Musical Ignoramus Question Time:
Does that mean that a piper and (e.g.) a banjo together cause a dissonance? The pipes being in JI and the banjo in ET? Or is that a dumb question? Does that limit the possible combinations of musical instruments?
It seems in your computation you equaled C# with Cnat.
Cnat in just intonation may be 16/9, which is -4 cents.
So the discrepancies are only somewhat bad for the sixth, but not the fourth.
I II III IV V VI VII- VII VIII II III IV
0 +4 -14 -2 +2 -16 -4 -12 0 +4 -14 -2
D E F# G A B C# D'
I II III IV V VI VII VIII
G A B C D E F# G
normal.+2: 0 +4 -14 -2 +2 +6 -12 0
But if we would tune the second to a ‘minor’, using 10/9 instead of 9/8, which is -18 cents, then for G on a D whistle we get a sixth which is right, and a second which is right as a ‘major’ 9/8 second:
I II III IV V VI VII- VII VIII II III IV
0 -18 -14 -2 +2 -16 -4 -12 0 -18 -14 -2
D E F# G A B C# D'
I II III IV V VI VII VIII
G A B C D E F# G
normal.+2: 0 +4 -14 -2 +2 -16 -12 0
I quite like to tune for just intonation to lower, flatter intervals, then push any note up, to reach an equal tuning, when playing with fixed equal tuned instruments like a concertina.
Poor (or clever?) choice of example DrPhil, the banjo causes a dissonance any time it’s played. Especially if the G string is involved. And that from one who plays banjo from time to time.
Doh! 
Thanks for catching that.
Yes, the 2nds and 7ths are always a bit squishy. One nice thing about the Shaku tuner is that it marks the choice points for each: 10/9, 9/8 and 8/7 for 2nds, and 14/15, 17/18 and 20/21 for 7ths. So that makes it easier to experiment blowing in and out of those pitches.
Sure, but navigating those dissonances is something experienced musicians do almost automatically anyway. Look at the classic situation of a violin sonata (violin and piano) The modern concert piano is the 12TET instrument par excellence. Playing solo, a violinist will gravitate to Just intervals. But with piano accompaniment, they’ll need to compromise toward Equal Temperament in order to blend and minimize dissonance.
The same with symphony orchestras. 12TET rules as a general standard. Yet strings and winds are capable of retuning themselves on the fly, and they do exactly that throughout, depending on the harmonic context, for maximum sweetness and richness of sustained chords, cadences, etc. As listeners, we’re all so used to the “squishiness” of pitches that we don’t even notice if it’s done well.
As I said before, characteristic dissonance can be normal in any folk music. Good session players are constantly listening to each other and adjusting when it matters. Even fretted strings can sharpen their pitches. In practice, good intonation is never as mechanical as it looks on paper. 
Thanks MTGuru, I know so little about music theory (ok, and practice ) that sometimes I need a second slightly different explanation of a point already covered. You have just done that nicely.
I can understand a fretless string adjusting to 12TET, but a fretted one? But perhaps this is the wrong forum for that question.
it is difficult to find a good fretted whistle, innit?
didn’t MT say adjust up when he mentioned fretted?
Well, fretted strings used in ITM are always fretted for 12TET*. So they can be considered fixed pitch ET instruments for all intents and purposes. Their problem, if any, is adjusting in the opposite direction, toward JI. It’s easy to pull pitches high by finger pressure or pushing strings sideways. But not to flatten pitches - unless you’re Tommy Emmanuel, and you grab the headstock and bend the whole neck forward (quite spectacular to see, and not recommended!).
- Even Buzz Feiten fretting is 12TET, and just compensates for the inharmonicity of string thickness and tension.
In theory, it’s perfectly possible to fret a guitar etc. for JI using split frets. But the trade off in limiting your ability to modulate keys, and having to deal with chord dissonances would hardly be worth it.
Most guitarists nowadays just tune to a 12TET tuner and leave it at that. But you can also pick, say, an open D string as your root pitch and tune the other strings to pure intervals from there. This is basically what tuning by harmonics does. The result is a kind of hybrid intonation - JI across the strings, and 12TET up the neck. In practice, you might then tweak the tuning for the best chords in the key you’re playing in.
I don’t know about dissonance, but a piper and a banjo player in the same room may cause rioting. 
Sorry, couldn’t resist.
Doc
But what happens when, as at our session last night, the piper and banjo player are the same person? No wonder he looked so … conflicted.
Here’s one example: http://users.rcn.com/dante.interport/justguitar.html
Yes of course if a banjo is tuned normally (to ET) and the uilleann chanter has its B tuned to the strict -16 JI interval it’s going to sound pretty awful every time a B occurs. And unlike a violinist, a piper can’t make his B 16 cents sharper on the fly by good musicianship. (Pushing harder on the bag to fix a flat note works with some notes but not with others… some notes actually go flatter if you blow harder.)
So a piper who is playing regularly with ET instruments can either 1) tune his chanter more or less to ET or 2) play out of tune with his mates.
Listen to any good uilleann player on an album or whatever and you’ll hear their B spot-on in tune to ET when they’re playing in an ET context.
The reason I’m focussing on B (the 6th) is because it’s the most deviant JI>ET.
The 4th and 5th are nearly identical.
The 3rd is the next-worst and I have the F# on my uilleann chanter at the ET interval.
The 2nd, E, is an odd one on the uilleann chanter because it’s nearly always sharp in the low register but flat in the high register. It’s the bogey note of the pipes.
To hear what an ET uilleann chanter sounds like playing in an ET context here’s me playing with a Pipe Organ…
Thanks MTGuru and PancelticPiper, I am starting to get out of my depth now, but I have learnt a little more. Maybe next time this topic comes up I will absorb a little more. There seems to be no end to the amount of musical knowledge to learn, but I guess I should concentrate on one small corner for now.
The guitar was weird… The pipe playing was nice - I am not normally drawn to pipes (except the gurt big ones played solo outside on a mountain) but perhaps my tastes are changeing as I learn more about music.
I might have to get the dots and try that tune now. (Is no tune safe from my mauling?)
A thread that got a bit more into the “whats and whys” of it: Just Intonation, Equal Temperment, and Irish Flutes
–James
Boy these threads can grow fast if take a day off!
A couple of observations about ‘ET’ instruments:
I knew a young woman at music school who shared duties tuning the harpsichord for concerts - it always sounded better (at least to my ears) when she tuned it (and retuned it at intermission), I asked her why one day, she tuned it to sound good in the key of the ‘main’ piece on that section of the concert - in other words she at least shaded it towards just tuning.
I’ve known guitarists (I’ll admit I don’t know much about tuning guitars) who after ‘tuning’ retune for a particular piece based on the chords they feel are most important - so tuning away from ET to make the chords they are about to play sound better and as close to JI without creating bad dissonances.
edit - this is getting too long - I’ll make a second post on my thoughts about whistles
Some thoughts on whistles:
Tuning is very situational - I find that I find a particular note’s tuning becomes most noticeable to me as a player when I play a tune which keeps going back that note. What might be easy enough to correct as it goes by quickly becomes a pain when repeated or perhaps my ear can deal with it being off briefly but if it keeps coming back it starts to grate.
I find it more satisfactory to blow notes up in pitch than to back off them to get them to come down, so I feel it is better to have a whistle in JI which one can blow into ET as needed. So the whistles I have made have been mostly JI - though I aim for A=440 and so D is 2 cents flat (in reality 2 cents is kind of splitting hairs) and G is 4 cents flat compared to an ET G which to me is definitely noticeable but not hard to raise as needed.
The hard note to figure out what to do with is the E - if you go with making it sound like a fifth from the A (I’m assuming in tune octaves !?!?) it has to be 2 cents sharp to ET A=440, if you go with a sixth from G that would be -18 cents compared to ET A=440, you could work something with E dorian which if you make E-B a perfect 5th would again be -18 cents. I’ve tried various things and am still not sure which I like best, except that ET does not sound right to me and -18 cents is way too flat to sound right in with or after an A - I’ve been trying something in between so the player ( just me so far) can go up or down as needed.
Of course if the E octaves are not in tune with each other one can play games with that too.
This can also be expanded into the whole oxx ooo C thing too - but I’m not going to open that can right now. If you have both a D and a Bb generation check out B, C, C# on both whistles (with oxx ooo C) what do you notice?
And to maintain a sense of perspective try playing a whistle into a tuner, and also record the whistle and play that into a tuner (so that you don’t have the feedback) - sometimes I think it amazing anything sounds in tune.
One more thought - maybe two…
Some are much more sensitive to tuning - what sounds fine to many people will send some of us screaming from the room (well usually my upbringing stops me from being so rude but …). I like to think that the many would notice if they heard the same music played in tune but I’m not sure.
An observation: a harp tuned by ear to JI rings and sounds more beautiful, more harmonious that when tuned in ET, this is not just when a chord is played, a plucked string starts other strings vibrating and when the harp is well tuned a single string sounds noticeably fuller (and better to my ears). Note I’m not a harp player (yet) - just made a couple and working on a couple more.
With apologies to those who just want to play the whistle, and believe me I want to get back there too…
Thanks James for that reference. Now that I have read more I am either more confused or have discovered something a little unsettling. I hope it is the former. Let me explain.
Octaves are very important harmonies. An octave interval represents a frequency doubling, so if we start at a frequency and go up an octave we get double the frequency. If we go up another octave we double that new frequency.
so the (relative) frequencies go 1, (2), (22), (22*2), …
ie 1, 2, 4, 8, 16 etc. These are all whole numbers.
Perfect fifths are important harmonies. A perfect fifth interval represents an increase by half of the frequency. So to go up a perfect fifth the frequencies are 1, 3/2. To go up another perfect fifth we multiply the 3/2 by another 3/2 and get 9/4.
So the relative frequencies go 1, 3/2, (3/2 * 3/2), (3/2 * 3/2 * 3/2),…
ie 1, 3/2, 9/4, 27/8, 81/16
Since these fractions will never be reducable none of them are whole numbers.
So if the octave ratios are all whole numbers and none of the perfect fifth ratios are whole numbers then there will never be a count of fifths that exactly matches a (different)count of octaves.
So the ‘cycle of fifths’, where a number of fifths matches (another) number of octaves, is based upon a mathematically provable falsehood.
The discrepency between the different tunings (eg equal and Just) is caused because both are inaccurate approximations to some mathematically ‘correct’ musical system.
Yes, I know there are good reasons for the approximations and juggling, (e.g. transpositions). But is my reasoning/understanding essentially correct?
Yes, it is! In fact, you’ve hit a key point squarely on the nose. Namely, that Just fifths and Just octaves are incommensurate (they don’t match up).
The discrepancy is called a comma. And one of the tricks of temperament is to hide or spread the comma among the other scale notes without detuning them overly much, in order to preserve the perceived consonance of the important intervals (3rds, 5ths, etc.) in the keys you’re playing in.
The science, mathematics, and magic of constructing the many, various temperament schemes is a black art indeed. There be dragons there along the road to madness …
I know enough about this stuff to understand the basic math and principles involved. Beyond that, the details are serious overkill for the performing musician who just wants to tootle some chunes.
Thank you MTGuru. Now I know where the discrepancy lies I can ignore it…
…and get back to tootlin’, which is where I should be.
(key point indeed! 
)