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[Owen Woods]

I wrote a blog post the other day trying to explain temperaments. I'd appreciate any feedback and if you don't know anything about it then it might be worth a look.

http://ukebert.wordpress.com/2012/12/16/an-illustration-of-the-non-existence-of-god/

[Theo]

Congratulations Owen for a lucid explanation of the tuning problem. 

[Steve_freereeder]

Congratulations Owen for a lucid explanation of the tuning problem.

Hear, hear! It is probably the clearest and most succinct account I have come across. I have added it to my bookmarks. Well done!

[pikey]

Excellent stuff.

[brianread]

Ollie - that is very well put, highly understandable. 

My mathematical instincts tell me that the "answer" lies in the properties of "irrational" and/or perhaps "transcendental"  numbers - perhaps pi or e.  Which would lead to the errors due to approximations, I guess we stray into physics and biology at this point, linked both to the properties of vibrating strings and also the makeup of the human ear and brain!

Got any thoughts?

[Theo]

I have even heard of anglo concertinas being tuned to quarter comma, which would be a fascinating experiment.

It was quite common for anglos to be tuned to a version of quarter comma meantone.  I've been privileged to renovate several Jeffries that were tuned like this.  Some players who understand the difference are very keen to get them in original tuning.  Makes a lot of sense for an instrument that is unlikely to play in all keys, and does give you rich smooth chords.  And there is some big attractions on tuning a melodeon the same way.  There is only one wolf interval, so its pretty easy to "hide" it where it will never be heard. The more significant downside is that as you go further from the home key you gradually get more sharp or flat wrt A=440, but for solo playing I think there are no drawbacks.

[Anahata]

My mathematical instincts tell me that the "answer" lies in the properties of "irrational" and/or perhaps "transcendental"  numbers - perhaps pi or e.  Which would lead to the errors due to approximations

Equal temperament semitones are ratios of 12th root of 2. That's an irrational number but not transcendental, for what it's worth.

The real problem is that we take a perfectly good pure harmonic scale and then pretend we can use the same notes to play in other keys. We can't, but we are tempted to think we can because the semitone intervals are so nearly equal. Forcing them to be equal is terribly convenient for making certain kinds of musical instruments with fixed tuning able to play in all 12 keys, but it's a completely artificial short cut and it's unmusical.

I have to add my thanks to Owen for his blog article, from which I learned a lot. I knew about the circle of fifths and equal temperament, but while I'd heard of the other "compromise" tunings, I didn't know exactly how they were constructed.

[mory]

Chaos and Order 

[pikey]

Pablo Casals, the famous cellist, allegedly used even tempering when playing an upward scale, but well tempering when playing a downward scale!

I can't vouch for it, all I know is that his playing tugs at the heart strings like no-one elses does 

[JohnAndy]

Interesting stuff.

A few random thoughts...

Firstly, I'm not sure that your average piano is actually tuned to equal temperament. The piano tuners who have tuned our piano seem to work by ear and seem to spend a fair amount of time tuning the fifths, if I remember right. I'm not sure what the resulting scale from this process might be, or even how consistent it is between different tuners - is there an agreed standard that they are all working towards or is it all really a bit random?

Then, I remember hearing a radio program about 20 or 30 years ago discussing the tuning the Bach would have used for his Well Tempered Klavier, which is a series of pieces ranging through all the possible keys. The argument of the program was that Bach was promoting a new temparament which avoided some of the worst problems of tuning in the keys round the back end of the cycle of fifths, but made the compromises in a way which still made some keys sound rather jangly and discordant and others more smoothly harmonious. The argument then was that Bach actually used the different characters of the keys to musical effect - so the fact that there is some discordance in certain keys is not necessarily a disadvantage but can in fact be used for expressive effect.

Final comment, as a flute and saxophone player used to playing in various amateur bands and orchestras, the overriding concern is not worrying about what temparament to tune to, but simply to not be horribly out of tune. There's an enormous amound of variability in the pitch of each note on wind instruments depending on lip pressure, air pressure etc., and in the case of non-fretted string instruments the pitch is entirely determined by the player's choice of finger position. Any decent player will continually be listening to the pitch of the notes they are playing and adjusting to fit in with the notes played by other instruments. This flexibility can also be used to bypass the problems of temparament, e.g. you can play the same note with slightly different tunings to fit in with two different chords.

One of the nice things about playing the melodeon is not having to worry about intonation!

[Anahata]

Firstly, I'm not sure that your average piano is actually tuned to equal temperament. The piano tuners who have tuned our piano seem to work by ear and seem to spend a fair amount of time tuning the fifths

They make the fifths slightly out from perfect 3:2 ratio, by ear. It's quite possible that they don't all tune to exactly the same scale. There is also the business of "stretching" the intervals at the low end: the issue there is that the harmonics of the string aren't exact multiples of the fundamental, so there is the question of which of the various overtones from each string to match against the others.

Bach was promoting a new temparament which avoided some of the worst problems of tuning in the keys round the back end of the cycle of fifths, but made the compromises in a way which still made some keys sound rather jangly and discordant and others more smoothly harmonious. The argument then was that Bach actually used the different characters of the keys to musical effect

This seems to be similar to Owen's description of the result of tuning to quarter-comma meantone.

Any decent player will continually be listening to the pitch of the notes they are playing and adjusting to fit in with the notes played by other instruments. This flexibility can also be used to bypass the problems of temparament, e.g. you can play the same note with slightly different tunings to fit in with two different chords.

Yes, and string players in a string quartet will do this, while they have to play in equal temperament if there is a piano in the ensemble.

One of the nice things about playing the melodeon is not having to worry about intonation!

...until you take it in for tuning and your favourite fettler asks whether you want ET, quarter comma meantone, Just or something else....

[Inventor]

1) With regard to mean tone scales, this is a minefield and books have been written about this to get round the compromises involved. Many years ago when setting up the standards for Hayden duet concertinas with Steve Dickenson of Wheatstone's, we decided that the larger instruments should have both G#s and Abs on separate buttons and also D#s and Ebs on separate buttons too, so as to give no less than 8 easy-peasy scales. These repeated sharps and flats are also found on English (system) concertinas, and some were tuned to a mean tone scale. In those days I knew very little about the problems of mean tone scales, and naively suggested that he might offer the same on my duets. To which he replied that no way was he going to get into the problems of mean tone scales.  I am sure that this was the best decision.
2) With regard to Just intonation, I see that you Ukebert have spotted the problem of the supertonic note (i.e. for example the D in the key of C) There is a solution to this and it would be possible to set up a bisonoric free reed instrument of the Melodeon Concertina Bandoneon type where all the Major and Minor chords in several different closely related keys are perfectly in tune. As you found my suggestion in another thread on how to set up a 31 button instrument to play the scales of no less than 5 different keys "interesting", I am sure you will find that only a small modification to this will give 5 different keys in double just tuning even more interesting.
I will show my solution in a reply to your Blog Site shortly.
Inventor.

[Chris Brimley]

I don't know why I've never thought of this question before, but in the case of the inexpensive digital electronic tuners which tell you the note you're playing, what temperament are they normally set to?  And are the frets on fretted instruments usually spaced to Pythagorean, Just, or ET?

[Anahata]

Inexpensive digital tuners would be equal temperament.
Frets are probably best set to ET too, or they wouldn't be straight and would have to move depending on what key you were playing in.

I think there are fancy digital tuners that offer different scales.

Also the much-maligned "autotune" tool (over)used in many recording studios can be set to a choice of scales, including one called "Indian" which apparently works well for unaccompanied folk singers. Allegedly.

[Andy Simpson]

An interesting read. Since I'm going down the road of having several 2 rows rather than trying to play outside the home keys on one or two big boxes I've been thinking about trying something like Meantone...

  And are the frets on fretted instruments usually spaced to Pythagorean, Just, or ET?

There have been some experimental instruments with other temperaments but 99.9% of modern fretted instruments are ET. It's not uncommon for historical fretted instruments like lutes and vihuelas to have them set to any of the panoply of temperaments that have been tried throughout history.

[Stiamh]

I don't know why I've never thought of this question before, but in the case of the inexpensive digital electronic tuners which tell you the note you're playing, what temperament are they normally set to? 

Yes, as Anahata says, ET. I sigh inwardly whenever I see young fiddle players at a session tuning (not just the A string but all the others) with one of those gizmos that clips onto the pegbox. If there's one place in the western musical universe left for a perfect fifth, it's on members of the violin family*. And if after a year or two at the fiddle you can't tune your strings to each other in perfect fifths, IMO you shouldn't be playing one...

* Except of course in specific contexts, as on cellos in string quartets. Apparently the error in the circle of fifths is sufficiently large over a couple of octaves that quartet cello players have to tune their C string sharp...

[Anahata]

If I'm playing my cello with Mary's concertina I have to tune each open string to its note on the concertina. Starting with an A and tuning fifths doesn't work, nor (for some reason) does doing each string with an electronic tuner.

[Owen Woods]

Ollie - that is very well put, highly understandable.

Owen, not Ollie!

My mathematical instincts tell me that the "answer" lies in the properties of "irrational" and/or perhaps "transcendental"  numbers - perhaps pi or e.  Which would lead to the errors due to approximations, I guess we stray into physics and biology at this point, linked both to the properties of vibrating strings and also the makeup of the human ear and brain!

Got any thoughts?

Not sure what you mean by this. The problem is that (3/2)^12 =/= 2^7. This is just a fact of arithmatic.

A more interesting question is why the harmonic series at all. I've always thought of it geometrically, thinking about wavelengths. With that thinking, it make sense that whole ratios are consonant. As for why strings and so on are harmonic (or very nearly), I can do the maths to prove it, but I don't know why. Science doesn't really do why. The real question is whether it could be otherwise in a differently designed universe - I'm not convinced that it can.

I have even heard of anglo concertinas being tuned to quarter comma, which would be a fascinating experiment.

It was quite common for anglos to be tuned to a version of quarter comma meantone.  I've been privileged to renovate several Jeffries that were tuned like this.  Some players who understand the difference are very keen to get them in original tuning.  Makes a lot of sense for an instrument that is unlikely to play in all keys, and does give you rich smooth chords.  And there is some big attractions on tuning a melodeon the same way.  There is only one wolf interval, so its pretty easy to "hide" it where it will never be heard. The more significant downside is that as you go further from the home key you gradually get more sharp or flat wrt A=440, but for solo playing I think there are no drawbacks.

I think that at some point, when I have a bit of cash, I will get my quint boxes retuned to quarter comma. I can't see a disadvantage either. I know that I like to play in funny keys, but if I'm honest, a quint box is best playing in a limited range of keys, if you stray much outside that then you begin to lose what makes a quint box special.

Interesting stuff.

A few random thoughts...

Firstly, I'm not sure that your average piano is actually tuned to equal temperament. The piano tuners who have tuned our piano seem to work by ear and seem to spend a fair amount of time tuning the fifths, if I remember right. I'm not sure what the resulting scale from this process might be, or even how consistent it is between different tuners - is there an agreed standard that they are all working towards or is it all really a bit random?

Piano tuners tune by ear by counting beats, or so I have always been told. You can't count beats in just, as there aren't any! In actuality though I believe that tuners do tend, subconsciously, to tune the common keys a little purer than the uncommon ones. I seem to remember a study about that. But as Anahata said, the overtones of the strings are not quite harmonic due to stretching and boundary conditions and things - strings are not 'ideal'. So in order to make the overtones consonant with other notes, the fundamental is tuned out of tune. So even with fixed pitch instruments, tuning can be tricky.

Then, I remember hearing a radio program about 20 or 30 years ago discussing the tuning the Bach would have used for his Well Tempered Klavier, which is a series of pieces ranging through all the possible keys. The argument of the program was that Bach was promoting a new temparament which avoided some of the worst problems of tuning in the keys round the back end of the cycle of fifths, but made the compromises in a way which still made some keys sound rather jangly and discordant and others more smoothly harmonious. The argument then was that Bach actually used the different characters of the keys to musical effect - so the fact that there is some discordance in certain keys is not necessarily a disadvantage but can in fact be used for expressive effect.

There has been much disagreement over what temperament Bach wanted for the Well Tempered Klavier. It is thought that either he used ET or used an irregular temperament. This means that the pythagorean comma would be distributed amongst some of the fifths, but not all and not always by the same amount. Temperaments like these, known as Well Temperaments, are still used today for organs. It all gets very complicated though and I don't know much about it. But as you say, the idea behind Well Temperaments is that every key is usable, but every key is distinct, with intervals that sound nice and intervals that don't, but without a truly horrible wolf, which you do get in quarter comma.

Final comment, as a flute and saxophone player used to playing in various amateur bands and orchestras, the overriding concern is not worrying about what temparament to tune to, but simply to not be horribly out of tune. There's an enormous amound of variability in the pitch of each note on wind instruments depending on lip pressure, air pressure etc., and in the case of non-fretted string instruments the pitch is entirely determined by the player's choice of finger position. Any decent player will continually be listening to the pitch of the notes they are playing and adjusting to fit in with the notes played by other instruments. This flexibility can also be used to bypass the problems of temparament, e.g. you can play the same note with slightly different tunings to fit in with two different chords.

This is quite true. And it is something which the string quartet excels in. Unfretted stringed instruments have perhaps the finest control over pitch variability of any instrument, so they can use consonance through just chords and dissonance through tempered chords to their advantage, making the music ebb and flow. It makes me wish that I played a stringed instrument.

2) With regard to Just intonation, I see that you Ukebert have spotted the problem of the supertonic note (i.e. for example the D in the key of C) There is a solution to this and it would be possible to set up a bisonoric free reed instrument of the Melodeon Concertina Bandoneon type where all the Major and Minor chords in several different closely related keys are perfectly in tune. As you found my suggestion in another thread on how to set up a 31 button instrument to play the scales of no less than 5 different keys "interesting", I am sure you will find that only a small modification to this will give 5 different keys in double just tuning even more interesting.
I will show my solution in a reply to your Blog Site shortly.
Inventor.

Thanks for your, as always, creative response! I will reply in more detail on the blog post when I have had some sleep...

Yes, as Anahata says, ET. I sigh inwardly whenever I see young fiddle players at a session tuning (not just the A string but all the others) with one of those gizmos that clips onto the pegbox. If there's one place in the western musical universe left for a perfect fifth, it's on members of the violin family*. And if after a year or two at the fiddle you can't tune your strings to each other in perfect fifths, IMO you shouldn't be playing one...

* Except of course in specific contexts, as on cellos in string quartets. Apparently the error in the circle of fifths is sufficiently large over a couple of octaves that quartet cello players have to tune their C string sharp...

Quite. For those that can't work it out, try the following. A viola is tuned CGDA, a violin is tuned GDAE. If we assume that the GDA of both instruments are the same and just tuned, then the E string of the violin should be (3/2)^4 above the C string of the viola. If we bring that down by two octaves by dividing by 2, then we get 1.266. When we compare to the just third of 5/4 (1.25), we find that it is out by a syntonic comma, which is noticeably out of tune. The same goes for the cello. So yes, in certain very specific circumstances, the strings might need to be purposefully detuned in order to get a consonant chord.

Thanks very much for the kind comments everyone! I'm glad that the post made sense to people.

[Jono]

I thought that was a very interesting and well written blog article.

I play the violin at an advanced student level. I'm in the first violins of two orchestras with volunteer members, which are of good standard, one might say 'semi-professional' as the main body of players are instrumental music teachers who studied at conservatories, with advanced students making up the others. I studied harmony and theory to BA level.

When tuning our violins we start with the A string, and get that perfectly in tune. Then tune D to A, and G to D, and then E to A. I find that I like to tune the D and G strings slightly sharp, because if I don't the open G will tend to be flat. In practice I find this is not a problem at all. I've thought about this many times, especially when I was in my twenties (I'm 40's now), about the God and equal temperament/just temperament question that you write about. Bach, a great man, was clearly not put off his faith by such things.

I've come to the conclusion that the tuning characteristics you refer to are not a flaw in music. I see them as requiring of music a more flexible, more subtle, more searching and compromising/listening and seeking way of making music alive; of coming up with solutions that work despite all else. If music were a 'perfect' thing, in the sense that a perfectly formed crystal is perfect, might it not be boring? I think it would be too cut and dried, so to speak. It might be like living in a world with no mountains to climb, or no sea to swim in.

[Owen Woods]

Jono - this is the conclusion that I have come to. It's just a conclusion that is difficult to live with as a fixed pitch instrument player!