

In addition, I have spent years working on the DIY Sustainer project. My tele has this too. However, although I don't use it often, and it will of course hold any note indefinitely, but the thing I like most is that it can produce automatic harmonics from the overtone series from any note by dampening the fundamental electromagnetically...much like you use your finger to lightly touch a fret at a strings nodal points to bring out the harmonics hidden within...so the harmonics produced are all 'just intonated' in relation to the original note.

...

The other thing is the placement of pickups. A vibrating string contains the fundamental as the biggest and strongest and lowest 'naming' vibration. Generally, the neck pickup will emphasize this part of the guitars 'tone' over the harmonics beneath it. As I mentioned, the harmonics get closer as the string gets shorter, so if you fret a note close to the neck pickup, you increase the brightness and harmonic content.

Now, the bridge pickup is always near the end of the string and picks up a lot more of the higher harmonic content or "tone" behind the fundamental...a more complex wave. You will note how much easier and louder natural harmonics are with the bridge pickup vs the neck.

I tend to use the middle selector position to combine both pickups, giving a strong fundamental and the harmonics below. Often, the bridge pickup will have a lot of complex harmonics that being from the overtone series, may sound a bit "off" with a chord that contains other notes that are at odds with the natural ones...if that makes sense.

So, an A note contains behind it the octave and fifth and 3rd, etc...but as budda mentioned, that C# 4th fret harmonic is flat compared to a the fretted note...so if you play a chord with both and with a high harmonic complexity on the bridge pickup, there will be subtle background conflicts of tuning. Hence, the neck pickup is often referred to as the "rhythm pickup".

The tele typically has this problem in spades with the bridge pickup as it will bring out more harmonics (jangly) than many guitars. My bridge pickup is a wide HB and samples more of the string.

...

So, yes, extremely important part of the way I play and work with the guitar, and design them. I'm almost finished with a new guitar that is largely replacing the tele at the moment, and have another one, almost a twin to it in the planning, perhaps for open tuning and a spare...it's not a tele btw...

...

While, I tend to shy away from open tunings, one should note that one of the reasons that they can sound so sonorous is that you can tune the thirds to be more 'sweeter' between string sets instead of in the typical 4th that standard tuning offers...something I am thinking of exploring more.

...

So, yep...it influences me a lot, but most of this is intuitive once you understand the 'problem' and the solutions possible.

[Budda]

See, I have been using this material since I first made any type of Music. I just didn't know how to label and/or explain it all.

But yes, every time I "go from the Minor 3rd. to the Major 3rd." on playing a Blues Lick, I'm actually either going to the Just Major 3rd., or approximating it.
And I would bet many here also do this, whether they think of it in this way or not.

I also will Bend the Minor 3rd. just a little, and there is the JI Minor 3rd. It's also between the Fretted Min. 3rd. and the Fretted Maj. 3rd.
So, I found myself always "Tweaking" that Minor 3rd. And I'd never realized just how much I actually had been doing it.
When you hit those JI 3rds. though, they just ring so fully! Especially with some OD and some Volume behind you.

Kinda like riding a wave.

[warmingtone]

Yeah...a lot of this stuff you do cause it "sounds good" intuitively...the more you know about it, the more intuitively you can address it...

Note that Distortion compresses and brings out even more of the background harmonic content...that's why chords distorted can sound so bad...as I was getting to in part above as well (though I almost always play 'clean' these days)...

....

Yes, you will find the flattened maj3rd 'sweeter' and so too bending into an octave with a lot of vibrato.

Try a heavy vibrato, perhaps wide on a fretted octave (say an A note, against an A chord) and you will hear that it is wavering between the octave and sharply out of tune...

Now, bend to an octave from the G note into A, then vibrato as wide as you like. The tendency is to, and the 'singing sound' comes from vibrato-ing below and up to pitch....you can make the vibrato as wide as you like!

...

Still, some sharp notes can be expressive...BB king and Buddy Guy come to mind as players who will often make a "stinging' sound by playing notes a little "sharp"...I find it more difficult and less to my ear, so I tend to like slightly 'flattened' note and vibrato for a 'sweeter' sound.

Part of the reason I find country music to often sound grating to my ears is that it does play with a lot of sharpness and major thirds and a jangly sound rich in high order harmonics and clashing of sounds with the conflicts between different notes being played. Still, it can be effective and is an acquired taste.

But it doesn't do to over generalize, many 'country music' instruments like fiddle and pedal steel and vocals are of course not bound by intonation compromises, much like 'horn players' are free in jazz...just listen to the sweet 'flat' tones of miles davis, another big influence on me in this aspect of things...

...

Singers are acutely aware of this kind of thing, well good ones, bad ones will often vibrato above a note and sound out of tune for it...

...

The blues and music like it, or unaccompanied lines, such as in a vocal quartet is all very well...but there is more to music for many than just the 'blues'...

Remember that the adjusted intonation allowed the playing of musics in different keys. Early guitars and such instruments, had movable frets and were not in equal temperament, but they had to be adjusted to other keys and musics.

It also made it hard for musicians to play together and for music of more complexity.

So...equal temperament goes back only to around bach and in particular the "well tempered clavier' that showed you could play with this system complex modulating music and in every key...this kind of thing sealed the deal, but it took a while and a lot of tweaking to get where we are with today's tuning and 'ear'. We did trade off this versatility for 'in tune-ness' and functional harmony.

As a result, generally you do need to deal with it and make compromises...still

In simple music like the blues, you can get away with more, in more complex harmony like a babershop quartet or a string orchestra or wind instruments, you can adjust the intonation to sound amazing.

...

There are things that you can do, it will always be a compromise on the guitar though we see tuning systems like the Buzz Fietlen thing and evana nuts and even more complexly fretted guitars, or even unfretted guitars.

If you are into a bit of mechanics, there are things you can do to make your guitar 'sound better' with intonation and set up and in the tuning of the guitar. Although tuned in 4ths, there is a third between the g and b strings and the b and e strings...this can be subtly tweaked to give a nicer sounding third effect by adjusting the B string to sound sweeter.

...

Also, these things are major influences in acoustic guitar design and quality. As mentioned, the harmonics get more complex as towards the end of the string and so the bridge which activates the top will have the highest potential content. An acoustic sounds more "jangly and complex" and higher in background harmonics generally. So, a great acoustic guitar will provide a balance to these things to suit it's purpose...some just sound 'sweeter'. A classical guitar with low tension nylon strings tends to be far more true to the fundamental content, giving it it's warm and round less complex sound.

It is also important to understanding the way you pick the strings, picking closer to the bridge will produce a lot of harmonics, nearer the neck a warmer rounder sound...

...

All this is the science of such things, but in the end once it is understood and adjusted for, it just comes down to an intuition thing and the ear. If it sounds good, you will tend to gravitate towards using it, if it sounds bad, you will avoid it.

By building guitars for myself, I can provide more options to intuitively mess with these things, in particular, the ability to flatten notes or vibrato below a pitch and to play 'close harmony' without a lot of clashes behind the notes being played.

[ddewerd]

Originally Posted by Budda

I also will Bend the Minor 3rd. just a little

You know, I actually do the same thing - so maybe I have internalized this more than I think.

I really respect you folks here who really know your music theory - and more importantly, know how to put into practice. That's always been my problem - I can learn some theory and say " oh that's interesting and I understand the why" but I don't know the "when". But every teeny little bit adds up over the years, especially when I have the attitude of " the more I learn, the more I realize how little I really know"

Cheers,
Doug

[klasaine]

Except for ABSOLUTE pitched instruments (piano, vibes, marimba, xylophone, etc.) all decent instrumentalists do it - whether they realize it or not. I call it playing "in tune". Some instruments have to adjust more than others but none of them are perfect in the equal temperament system (which I think is genius by the way and I would hate to change it). That's the compromise for being able to play in all keys with the least amount of seriously out of tune pitches. I feel it's a great compromise.

[Budda]

Wow! You Guys are great!
What a lot of solid points made by everybody.

Warmingtone, you should write a book.

Oh, too late! :) LOL

Just kidding!

You brought up some really good insight.

And one thing I have always played off of, is the two Schools here. The JI Stuff is SMOOTH and FLOWING. And yet, the acidic/tangy Timbre of ET gets my juices flowing, just like stepping on an Accelerator Pedal. Part of the more Rock And Roll Spirit in me, I guess.

I would in no way want to abandon either.
Good thing I don't have to!

[warmingtone]

Yeah, I know...too much information, still lucky I can type!

Yeah, in context ET was the best thing to come along since sliced bread and made all these things possible.

...

At Uni, we studied jazz and so blues history all the way back, but there is lots to be gleaned from the worlds music and the real heritage of music like the blues, all the way back to Africa and the different influences of different transplanted societies and the pressures on them. The french influences in Cajun and Creole musics, the rhythmic influences that remained in southern america and elsewhere in the world. A fabulous rich history...and there are continuous if not so momentous cross pollination's, like rock, country and jazz back into the blues of course.

I love to hear those people who apply their own 'pressures' on the form as well, or take a few steps back and start from there and keep the genre alive.

But I think it is worth delving deeper than the simple overtone series to understand where some of this stuff comes from though it is no doubt a part of things.

...

But yes, there is some 'science' to these things...but there has to be compromise to make more possibilities possible.

The guitar is an instrument that requires a bit of compromise, there are ways to help it along though, and it is well worth thinking about a little, making adjustments, then carry on playing. It is not as "technical" as the explanation and much of this stuff is 'intuitive' but sometimes elusive.

So, I certainly would keep an ear to such effects, see if your guitar is optimized to be 'in tune', see if you can through technique limit effects, see if you can find why something like the "JI 3rd" sounds good, and find other "good" sounds too.

Every now and again you see someone that just "sounds good"...I remember a 'rock star' down here used to have two primary minimum criteria for hiring people for his band (that he unusually had on a wage, not a cut)..."In Tune and In Time" ...his bands always sounded phenomenal even compared to far more technically proficient players that he could have had his pick of. Plus, he applied this to himself as a vocalist, amazingly professional and attentive to detail.

Anyway, interesting thread, obviously touching on some pet ideas of mine...cheers

[Tim Bowen]

Originally Posted by Budda

So, does any of this information filter down into your playing?

Sure. Even if I have a Peterson Strobostomp on hand, I still wind up tuning "by ear"; maybe not so much for live, but definitely for recording. There's an old trick that dobro players use, which is to slightly flatten the B string as functioning as major third. It's a horrible choice if you need to play a C chord, but for G-based slide playing, it's a good call. I'm guilty of tweaking intonation via fretting hand pressure; I'm definitely going to apply different pressure for Andy Summers-approved add9 chords than I will for cowboy chords.

But yes, every time I "go from the Minor 3rd. to the Major 3rd." on playing a Blues Lick, I'm actually either going to the Just Major 3rd., or approximating it.
And I would bet many here also do this, whether they think of it in this way or not.

I call it a "teaser". Unless it's a minor blues, the chord at hand over a bare bones progresssion is *usually* a dominant seventh with a major third. What usually sounds best to me over this chord is a sound that sits somewhere between a minor and major third. Nothing new though, people have been choosing this sound since long before I was born.

[68thinline]

That's an interesting point about tuning by ear. I use alot of drop tunings on my acoustic and find myself naturally wanting to flatten the third.

[Larry F]

Originally Posted by ddewerd

You know, I actually do the same thing - so maybe I have internalized this more than I think.

I really respect you folks here who really know your music theory - and more importantly, know how to put into practice. That's always been my problem - I can learn some theory and say " oh that's interesting and I understand the why" but I don't know the "when". But every teeny little bit adds up over the years, especially when I have the attitude of " the more I learn, the more I realize how little I really know"

Cheers,
Doug

You know, I think you should continue to open yourself up to other ideas about music. Sometimes you will find something that directly applies to what you are working on at the moment. More often, some of those ideas seem to consist in a world apart from daily musical experiences. New terminology is used, sentences are densely filled with technical terms that seem mathematical or academic. In this kind of world, people are writing for a different audience than the daily person. But it is not an exclusive club and people writing this way are not putting on airs (except for one guy: Charles Wuorinen. He pisses everyone off).

I studied tuning with Easley Blackwood for two semesters. He had a Motorola keyboard(s) that could be programmed to play all kinds of tunings. He would demonstrate classical works that he believed to have had a particular kind of tuning. Sometimes he would play 4-part lines by spreading his fingers over three keyboards. One of his ideas was considering how far off a tuning could be before the music became unrecognizable. Now, here is an idea that seems to live in its own world. How much does this idea have to do with our daily music making? OK, if that bothers anyone, they don't have to read on. But some people find these kinds of questions interesting, providing the amount of work you have to do before you can appreciate what's going on.

In one example, he took a Bach 2-part Invention and had a computer programmer make a program that would play the piece. This was in the 70s, so the sound was very dull and mechanical. Then he considered the ratio of the whole step to half step. In 12-note equal, the whole tone is twice the size of a semitone. You can see that right away on the guitar. So had had the programmer vary the size of this ratio from 2:1 to 1.9:1, 1.8:1, etc. I don't remember the actual values. The results were astounding. As the ratio decreases, you could still hear the composition, although it was obviously way out of tune to our ears. Then suddenly, once you crossed a threshold, say at 1.1:1 or something, the melody disappeared.

Worthless theorizing for other theorists? So what? He did this out of curiosity and did the research in a way that was interesting to follow, although you had to read while working out the steps yourself on paper and pencil. Anyway, this is just one of my little pitches for the value of theoretical research and its place in our lives.

[LeopoldPlumtree]

Originally Posted by Larry F

In one example, he took a Bach 2-part Invention and had a computer programmer make a program that would play the piece. This was in the 70s, so the sound was very dull and mechanical. Then he considered the ratio of the whole step to half step. In 12-note equal, the whole tone is twice the size of a semitone. You can see that right away on the guitar. So had had the programmer vary the size of this ratio from 2:1 to 1.9:1, 1.8:1, etc. I don't remember the actual values. The results were astounding. As the ratio decreases, you could still hear the composition, although it was obviously way out of tune to our ears. Then suddenly, once you crossed a threshold, say at 1.1:1 or something, the melody disappeared.

As the diatonic semitone reaches the size of the whole tone, you arrive at 7-tone equal temperament (or 5-tone if the chromatic semitone is made the size of the whole tone). It'll sound strange, but the melody won't really disappear. You'll most likely have no problem identifying this (tuned in 7-tet)...

http://www.io.com/~hmiller/midi/canon7.mid

[klasaine]

The oldest unambiguously notated piece of music we know of, is the "First Delphic Hymn to Apollo", from 138 B.C. It's in a quintuple meter...it's in the Phrygian mode...it utterly rocks...
http://www.youtube.com/watch?v=XWvHk...layer_embedded

[Budda]

Originally Posted by klasaine

The oldest unambiguously notated piece of music we know of, is the "First Delphic Hymn to Apollo", from 138 B.C. It's in a quintuple meter...it's in the Phrygian mode...it utterly rocks...
http://www.youtube.com/watch?v=XWvHk...layer_embedded

"Just let me hear some of that old time Rock and Roll......."

:)

Kinda sounds like an old Fugs record my brother had.

Now, if the Fugs and the guys from Monty Python's had a jam over at the Oracle of Delphi..........

[Budda]

Seriously, though, it was kinda cool. I like how it had the dramatic pause, the different layers, Vocal, Vocal/Strings, etc. Some real arranging there!

So, that's the oldest one they can read for sure? Pretty neat.

[Budda]

P.S.

What was the B Side?