Why JI tuning works on the E9

[Phil Halton]

I'm no expert by any means, but I've done alot of reading, experimenting, spreadsheeting, and just plain head-scratching on the subject of Equal temperment, Pythagorean, Meantone, and various forms of well-temperment tunings,and have been especially interested in just why Just Intonation works on the E9, when, in fact, it really shouldn't.

First of all, this table shows the JI ratios, and the cents offsets from ET those ratios produce for the 12 chromatic pitches.

Pitch, Ratio, Cents from ET
m2, 25/24, -29c
M2, 9/8, +4c
m3, 6/5, +16c
M3, 5/4, -14c
p4, 4/3, -2c
dim5, 45/32, -10c
p5, 3/2, +2c
m6, 8/5, +14c
M6, 5/3, -16c
m7, 16/9, -4c
M7, 15/8, -12c

Note: On the m7 I use the Pythagorean ratio of 16/9 instead of the simplified ratio of 9/5 for purposes of shifting one of the wolf 5ths to another pitch. The JI ratio 9/5 would have produced a cents offset of +18c instead of the -4c.

For the sake of brevity, I'll skip the details and simply state that:

Using the cents offsets produced by these JI ratios, any chord built from the m2, M2, m3, and dim5 pitches will have a wolf 5th narrow from pure JI of 22 cents, and one (built from the m2) will be wide of pure JI by 44c.

Everything else will sound Just great (pun intended), but chords built from these tones will have howling wolf fifths, and will sound off to say the least.

Incidentally, if I use the JI ratio of 9/5 for the m7 pitch instead of the Pythagorean ratio of 16/9, it simply moves one of the wolf 5ths off the M3 and puts it on the m7.

Now, from the perspective of E as the unison pitch, this means that any chord built on the F, F#, G, and A# pitches of the tuning will have sour fifths. Fortunately for E, the only natural chord that suffers is the II minor chord, or the F# minor played on strings 7, 6, 5 with A&B pedals down. The 7th and 5th strings will produce a 5th interval narrow of JI by 22c. As far as I know, most copedents don't build chords off an F, G, or A# pitch.

If we shift the perspective to another key, that is, make some other pitch the unison and reassign the JI ratios accordingly, the relative position of the Wolf 5ths change. For example, if we make A the unison pitch, and assign JI 4/3 to D, JI 3/2 to E etc, then:
The m2, M2, m3 and dim5 of A will have the wolf 5ths which means that any chords built off the A#, B, C, D# pitches of the tuning will be sour. So, a B6 chord would be sour, while most all other chords sound fine--not an acceptable trade-off.

I've looked at virtually every key like this, and I found that E is really the only key that JI will work well in for the most common opedent, and even E isn't perfect because of the wolf on the II minor chord.


Well, that's it! I've bent my mind as far as I care to on this topic. I have to say that I haven't actually re-tuned my guitar and checked these intervals out--its strictly headwork. For that reason, I could probably be way off on some things I've presented as fact. I'd be glad for anyone to show me exactly where and how I'm wrong on this.

P.S. The one thing I like about the meantone tuning given to me by Ernest Bovine, and presented in an earlier post, is the fact that there aren't any wolf 5ths, and virtually all intervals (with the possible exception of the M2) has no perceptable, or at least troublesome, beating. Also, it retains a little of the "bite" of ET and, while I'm not sure on this, I think that would make a difference when playing with keyboardists and 6-stringers etc.

[chas smith R.I.P.]

I use the Pythagorean ratio of 16/9 instead of the simplified ratio of 9/5

And 16/9 is the same distance, or symmetry, from 1/1, 2/1 as 9/8. You can also view it as the "reciprocal" of 9/8.

Using the cents offsets produced by these JI ratios, any chord built from the m2, M2, m3, and dim5 pitches will have a wolf 5th narrow from pure JI of 22 cents, and one (built from the m2) will be wide of pure JI by 44c.

Not true, I have a chime, that I made, using the Partch scale plus the 45/32 and 64/45 tritones. If I play the dim chord, 1/1, 6/5, 45/32, 5/3 there are no beats.

I've looked at virtually every key like this, and I found that E is really the only key that JI will work well in for the most common copedent,

That's because you're only thinking of the "open tuning". Once you place the bar on the, say, 5th fret, all of the "relationships" will transpose.

What works the best for me is a kind of "hybrid just" based on the "statistical likelihood" that an interval will occur, meaning that, how more often am I going to play my 7th string F# with pedal-up B or pedal-down C#. Which will determine who's going to have more beating, but then, keep in mind that I'm moving the bar around and if you're listening for the beating interval, you're not listening to what I'm playing. Or, if my playing is that boring, I should be a welder....

[Phil Halton]

Hi Chas,

Thanks for the response. My post was kind of "quick and dirty", and, like I said, this was all on paper so I'm sure I made errors.

I saw that the distance from F#(+4c) to C# (-16c) would produce a fifth only 680c and that's what got me thinking on all of this. I then started looking at other fifth intervals and also saw that the flat 5 (-10c) to the m2 (-29c) also produced a narrow fifth by around 20c or so. Maybe I'm missing something there-wouldn't doubt it.

Then there's the fifth from m3 (+16c) to m7 (-4c) narrow of an ET fifth by 20c. Finally, there's the fifth built from the m2 (-29c) to the m6 (+14c) wide of an ET fifth by around 44c.

I'll be embarrassed if I'm making some obvious math error here, but I don't see where I'm wrong.

[chris ivey]

if you guys keep talking like that you'll go blind!

[Kevin Hatton]

Tune by ear and you will be a better player. Use your meter for reference when you need to.

[Marc Jenkins]

Phil, the math has slipped from my brain, but I do recall that part of what got me interested in an alternate to JI was the impossibility of the F# as root, and any interaction between the C# and either F#. The C# (A + F) chord was always a nuisance for me as well. I found a meantone tuning that while not as sweet on major chords, especially when playing solo, blended much better with my band. We don't play country, and use a lot of extended chords and suspensions. I'm also playing a lot of 'harmony section' stuff with an accordion, Wurlitzer, and/or Mellotron.

[chris ivey]

that's interesting...mellotron, accordian, wurlitzer....none of these deal with a true pitch center, do they?

[David Doggett]

On my E9/B6 Universals, I solve the iim problem by putting an 8th string raise to F# on my C pedal. That stop can be tuned pure JI as the tonic of the F#m chord (B and C pedals). It makes the C pedal a little stiff, but I love the sound of that raise. The 7th string remains tuned JI as the 5th of the B chord (open strings 7 5 2 1). No wolf whistles. E9 is an amazing tuning for keeping everything JI. But as Earnest says, the intervals are all transposable to any key by tuning to a different root, or taking the bar up the neck.

[b0b]

Everything works perfectly until you get to the F# note. Then you have to choose either the 9/8 to be in tune with the B, or the 10/9 to be in tune with the (pedaled) C#. So you are correct - the F#m is out of tune.

Cabinet drop when pressing B+C helps a little bit, but not enough on most guitars.

I tune my F# to the pedaled C#, and use a small compensator pull to raise the F# on the lever that lowers the E strings.

Most players do it the other way around. They tune the F# to the B, and then lower it a bit on their A or B pedal.

You wrote "E is really the only key that JI will work well in". The bar makes all of the keys in tune. If the intervals of the A chord are in tune on open strings, then C will be in tune at the 3rd fret, and Eb will be in tune at the 6th fret.

If you haven't seen it already, please take a look at my page on the subject at: b0b.com/infoedu/just_e9.html.

[Larry Bell]

Do ya ever play that thing or do you just spend all yer time trying to figger out how to tune it?

Honestly, I've listened to players whose tuning methodologies are all over the map and a good player always sounds in tune, no matter how the guitar is tuned open. Once the bar hits the strings and the strings warm up unevenly, all the precision in the world goes out the window.

I have a buddy who loves to keep his guitars spotless. Before and after every gig he wipes it down from one end to the other - several minutes each time. I remember some old codger coming up to the stage b4 the band started and asking, "do ya play that thing or just polish it?"

[Phil Halton]

So many great responses--where to begin?

First to Chris--guess what--I went blind almost 45 years ago! Ha!

Kevin, Its more for the scientific and mathematical interest that I investigate this stuff. I don't mind a slightly out-of-tune guitar. Heck, I'll go for a week without tuning sometimes.

David, that F# raise on the 8th string is a brilliant solution. Who'd a thunk it--not me, that's for sure.

Hey B0b, I never understood the workings or purpose of those compensator pulls before now. Its interesting because I got those JI ratios from a previous post of yours to another question of mine on tuning long ago. In there, you mentioned the different ratio for the '"compensator" but I ignored it for lack of understanding of what they were.

As far as E being the optimum key... what I meant was shifting the unison ratio from E to another key, say A for example, in which case, the wolf fifth of the F#minor chord (M2 of E) would then appear on the B chord (M2 of A).

Hi Larry, like I said to Kevin, I use the guitar equally as a platform to understand the Harmonic series, and Pythagorus etc as I do to enjoy playing Faded Love. Its funny though, because I do spend less time playing and more time thinking and writing when one of these ideas hits me. I consider myself a musician, but I also like science and mathematics. I'm investigating to learn the Physics of sound, and the ages old art of temperment as much as to be able to have an in-tune guitar--whatever that means.

It occured to me last night that I might be inadvertently starting another tuning war with this topic. That's not my intent. I played banjo for thirty years and never had to give much thought to tuning (man, is that a slap to banjo players or what;-). Now I got ten strings 8 knees and pedals and a world of possibilities. Learning to tune the thing is as much fun for me as learning to play a tune.

[Larry Bell]

Hi Phil
Glad you took my little jab in the spirit it was intended -- just yankin' yer chain

In reality, I am a scientist (well, not a physicist, a synthetic organic chemist) and managed a group of engineers for many years. I know from whence you speak. I just accepted the disconnect between the theoretical and the practical when it comes to pedal steel guitar many years ago.

I will never (consciously) participate in another tuning war -- been there -- done that -- got the prescription for Prozac to show for it.

[chas smith R.I.P.]

Then there's the fifth from m3 (+16c) to m7 (-4c) narrow of an ET fifth by 20c. Finally, there's the fifth built from the m2 (-29c) to the m6 (+14c) wide of an ET fifth by around 44c.

Phil, there's 5ths all over the place, and not all of them are in tune. A good example is why, in the "traditional" just scale, there are 2 major 2nds, 9/8 and 10/9. Reason being, the 9/8 is in tune with 3/2, the p5 and 10/9 is in tune with 5/3, the 6th. You've probably noticed that 9/8 and 3/2 have friendly math, in that the numerators and denominators are easily divisible, same with the 10/9 and 5/3 and 5/4, 15/8.

Back to your scale, I would use 16/15 instead of 25/24 for the minor 2nd, that way it's symmetrical with the 15/8.

I consider myself a musician, but I also like science and mathematics. I'm investigating to learn the Physics of sound, and the ages old art of temperment as much as to be able to have an in-tune guitar--whatever that means.

One of the pleasures of working with music is that it can be enjoyed on a lot of different levels. Whether I'm just listening to the radio, in the shop or truck, playing on various venues or composing in my studio. Part of composition, like working in any medium, is understanding the stuff you're working with, what it's made of and understanding the various ways on how to organize it. In that regard, music is a huge playing field and the deeper you want to get into it, the deeper it gets. The thing about tuning systems, and "alternate" tuning systems is the math is very "friendly". For myself and my own compositions, sometimes I adhere to just scales and more often than not, I don't use a formal tuning system.

[Phil Halton]

Chas, you wrote...
Phil, there's 5ths all over the place, and not all of them are in tune. A good example is why, in the "traditional" just scale, there are 2 major 2nds,
9/8 and 10/9. Reason being, the 9/8 is in tune with 3/2, the p5 and 10/9 is in tune with 5/3, the 6th.
...
I guess I'm coming at this from the Pythagorean perspective where all fifths were in tune with each other as they were an unbroken chain of perfect fifths, but as a result, with very "active" 3rds. As I understand it, the ratio 5/3 was a simplified approximation of the pythagorean ratio of 27/16 which was a perfect fifth from 9/8 which in turn was a perfect 4th down from 3/2 etc. Same with 5/4, a simplified approximation of the pythagorean 81/64, which was a p4 down from 27/16 etc.

As I understand it, they approximated and simplified the ratios to tame the 3rds. By approximating and simplifying the origional Pythagorean ratios, they broke the chain of perfect fifths, so that they're no longer always in tune with each other. Hmm... I think I get it -- that's what they're calling the JI scale? Didn't realize there was a "traditional" JI scale. It seems there must be many, or at least some, variations of JI depending on the alterations of the Pythagorean ratios that are chosen. Like 25/24 for example, approximated and simplified from the Pyth 256/243 which was derived from the octave (2/1) minus the M7 (243/128). That M7 in turn being derived as a perfect fifth from the M3 (81/64 X 3/2 = 243/128).

You know, I think that fellar was right, you can go blind talking like this--better stop now.

[b0b]

Phil, you should get a copy of Harry Partch's book, Genesis of a Music. You would probably enjoy it.

JI didn't evolve from Pythagorean - it's an expression of pure harmonic ratios. The fact that thirds sound wrong in Pythagorean underscores the folly of taking the circle of fifths to extremes.

[Archie Nicol R.I.P.]

Where's me kitten' banjo?

Arch.

[chas smith R.I.P.]

where all fifths were in tune with each other as they were an unbroken chain of perfect fifths,

Sounds like meantone and when it comes back to the octave, it's out by 81/80, the wolf-tone.

It seems there must be many, or at least some, variations of JI

There are and it has to do with the limit you're working with and this isn't the place to get into that. I was recently at a party where there were a bunch of microtonal composers. The first question they asked was, what was the limit I was working with. I told them I wasn't working with the matrix, but rather with events and texture. Good thing we were all drunk.

, approximated and simplified from the Pyth 256/243 which was derived from the octave (2/1) minus the M7 (243/128). That M7 in turn being derived as a perfect fifth from the M3 (81/64 X 3/2 = 243/128).

You're going to make yourself crazy with this stuff...

Phil, simply put, Just intonation comes from the relationships of the harmonics in the harmonic series, which is why they all relate to the Fundamental pitch, and each other. Again briefly, because it's raining outside and I can't work. If the fundamental is C, like on a piano. The reason the piano, or any instrument, sounds like it does is because of the relative loudness of all of the harmonics in the sound when the sound is happening. the reason a sine tone sounds like it does, is because it doesn't have any harmonics.

Starting from C, the fundamental or 1st harmonic, the 2nd harmonic is a C an octave above. The ratio of their interval is 2/1, which coincidently is their "placement" in the series and is also the ratio of their respective frequencies.

The 3rd harmonic is a G, above the 2nd harmonic C. Their relationship is 3/2, the 4th harmonic is another C, an octave above the 2nd. 4/2 = 2/1. And the interval of a C over G is a p4th, 4/3.

The 5th harmonic is an E and the interval between E and C is 5/4.

The 6th is another G. G over C, 6/4 = 3/2 and G 8va G, 6/3 = 2/1.

Continuing up:
7 is a Bb
8 is a C
9 is a D
10 is an E
11 is an F#
12 is a G
13 is an A
14 is a Bb
15 is a B
and 16 is a C

As you go further up, things get a lot more microtonal.

Like b0b said, I think you would get a lot out of, Genesis of a Music.[/quote]

[Phil Halton]

Okay Chas, Got it. I was coming at it backwards. They come from the overtone series--not from the pythagorean ratios.

I'm gonna check out that book.

[Ken Adkins]

Holy Holyoke Phil!! You lost me.

[chas smith R.I.P.]

Holyoke Massachusetts. I was born in Hardwick and grew up in West Brookfield, a bit east on Rt9. Good luck -- cs

[Stan Paxton]

I don't speak this language at all, nor do I understand it. But it has piqued my curiosity: what is the Wolf tone? What does it sound like, or is it something I have heard but did not recognize it?
I'm too old to wrap my mind around all this stuff now.
Thanks

[b0b]

The wolf is a note that sounds way out of tune when everything else is very nicely in tune. On the E9th it usually shows up on the F# strings, when you compare them with the pedaled C# notes.

[chas smith R.I.P.]

I'm too old to wrap my mind around all this stuff now.

You're never to old, besides, it's good for you....

To give you an idea of how long this has been around, the wolf tone is also known as the Comma of Dydimus, the Ptolemaic comma and the Syntonic comma, plus some other names.

[Bo Legg]

I missed some of those lectures on Pythagorean tuning, marvel temperament, miracle temperament, equal temperament and etc... However that was back in the day when Oscilloscopes were big bulky expensive things and you couldn’t carry one around with you to tune by.
I do remember being told that this whole subject came under the heading of near useless info because back in those days they had no Oscilloscopes or Audio Spectrum Analyzers to measure those frequency ratios so in reality they were TUNING BY EAR.

[Don Brown, Sr.]

A question! At the end of the day, which would you rather play:

A steel that's scientifically out of tune, but sounds in tune when played, or one scientifically in tune, but sounds out of tune when played?

I'll play the one that is scientifically out of tune, but sounds In Tune when played. That's where the ear enters into playing..