A short return to the just vs equal temperament discussion

[Wombat]

Jerry's scale above seems about right. Notice how the 3rd and the 6th are off about the same in equal. It's because the 3rds and 6ths are related. mathmatically. Peter is right about equal being a compromise. Like someone else said, just is only good for one key. Just makes the F# sound great in D. I prefer the compromise though since I play in about 4-5 keys on the D whistle.

The compromise is, I think, what gives different keys their distinctive colour. For heavily harmonic music there is no question that equal temperament opens up a whole world of possibilities that just temperament would not make possible, at least not with practical choices of instrumentation.

For music that is less harmonically based, just temperament often sounds better. I note with interest that the notes we like to hear a bit flat are mainly the third and the seventh and many of us deliberately vary the intonation of these notes as we play. This is true of blues as well as Irish music. The old (patronising) theory is that blue notes were the result of African Americans used to pentatonic scales lacking the third and seventh having no ear for correct intonation for these intervals. I think that a preference for just intonation and floating intonation is a much more plausible explanation. Variable intonation grabs me in much the same way in both musics.

One interesting thing is that there is a dispute about how well harmony sounds superimposed on Irish music. Not only do blues musicians play and sing modal melodies against the background of pianos, guitars and horns playing full ninth chords and even major sevenths in equal intonation, so too do Romanian gypsy bands use just temperament for melody and equal temperament for accompaniment. Both are an acquired taste and I imagine that if harmony really catches on in Irish music it will be an acquired taste there too. It's still great to hear a blues with little or no harmony though but the tradition is broad enough to allow both strands to flourish. Playing blues you need to make slight adjustments for even intonation, just as you do when playing Irish music with instruments like concertinas.

[PhilO]

Certainly don't have my arms wrapped around this yet, but am getting an inkling; thanks to all for the explanations.

Philo

[spittin_in_the_wind]

Tone generators usually only generates very pure sinus waves. The pitch of these pure notes is actually percieved differently depending on the strenngth of the note. This does not happend when you have a note with many higher harmonics in it. Therefore, a tone generator might not be the best thing to tune against.[/i]

True. However, it is really useful for training yourself to blow notes in and out of tune, high and low. I used one in college a number of times with my saxophone, which required a lot blowing high and low, by the nature of the instrument. It really gets your ear where you can hear teeny differences in intonation, which is kind of cool.

Robin

[Darwin]

I note with interest that the notes we like to hear a bit flat are mainly the third and the seventh and many of us deliberately vary the intonation of these notes as we play.

I play a number of D mixolydian tunes on guitar, and I find myself wanting to play something between the F and the F#, so I often slide between them, as the style doesn't lend itself to bending the F by stretching the string. Back when I played these particular tunes on fiddle, I'd just hit the note I wanted without the slide.

Given how bad my ear is, it's hard to believe that I can even hear the difference, but now I think that I've been trying to play a just-temperament F# on what is an inherently even-tempered instrument. How strange.

[chas]

Jerry's scale above seems about right. Notice how the 3rd and the 6th are off about the same in equal. It's because the 3rds and 6ths are related. mathmatically.

How are the third ans sixth related mathematically? It seems to me that the octaves and fifths are closely related, but getting to a third or sixth takes a few steps. (I'm not challenging, just wondering)

[Wombat]

I note with interest that the notes we like to hear a bit flat are mainly the third and the seventh and many of us deliberately vary the intonation of these notes as we play.

I play a number of D mixolydian tunes on guitar, and I find myself wanting to play something between the F and the F#, so I often slide between them, as the style doesn't lend itself to bending the F by stretching the string. Back when I played these particular tunes on fiddle, I'd just hit the note I wanted without the slide.

Given how bad my ear is, it's hard to believe that I can even hear the difference, but now I think that I've been trying to play a just-temperament F# on what is an inherently even-tempered instrument. How strange.

The trick you describe is commonly used by blues pianists on both the third and the fifth which I also think is a blue note although not one you could explain by reference to just intonation. To get the effect of a 'blue' third or fifth I'll play the flattened third or fifth followed by the natural so quickly that the first note sounds like a grace note with about the same value as a note you might cut or tap with. It doesn't work so well for the seventh probably because it sounds best ascending and you don't want to come to rest on a major seventh playing blues. Another trick that gives a nice dissonant blue note is to play major and minor third, fifth or seventh simultaneously—Thelonius Monk uses this trick. That's as close as you'll ever get to getting into the cracks on a piano.

Fretless banjo might be good for teh tunes you describe. I imagine the problem on guitar is that you are playing too fast to bend strings without making the slide sound. (Maybe not since you use a slide anyway.) It is possible to play a nice justly intoned third on guitar by bending F up a few cents, not hitting the note until you've completed the bend and then dampening before release.

[Lorenzo]

How are the third ans sixth related mathematically? It seems to me that the octaves and fifths are closely related, but getting to a third or sixth takes a few steps. (I'm not challenging, just wondering)

In one sense, I think you are right. The relation I was trying to get at was the cycles, or beats p/sec., or p/5 sec.

After I posted yesterday, I poured a bunch of cement, then went to the mountains to build a road into the cabin site, and then stopped back by Caledonian Days to hear a great concert by Raven--a fine celtic band out of Portland! So, this morning I finally got around to looking in the book.

In the equal temperament scale, C28 (acsending), the major third above is widened by 5.19 beats p/sec. The major sixth is widened by 5.93 beats p/sec. This is what I was trying to guess at (just counting in my head the beats I hear in major 3rds, equal) which I thought was around 20-25 beats in 5 seconds. I was roughly trying multiply 5.19x5. It's approximately the same for a major 6th because in some sense the 3rd and 6th are related. IOW, the E above C is a major third and the E below C is a major sixth. For all practical purposes, they produce the same beats by ear...because the E octaves are perfect, but technically a little different when measured by machine.

In contrast, the fourth is widened by 0.59 beats p/sec., and the fifth is narrowed by 0.43 beats p/sec. So for all practical purposed, tuning by ear, the fifth is slightly more pure than the fourth.

Bottom line: any good piano player would consider a just tuned piano to sound extremely sour (out of tune), except for one chord. Like someone else said, it's the compromise (impurity) of equal tuning that adds richness to the chords. It produces waves/beats kinda like vibrato in a vocalist whose voice is not completely steady.

[Jerry Freeman]

With all respect to Wombat and Lorenzo, it simply isn't the case that equal temperament makes chords sound better/richer.

The only reason for equal temperament is practicality.

To have accurate just intoned chords, the exact pitches of the various named notes (C, D, E, F, G, A, B) in every chord would have to vary according to the base note of that particular chord, with each note of the chord in precise mathematical harmonic relation to the base note of that particular chord.

In other words, the notes of a Cmaj chord would have to be built from the just intonation Cmaj scale; whereas the third (Emaj) chord would have to be built from the just intonation Emaj scale; the fifth (Gmaj) chord would have to be built from the just intonation Gmaj scale, etc. For some of the notes needed to make chords, the just intonation C of one scale will be different from the just intonation C of another scale; the just intonation D of one scale will be different from the just intonation D of another scale, etc. If you're limited to only one C, D, etc., you simply can't produce just intonation chordal music. You don't have enough notes to work with.

Built the way I've described above, with every chord in exact harmony with itself, the chords sound very sweet and rich. In fact, this is exactly how well executed choral music is performed. I believe this is also true with well performed music of brass and string ensembles. Here's an example:
http://www.justintonation.net/sound/sequence_II.mp3

Chromatic instruments capable of playing chords don't have the ability to supply variations to each note according to what chord the note occurs in that would allow each chord to be harmonically exact (just intonation), so as a matter of practicality, the task of playing chordal music falls to equal tempered instruments, not because they sound richer, because they provide a close enough approximation to make it possible at all.

The idea of a just intonation piano that can play accurately all chords and all keys is a practical impossibility, as Lorenzo has stated, although piano tunings based on just intonation have been achieved that allow for some interesting, and in my opinion, sweet sounding music. Here's an example:
http://www.newmusicbox.com/32/audio/harrison1.ram

Best wishes,
Jerry

[Lorenzo]

Apples, oranges. I agree that just is better for chorals and choirs. Some of the better directors have discouraged the soloist's vibrato when singing with a group...it just adds to the confusion, the clutter of multiple vibrations. A choir can correct itself to just tuning as it goes...only if it doesn't go too fast! But let the choir try and adjust when singing Irish reels up to speed!

Fixed instruments don't adjust..not even a violin (unless played completely w/o open strings). No ones talking about the exception to the rule...the human voice. We're talking about fixed instruments. Of course perfect is better for a pure three-note chord in any instrument. It just isn't pratical, like you say, or even possible with fixed instruments. So we tolerate imperfection and find richness...which is better to many peoples taste. I'd go so far as to say that even a wayfaring man, though a fool, would agree that a just-tuned piano needs tuning...and badly, if played in more than one key (which is nearly always the case). General consensus doesn't always prove everything, but listening to the experts, you'll not find a fraction of one percent of piano players who have the opinion that a just-tuned piano sounds good (if played in more than one key (or chord). Examples abound. You can't get the full picture of a just-tuned piano by listening to selected samples. Show them a just-tuned piano that sounds good, selecting the best key and chords, and any good piano player will show you how that same paino sounds terrible in another key (or other chords). But, it's a matter of taste, and I would never claim everyone dislikes sour notes.

It's a mater of taste as to whether a soloist playing the violin sounds better/richer without vibrato (or the entire orchestra). I personally would say that a violinist sounds richer with vibrato, and adds depth. In essence, that's what equal tuning does...it builds-in the vibrato for you on each chord, because each note has been tuned equally off from perfect.

Another start difference (apples and oranges) is chord instruments vs. solo instruments. Here, when talking about just vs. equal, we're talking about a limited, fixed, whistle (or flute. or pipes).

[glauber]

if you read the flute board you often hear the comment that old flutes need to have their Fs sharpened (to get the tuner to tell you it's right). the old flutemakers were no fools and I think the good ones chose their tuningwell.

In the one-key (Baroque) flutes, there is a compromise between the F# and F natural. You can't get both of them right. In most flutes, the F# is a little flat (and has to be "lipped" up) and the Fnat is way sharp (has to be "lipped" down). The recorder has a better system, but at the price of having more complex fingerings for both F# and Fnat.

To add to the complication, the idea of what constitutes "in tune" seems to have varied a little with time. The slightly-flat-of-equal-temperament F# in the old flutes is very easy to play correctly as part of a D chord in equal temperament, so the theory (probably right) is that old flutes were played in just temperament. You can do this with a flute more easily than a whistle, because the flute allows for a certain amount of fine tuning of the notes. A good Baroque flute will allow you to play every note at least a full semitone sharper or flatter. The small fingerholes and small embouchure hole help.

In just intonation, the seventh note of the scale (e.g C# in a D scale) is slightly flatter than in equal temperament, and this is the way it seems to have been in the Baroque period. But in the Romantic period, people were favouring a "supersharp" seventh, which would make it sharper than in equal temperament.

Equal temperament exists because of keyboards. It's really a cop out. You can't have a keyboard instrument in just temperament because of what someone mentioned, the extra "coma" at the end of the scale. I won't even go there... PM me if you want more info.

Anyway, you can't have a just temperament keyboard, so there were all sorts of "tempered" tunings, some favouring the keys with sharps, some favouring the keys with flats. Composers in the Baroque period were able to write for the keyboard in every key, by carefully avoiding the intervals that sounded bad in whatever temperament they used.

Eventually equal temperament won, because it does make more sense for keyboard music, but tempered tunings are making a comeback, not only with people playing Baroque music, but with piano players too.

g

[Jerry Freeman]

A few comments.

1. A just intonation tuned piano is not a multi-key adaptable instrument to begin with, period. That's part of the definition. Saying that everyone would agree that a just intonation tuned piano would sound terrible if played in other keys doesn't help much, since that's already a given when you select just intonation. It's kind of like saying a key of D whistle is no good because it's too hard to play in the key of Bb. The entire reason for equal temperament is so you can play in more than one or two keys with a single instrument. However, the tradeoff is that no scale or chord is ever really in tune.

2. I disagree with your statement that voice is the single exception to fixed pitch instruments.

Defining strings as fixed tone instruments because open strings are tuned to a fixed pitch doesn't match what I've been taught about them. The man who taught me about just intonation was a choir director who was also a master of violin and trumpet. He used the example of stringed instruments to illustrate the concept. I'm sure a good violinist would be able to select an alternative fingering if an open string happened to be dissonant for a certain chord the ensemble was playing.

Similarly with brass instruments. I've seen webpages discussing the importance of playing notes up or down to fit exact harmonies in ensemble chords when playing brass instruments, including exercises to train the ear and develop the technique.

3. I also disagree with the idea that the clash of notes in equal temperament chords adds richness in a way similar to how vibrato does. If you listen to various chords played in equal and just tuning, the just tuned version of the chord sounds cleaner, richer, more musical, and I believe most listeners would agree that this is the case. The clash is completely unintentional. It's an artifact, not an embellishment, and I'm sure if the developers of equal temperament could have avoided it, they would have.

Best wishes,
Jerry

[Martin Milner]

Most fiddlers I know tune with an electronic tuner first (Even Intonation), but then microadjust by bowing the strings in pairs and finding the perfect fifth (Just Intonation) by ear.

If they tune the A only, with a tuning fork maybe, and then tune the other strings in fifths, again they end up with just intonation.

May this explain why fiddlers are occasionally accused of being out of tune?

But since the tunes are mostly in D, the Dstring should be the one tuned to (electrical) pitch, and then the other strings tuned just from that one, right?

I follow your logic Andreas, but that's not how people do it. Not sure why. I've never even seen a D tuning fork. To

Apparently fiddlers can also tune using harmonics, but I've never seen it. Harmonics have to be "just", because "just" is nature's way, and even is man's compromise.

[glauber]

A while ago,in the High end or just expensive thread, wanderer wrote:

It was when someone tried to use "just temperament" as a possible excuse for this brand of whistle to be so badly out of tune that I realized that there was just no rational discussion to be had with certain Generation apologists. :roll: I realize that there's no 'convincing' the established folks like Peter and others...

[...]
Whether this way of tuning is by design or by chance I don’t know but I think it is by design, if you read the flute board you often hear the comment that old flutes need to have their Fs sharpened (to get the tuner to tell you it's right). the old flutemakers were no fools and I think the good ones chose their tuningwell. WHistle makers quite possibly followed their system and some of the equal tempered thinkers are probably not considering the possibility that these makers used a system that might just be very suitable to certain situations, more so than equal temperament. Those of us working the elctronic tuner as the measure of all things instead of their ears can be easily led to think this tuning isn't right.[...]

For whatever it's worth, i think Peter is right. The old makers of flageolets and pennywhistles were not working with electronig tuners, but working with the scale that they heard in other instruments of the time. This is not necessarily just intonation, but a system of compromises that (at least in the case of the flute) allowed the player to play in just or whatever other intonation he had to cope with. This series of compromises was probably arrived at by trial and error, and not by math.

Has anyone done a study of the scale of uillean pipes, what it's based on? That and other kinds of bagpipes are a window to the past, instruments that are made more or less the same now as they were being made 200 years ago.

g

[Nanohedron]

With all respect to Wombat and Lorenzo, it simply isn't the case that equal temperament makes chords sound better/richer.

The best-tuned pianos sound off-tune to me, FWIW.

[Cayden]

You'd expect the pipes to have their scale tuned in such a way that every note harmonises with the drones. Geoff Woof has written about this issue (in Ceol na hEireann and the subject has been touched upon in the Journal of the Sean Reid Society). Early developers of the pipes were also involded in making other instruments. There are flutes, oboes etc known made by Kenna and Coyne. These men were no fools and it has been suggested that there are similarities between the design of early chanters and the oboe of the period.