Equal vrs. Just tuning

[Carey]

This is a just intonated scale with exception of the E, which is a compromise between the -18 cents flat 'minor Second' and +4 cents sharp 'major Second', and will allow to play both.

[bunch clipped out here]

The result is a very sweetly intoned whistle, especially lovely to play in minor modes.

Hmm... this is starting to shed some light on the key/mode issue, the part about the E especially. But then says I could play a minor mode on a JI whistle. Which again makes me think I don't have to re-tune the whistle to move between modes. Can it be said that changing modes "fer sher" requires different freq for some notes? I'm comfortable with changing keys requires different freq for some notes. Maybe someone will say changing modes IS changing keys, and E Dorian is really in the key of E not D.

[benhall.1]

Yep

[Carey]

Ok, now I get it.

[hans]

E Dorian is a mode in the key of D only if you assume equal temperament with (logarithmically) equal spaced semi-tones. In ET you can say that the tones in a mode are derived from a progression of single and double semi-tone intervals, starting from the tonic of the mode:
Ionian major: 2-2-1-2-2-2-1
Dorian minor: 2-1-2-2-2-1-2
Lydian minor: 1-2-2-2-1-2-2
Mixolydian major: 2-2-1-2-2-1-2
Aeolian minor: 2-1-2-2-1-2-2

But if the notes of those modes should be in just intonation, then simple ratios for their frequency intervals are used, and the progression of equal semi-tones goes out the window. you could just keep it for reference to show approximately the interval. But the ET model is not valid, and need to be replaced with a series of frequency ratios, in order to construct a correct just intoned mode from a reference base tone.

I really want a fretless whistle now!

[benhall.1]

Ok, now I get it.

It was only a matter of time ...

[Steve Bliven]

I really want a fretless whistle now!

Would that be a slide whistle? Like this http://www.youtube.com/watch?v=3Aa_0OnEwXM ? Or maybe a kazoo?

But back to the original question - some 90 responses later - can we really play whistles into tune with breath control or do they need to be temperament-specific for various modes (or keys or whatever they are really)? Or do we just play out of tune and "it's traditional, laddie"?

Is this some sort of plot by the whistle-makers to foster another massive bout of Whistle Acquisition Frenzy (WAF)? A whistle for each key/mode combination? One in brass and one with a plastic top?

Thanks to all for your erudite and informed input.

Best wishes.

Steve

[peeplj]

You can do a lot with breath control.

Also, at the speeds at which most Irish traditional dance tunes are played, a note being a few cents off this way or that will never have time to register to the ear.

Just my opinion...

--James

[Carey]

Thanks for being patient with me Hans. I might learn slow, but I never forget. (Ask my wife, we went to college together.) It feels to me that there is some small issue, perhaps a definition of a fundamental word like "key" or "scale" that is tripping me up and not allowing me to internalize what you are meaning. It's quite possible that I've latched on to the meaning of a term in the context of ET and by using it to talk about JI I'm mixing apples and oranges. (and by the way peep, I agree completely with the practical aspects of actual playing giving wide latitude, but sticking with theory for the moment...)

Let me try to say it this way, and if you could reply using the concepts and notation I've used that would help me greatly. Or perhaps point out where I've missed the boat in my constructs below.

Using my apparently flawed mental model, and assuming D Just for purposes of discussion, there would be certain ratios between the frequencies of the notes of the D Ionian scale. Let's say they are as follows:

8-tone scale: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1

And let's call them Do, Re, Me, Fa, So, La, Ti, and Do'.

Once I make a corresponding whistle, could I not play E Dorian Mode in the Key of D on this whistle by playing Re, Me, Fa, So, La, Ti, Do' and Re' and not needing to change any note's tuning? The ratios for that approach would be based on D and would be:

8-tone Dorian scale: 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, (and whatever comes out when I over-blow the 9/8 ratio)

If you say "no silly, you've spotted the problem with 'whatever comes out'", I would then have to take Re (or E) as the root and base frequency, and go back to the 12 tone scale ratios I pulled those 8 from notes, which is:

12-tone scale: 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 16/9, 15/8, 2/1

and form a new sequence of ratios for E Dorian, which based on 2-1-2-2-2-1-2 semi-tones would seem to be:

8-tone dorian scale: 1/1, 9/8, 6/5, 4/3, 3/2, 5/3, 16/9, 2/1

But that would mean that changing modes is indeed changing the root frequency, and in JI at least, E Dorian is not a mode of the key of D, but it's own unique (what? key?.) So I'd have to have a gazillion (maybe not quite that many) whistles in my roll to play in any mode of any key, (lipping ignored.)

Or it might mean (more likely) I've still got it wrong.

(edit to add: Or I should just work harder on the fiddle and just play what sounds good.)

[hans]

Carey - i think either approach is wrong
You cannot just use a different starting point in the sequences of intervals you mention to arrive at the 'right' intervals for the Dorian mode (and i write 'right', because there may not be just one right set of interval ratios, but i plea ignorance for the moment). I can't even give you a set of ratios for the Dorian mode, my head is spinning a bit from dividing ratios by ratios to get new ones.

Still, one main point was mentioned earlier:
On a D whistle the Dorian starting on E should have a perfect fifth B (3/2), but the B being the sixth on the D Ionian mode is 5/3, (and ca -16 cents flat from ET B), and if the E is tuned 9/8 of the D (root tone of the whistle), then we have an interval from B to E which is (5/3) / (9/8) = 40/27, which works out 22 cents flat from the JI 3/2 fifth, very bad! Now we can mend this by flattening the starting E, using not 9/8 over D but 10/9, a tone which is -18 cents flat from ET E, instead of +4 cents sharp as the 9/8 interval (there you got the 22 cents difference). Setting the E at 10/9 sorts out the fifth, and helps also all the other notes in the Dorian mode. With that shift in the E tuning the other notes are fairly okay (and i can't show this in theory yet, i just trust my ear on this).

The E on a D scale is really the problem note for JI, from a whistle maker's view, as we would really like two quite different E's, a 'major' and a 'minor' one. Therefor my suggestion to try a compromise for E on a JI whistle, by making it ca 8 cents flat from ET E. That should help to play both E's right with a bit of adjustment. If the E is tuned +4 cents sharp of ET, it would be really difficult to play in Dorian and other minor modes.

And i am most grateful for this insight to highwood who pointed this out on page 3 of this thread! A new whistle was born from that!

[david_h]

I only noticed the E dorian 'problem' on slow tunes. On flute I find it OK because there is that bit more pitch flexibility (and I have other things to worry about...). This discussion having firmed up my mind that E is the culprit has set me experimenting on whistle. I suspect that shading the hole to bring the E down a bit may be possible on slow tunes. Don't some of the experts shade holes to permit blowing a bit harder for volume without going sharp ?

[highwood]

Back to the L1 hole - some of what has been said here does not quite line up with my experience and so I feel compelled to add a little.

I would suggest to anyone making a whistle that they should experiment a little to find their truth.

The C# is tuned with the position and size of the L1 hole (for a chosen C# L1 also depends on pitch chosen for the B and bore perturbations also effect it as well) The combination of hole size and position chosen will effect the octave D (oxx xxx compared to xxx xxx) and will effect Cnat oxx ooo (or whatever fingering you aim for) How exactly you ask ... I am still playing with the possibilities, and I don't know the answer - in fact I suspect there is not one answer because it is a matter of trade offs. Say you choose a hole size because of the sound and discover that Cnat oxx xxx is unusable - you have to decide whether to try a different hole size (and position) or learn to half hole. In practice in the choice of where to drill L1 you will have to consider pitches of B and C#, sound of C#, how easy the second octave C# is to get tuning of Cnat, octave D oxx ooo v. xxx xxx and even size of all the holes if you want the whistle to sound consistent. It is a simple instrument but its design quickly becomes complicated if you aim high, partly because it is simple and the player is more restricted with what they can do while playing.

now back to the discussion on what the mode E dorian notes are pitched to ...
I love math and theory but in the end we are talking about music and what it sounds like is what matters, and tastes are different, and you can learn to love new things, and while there is no right answer there are wrong ones!

Bill

[Carey]

Carey - i think either approach is wrong
You cannot just use a different starting point in the sequences of intervals you mention to arrive at the 'right' intervals for the Dorian mode (and i write 'right', because there may not be just one right set of interval ratios, but i plea ignorance for the moment). I can't even give you a set of ratios for the Dorian mode, my head is spinning a bit from dividing ratios by ratios to get new ones...

Whew! It's a relief to hear you say that. I was beginning to feel like the only one with a spinning head. (I might have you on RPM tho. )

I get the "feeling" of JI, and have made a JI whistle with the help of a piper who like to play the whistle over drones. It took two or three iterations of me moving holes around but in the end he was pleased, and that's what I'll ship a person if they were to order a JI D whistle from me.

But being the engineering type, I like to (as much as possible) understand what it is that I am doing when I do things. So I was/am trying to combine my understanding of ET modes with my understanding of JI tuning. It would have been handy if I could simply substitute the JI tunings for ET tunings in my perspective on ET modes and there you go. But it doesn't seem to be that straightforward.

I think I dig out one of the JI D whistles I have around here and see what happens when I play the various modes over a D drone. I wish I had a piper handy to give me a D drone so it would have all the complexity of a piped D and not the electronic D I'm going to use.

Any piper's out there like to record a D drone for me to play against? I've found a Bb drone on the web:

http://www.freesound.org/samplesViewSingle.php?id=32878

But so far no D.

[Mitch]

a Bach peice on a just tempered keyboard compared with an equal tempered keyboard

There can be no such thing as a "just tempered" keyboard. It's a confusion, a contradiction, in terms.

"Just" is from "just intonation", a scale based on the harmonic series.

"Tempered" is by definition a departure from "just".
.
...

Ah well - many years ago, before I cared about this subject, I lived nearby to a Dutch fellow who had this wierd contraption he called an "Archefoon"(sp?). It looked like the progeny of an insane typewriter crossed with a hammond organ - I suppose there were about 8 rows of keys. The lower 2 rows could execute an equal tempered scale, the ascending rows had adjustments. He could play any Bach composition in any temper described on his contraption - the equal temper (as found on the parent Hammond) sounded OK - but the tempered version sounded divine. He only had to go up to the 3rd row to achieve this.

The sweetest whistle I ever tried was tuned to the 22 note Indic chroma. That is worth consideration.

And for those who insist on misbeleiving their own experience in the face of numerically supported disproofs of their senses, I would say this: numbers are full of singularities - they cannot always be trusted - certainly a useful digression, but the main game is life - I have come to respect mine - it helps with prioritisng what you hear

And hey - a whistle gets tuned (if the maker bothers) to a "sweet point". That "sweet-point" might be a linear ascending breath pressure when playing a scale .. but then it might get adjusted for the tunes that are likely to get played on it .. Remember - these are whistles, not R******rs - we are not trying to create a whistly version of a piano To my thinking, a whistle should be able to span much more - but it will depend (in more ways than one .. much, much more) on the player's effort and musical subjectivity.

(Edited to say: the thing about subjectivity: Legend has it that many of the great "punk" bands never had a music lesson in their lives - maybe some of that was true, and yet, they were world beaters in thier time. For those who consider themselves "n00bs" at the whistle - you have been inspired to play - that inspiration is valid - it is timely and it does not matter one jot whether your performance is traditionally, historically or mathematically correct in the eyes of those who will not be your audience - play it anyway - the world of the on-going present requires original expression )

[benhall.1]

I have to say, I reckon this has been the most fascinating thread I've seen on any music forum. Ever. Fantastic.

Meanwhile, I promised to get back to david_h on one point:

I read somewhere on one of the articles on JI that in the past (medieval times ?) string instruments where sometimes tuned to the just scale of the lowest note of the tune, not neccessarily the 'home note' (is that plagal modes ? Ben ?). So that would be like most irish tunes in G being played over a D drone or in a D scale.

A question I keep asking and maybe those gathered above can answer - do pipers play E dorian tunes over the D drone (or do they stop the drones for those tunes). If so does that have an effect on what irish E dorian tunes are 'supposed to sound like'.

I think you're confusing two concepts, David. I don't know your source reference, but I suspect it said that medieval stringed instruments were tuned to the lowest note of the scale, rather than of the "tune". So, in a tune which was in a plagal form of myolydian, for instance on, where the equivalent myxolydian would be on G, the tuning would indeed be JI based on D and within the confines of that mode, whereas if the tune were in the authentic mode of G mixolydian, the tuning would be JI based on G. But, I have to say, my own head is starting to spin now, as I'm struggling to see what the difference between those two would be. I think there is one. For instance, I think (cross fingers here ) that, because in the plagal form of the mode there would be more of an emphasis on the C (as opposed to the D in the authentic mode), that note would tend to be flatter in the plagal form of the mdoe than in the authentic. There may be others ...

I think I can smell brains frying ... Mine.

[hans]

I think i figured out now how these just "modes" work. Note i am using modes in hyphens, since i found that related notes in just "modes" have their distinct different frequencies. It may be better simply to call them just scales, like ladders with seven rungs per octave and each rung with a distinct frequency. Anyway, i will try to explain, including a bit of math:

The just ionian major scale:
This consists of the intervals
1. just major second: 9/8
2. just major third: 5/4
3. perfect fourth: 4/3
4. perfect fifth: 3/2
5. just major sixth:5/3
6. just major seventh: 15/8
7. octave: 2/1

Each ratio is the frequency ratio of the note to the scale root note.
To see the frequency ratios of adjacent notes we divide the higher note ratio by the lower note ratio. I put that into a table: first the note "rung" as roman numeral, then the ratio to the root, then in the gaps the ratios between adjacent notes, and lastly underneath each note's derivation from the corresponding closest ET note, in cents:

Code: Select all

I     II     III     IV    V      VI    VII      VIII
1/1   9/8    5/4     4/3   3/2    5/3   15/8     2/1
   9/8   10/9   16/15   9/8   10/9   9/8    16/15
0     +4     -14     -2    +2     -16   -12      0  

Now you can see that there are only three ratios for adjacent notes:
9/8 is the just major tone, 10/9 is the just minor tone, and 16/15 is the just major semitone. So that correspond somewhat to the ET scheme of 2-2-1-2-2-2-1.
If you multiply all the adjacent note intervals up to a note, you get that note's ratio to the root.

Now let's do the same for
the just dorian minor scale:
The difference to the major scale is that the third and the seventh are minor.
The just frequency ratios for a minor third is 6/5, and for a minor seventh 16/9. These are the only differences in note to root ratios. So we get a table like this for just dorian:

Code: Select all

I     II      III    IV    V      VI      VII    VIII
1/1   9/8     6/5    4/3   3/2    5/3     16/9   2/1
   9/8   16/15   10/9   9/8   10/9   16/15    9/8
0     +4      +16    -2    +2     -16     -4     0  

Again the adjacent interval ratios are 9/8, 10/9 and 16/15, and we got the correspondence to the ET scheme 2-1-2-2-2-1-2. The differences are in the third and seventh: a more in-tune with ET seventh, and a very sharp to ET third.

This last one surprised me, and i wonder if i ever played in just dorian on a whistle, with such a sharp (to ET attuned ears) third, since it corresponds to the major fourth, which is much like the ET fourth. On a D whistle it would be a sharp G. The second is also a lot sharper than the corresponding just major third (+4 cents rather than the flat third -14 cents). I can blow those notes sharp and get the just intoned feeling, like a wailing... So there is yet another "anomality" (from ET's view or ear).

I don't see how such "modes" can be combined on a whistle. There will be a need for compromises, and some may be bad, from a just intonation POV. Still, i think it is fun to try to get out of the ET interval straightjacket, even in theorising. In playing music it is done anyway, hopefully. And in light of JI a good whistle design seems to me one which allows lots of flexibility to adjust notes.

~Hans