Equal vrs. Just tuning

[david_h]

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.

Thanks Ben, I think the reference said what you are saying and I was mainly confused over the terminology. It had me wondring if G major irish tunes were like as in this quote from wikipedia but with the melodic range going a bit higher.

the plagal version of the Ionian mode, called Hypoionian (under Ionian), based on the same relative scale, but with the major third as its tenor, and having a melodic range from a perfect fourth below the tonic, to a perfect fifth above it.

(not supposed to be behind a computer at the moment so will try to find that reference later and get back to this discussion.

[hans]

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

Thanks Mitch! That lead me to look at some nice comprehensive pages on:
The Classical Indian Just Intonation Tuning System explaining a 22 tones per octave system.
Then i realised that my previous posted just scales for ionian major and dorian minor could be done with some different intervals, with the result that the dorian scale fits perfectly on top of the ionian, starting on the second.

Code: Select all

Ionian major
I     II     III      IV      V     VI       VII      VIII
1/1   9/8    5/4      4/3     3/2   27/16    15/8     2/1
   9/8   10/9   16/15    9/8     9/8     10/9    16/15
0     +4     -14      -2      +2    +6       -12      0   

      Dorian minor
      I      II       III     IV    V        VI       VII    VIII
      1/1    10/9     32/27   4/3   3/2      5/3      16/9   2/1
         10/9    16/15     9/8   9/8   10/9     16/15     9/8
      0      -18      -6      -2    +2       -16      -4      0  

You can see that the intervals are simply shifted. Each value for the Dorian minor can be derived by dividing by 9/8, or by subtracting 4 cents from the Ionian values.

Also quite apparent is the symmetry of tetra-chord notes, the lower tetra-chord notes I II III IV and the higher tetra-chord notes V VI VII VIII. So this should work nicely, possibly also in the other modes.

A whistle tuned to 0 +4 -14 -2 +2 +6 -12 (from foot end upwards) should deliver Ionian major, and the related Dorian minor by blowing 4 cents softer. Well I better try this out! But this is a known Indian scale.

Just intonation - Non-western tuning

In Indian music, the basic unaltered diatonic scale is considered to be 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1. This would appear problematic, since (27/16):(5/4) = 27:20 (a wolf interval), not 4:3. But Indian music uses melodies over a drone dyad (usually 1/1 and 3/2), so these two pitches (27/16 and 5/4) would seldom be heard sounding together. See sargam and swara.

[The just scale with the ratios 1/1, 9/8, 5/4, 4/3, 3/2, *5/3*, 15/8, 2/1 gives (5/3):(5/4) = 4:3 (a perfect fourth), and allows these notes to sound together in a consonant fashion, but then introduces another problem in that (5/3):(9/8) = 40:27 (a wolf interval), not 3:2. These issues prevented strict just intonation from becoming prevalent in the West, but it thrives in India, where they are largely irrelevant.]

Fromhere

[david_h]

But what would Pythagarous* with his simple ratios think of that 27/16 VI ? The nice neat 5/3 is 21 cents different (I found my spreadsheet )

* if he's watching I guess he's seen all this a million times

[hans]

But what would Pythagarous* with his simple ratios think of that 27/16 VI ? The nice neat 5/3 is 21 cents different (I found my spreadsheet )

* if he's watching I guess he's seen all this a million times

27/16 is called a Pythagorean major sixth. It is classed as 3-limit just intonation, or Pythagorean, meaning there is no higher prime factor than 3 in it [27/16 = 3*3*3/(2*2*2*2)]. So he may be pleased seeing such a simple ratio of powers of the two smallest primes, signifying good harmonic relationship.

See http://en.wikipedia.org/wiki/List_of_musical_intervals.
When you come to the list you can order it to show the just intervals first, by clicking the column with the 5 twice! If you click the 3 you can see the Pythagorean intervals first (in yellow).

[MTGuru]

To see the frequency ratios of adjacent notes we divide the higher note ratio by the lower note ratio.

Yes, it does seem odd to be dividing ratios. But that's the way to do it.

plagal ... authentic ... 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.

Be careful ... The aroma of frying brains will attract zombies.

Yes, I'm not sure how useful the original concepts of plagal and authentic are now, in the context of intonation or otherwise, since we no longer think in terms of tetrachords. Where I do still find them practically handy is in gauging the melodic contour of a tune.

For example, to me the A part of the Kesh Jig (G-authentic, spans G to g) and the A part of Tell Her I Am (G-plagal, spans D to e) have a slightly different finger feel. A high or low climax on the tonic feels different from on a mid-scale note. And if you're matching whistles in different keys to tunes, thinking plagal or authentic helps decide whether you want to put the tune tonic on the right hand or the left.

[david_h]

I stand corrected hans, I picked the wrong ancient worthy. Good list that. However, that is still as far away from the sixth note in a just scale as the E in D just is from where we might want an E to play E dorian.

And I am still not convinced that stacking up the ratios like that so you would be way up a harmonics series is not mathematics struggling for a solution (back to my grumbling about the circle of fifths earlier). When playing over a drone or trying to play one end of an interval those small number ratios are fairly easy to pick up. My untrained ears start struggling to find a match by the time it gets to the second and especially the seventh - too may ways in which those higher harmonics can be almost lined up I guess. I suspect by that point I would be trying to find a pure interval from a previous melody note or in an implied chord (that came up in a previous discussion like this, I will try to find it so I can acknowledge who gave a detailed comment).

Equal temperament seems to have good survival value as a fix.

Plagal - authentic. From memory (am still supposed to be doing something else) Cecil Sharp in his Folk Song in England said it was not important but his editor later added a foot note to say that other people thought it was. When I came to whistle as teenager with zero musical training and not a text book in site it did seem to me that that was the main difference between tunes in G and tunes in D - the ones in G almost always went down to D and if you tried to play them in D your ran out of notes. Folk singers don't have that problem, what with them being chromatic and all.

[hans]

To see the frequency ratios of adjacent notes we divide the higher note ratio by the lower note ratio.

Yes, it does seem odd to be dividing ratios. But that's the way to do it.

Since all what matters here are intervals as determined by frequency ratios it is actually quite natural to divide and multiply ratios. But if we use the cent values we can subtract and add these instead, since the cent scale is logarithmic. That should be even odder. Would be nice to make a musical slide ruler....

[david_h]

Would be nice to make a musical slide ruler....

Before computers crystallographers used to do something like that when working with X-ray diffraction data. They used an ordinary slide rule and pencilled on to it something very like harmonic series then slid it backwards and forwards 'till they lined up. I think it would work hans.

[MTGuru]

Folk singers don't have that problem, what with them being chromatic and all.

Except that when you only have a limited vocal range of an octave and a fifth or so like me, figuring out how to "center" a given melody in your good range is actually pretty important if you want to avoid imitating either a croaking frog or a strangled chicken.

[hans]

However, that is still as far away from the sixth note in a just scale as the E in D just is from where we might want an E to play E dorian.

I don't understand what you mean.
I am quite pleased that I showed to be wrong in most things i posted earlier in this thread, that modes seem to work in just intonation, also in the theory.
The major just scale consisting of intervals 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1 does not seem to pose problems on the whistle, works well with drones, has good fifths even in the Dorian minor mode, and seems to go well with ITM. Or what am I missing?

[david_h]

The major just scale consisting of intervals 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1 does not seem to pose problems on the whistle, works well with drones, has good fifths even in the Dorian minor mode, and seems to go well with ITM. Or what am I missing?

It may be a case of terminology but I thought a just major sixth was 5/3 of the root (as in that Wikipedia table you linked) not 27/16, and those are 21 cents apart. They may both 'sound good', but they sound different.

Where I am coming from on all this is that my intonation on flute still leaves a lot to be desired but I now have a better grip on it than on whistle. If I play along with a slow tune on a recording of, say, a piano accordian with my eyes shut then open them and look at the tuner the needle is usually within a few cents of the ET note. If I do that with a solo fiddle recording the needle is often 10 or 15 cents off ET, and major thirds near to Just are very common and repeatable. If its a recording of a respected fiddler I assume that it is all deliberate. If I play a slow tune solo I often look at the tuner needle and think 'Thats odd, whats it doing there ? Try again, similar result. Whats going on ? Does this sound awful to other people ? Am I copying something I have heard ? Have I drifted sharp or flat and this note just makes it a bit more obvious ? Am I doing what I became accustomed to on that out of tune whistle I had for years? Should I just turn the tuner off and do what Mitch suggested above ?

[david_h]

Except that when you only have a limited vocal range of an octave and a fifth or so like me, figuring out how to "center" a given melody in your good range is actually pretty important ...

Yes, sorry my mind had wandered slightly off topic. What I was thinking was that the singer can do all those modes and move the melodic range (in scale terms) around within the same pitch range, and also do things like lower sevenths as on GHP or r*c*rder.

[hans]

The major just scale consisting of intervals 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1 does not seem to pose problems on the whistle, works well with drones, has good fifths even in the Dorian minor mode, and seems to go well with ITM. Or what am I missing?

It may be a case of terminology but I thought a just major sixth was 5/3 of the root (as in that Wikipedia table you linked) not 27/16, and those are 21 cents apart. They may both 'sound good', but they sound different.

I meant a major sixth of 27/16, a Pythagorean major sixth, not a just major sixth of 5/3. The 27/16 forms a pure 3/2 fifth to the 9/8 second. It also sounds better against a D drone than the 5/3 sixth.

... Should I just turn the tuner off and do what Mitch suggested above ?

Perhaps a good idea For exploring your own just intonation it can be helpful to play along to a drone, on a D whistle a D or G drone. You can hear subharmonics appearing when playing some intervals to the drone note: a lower note than the drone or the note you play, which is generated by both waveforms mixing.

Otherwise you will adjust to whatever instrument you play along with: you play just intoned with a fiddler, and equal tempered with a concertina for instance.

[MTGuru]

<-- I say, Hans ... You've evolved!

[hans]

I say, Hans ... You've evolved!

Yep. Its quite a shock. Too much math they say. I may be able to revert if I keep chewing on this stick.