View Full Version : Temperament

Thomas Dressler
Jan-04-2006, 06:39
The subject of temperament came up in another forum, and it has come up once or twice here, recently in joking. I think it's a very interesting topic and one that needs to be explained for those who are not familiar with the idea. So what I'm going to do in this post is outline some of the basic ideas of temperament.

In reality, the subject is extremely complex, but I'm going to try to make it as simple as possible. (So those who are experienced in the subject may find it over-simple, but try to bear with me because it's also difficult to understand for those who haven't thought about it.)

When tuning a keyboard instrument with 12 notes to an octave, you quickly find that there is a puzzle which it is impossible to solve. That is that if you start with C and tune the fifth (C-G) perfectly in tune, then tune the fifth above that (G-D) perfectly in tune, and continue through the circle of 5ths until you arrive back at C, this ending C is going to be sharper than your original C. And not only that, but when you play thirds, they are going to be WAY sharp. So the basic puzzle is that if you want to cram all 12 notes into an octave, you have to flatten some of your fifths. The amount of "overshoot" when you tune all the fifths pure is called the "Pythagorean Comma." So you have to somehow flatten your fifths somewhere to make up for the comma.

One of the earliest methods, and it's called "Pythagorean," is to make ONE fifth really flat, the amount of the comma, and leave the others pure. You just have to make sure you flatten some obscure fifth that is not commonly used, and make sure you never use it. It's an interesting kind of temperament with its sharp thirds that are almost intolerable to our ears. It works best for music written with the assumption that thirds are dissonant and fifths are the important consonances--Medieval and some early Renaissance music. For any music that uses thirds as a consonance, it sounds pretty bad.

The next option, that used mostly during the Renaissance, is Meantone. It's a complicated concept, but it is much simplified if you use a particular kind of meantone called 1/4 comma. In the case of meantone, the comma is spread out rather than just being hidden in one particular fifth. The idea of spreading the comma out evenly among all of them, called equal temperament, was known to musicians and theorists, but they rejected it, because it makes all of the fifths equally flat, and all of the thirds equally sharp--nothing is pure. They liked pure intervals. So during the Renaissance, the solution to the puzzle was to put more of the comma on some intervals than others. So in some keys, you'd get really nice intervals. As a matter of fact, in Meantone, you get lots of pure thirds, but the fifths are mostly pretty flat. But for their methods of composition, those pure thirds were important. The caveat with it is that you can play in certain keys really well, but others sound pretty bad. From C major up to about three sharps or three flats sound good. They just didn't play music in the other keys.

During the late 17th century, a German theorist named Werckmeister invented another way of tuning which compromised a little more but allowed you to play tolerably in all the keys. His most commonly used temperament, which we usually just call Werckmeister, has better fifths than Meantone, but it doesn't have the pure thirds. And it has the interesting effect that the more sharps or flats you're using for a given key, the more jangly the key sounds. In other words, C major, and F major (the best key in Werckmeister) have a very restful, pure, peaceful sound, more than in equal temperament. As you move to keys like A flat, the intervals are more out of tune and the music jangles more. Werckmeister was a commonly known system of tuning in those days, and while there's no evidence I know of that proves that JS Bach or Buxtehude used it, my own belief is that Buxtehude used it and so did Bach, at least early on. (This is based on experience with their music and how it's written and how it sounds in different temperaments.)

JS Bach became very interesting in writing in all the keys. Remember, the concept of major and minor mode were being more fully developed in his lifetime. He eventually wrote the two sets of keyboard pieces called "The Well-Tempered Clavier" which has pieces written in all the keys. It has been argued on and off in the hundreds of years since his death that he meant for these pieces to be played in equal temperament. Most scholars disagree with this, though it was a common belief until the late 20th century. Recently, Bradley Lehman did an interesting study of the title page of the Well-Tempered Clavier, which seems to have a handwritten puzzle by Bach which explains how he meant the keyboard to be tuned. It does NOT indicate equal temperament. It's interesting, but somewhat complicated. To read more about it click here (http://www.larips.com/) . You can also listen to my own modification of this temperament on my website (http://www.thomasdressler.com) . Just look on the soundclips page. You can also hear various other temperaments there.

Not long after the death of JS Bach, there was quite a controversy over the introduction of equal temperament, which spreads the comma out evenly among all the fifths and thirds, so all the keys are equally in/out of tune. One of Bach's students, Kirnberger, adamantly argued that his teacher rejected equal temperament because he liked the differences in key color. Equal makes them all sound the same. This battle was being fought up into the 19th century, with the Germans, for the most part, adopting equal temperament early on. There is actually plenty of evidence that in England and the United States, some organs of English design were being tuned in Meantone up until the mid 19th century. This is a very interesting and possibly very important fact, because it affects the sound of the instruments and also affects what music could have been played on them. The Germans in America, starting with David Tannenberg in the 18th century, were tuning in something very close to equal and possibly real equal.

Music in the 19th century began to use kaleidoscopic changes of key and equal temperament became more common, until by the 20th century, it became almost ubiquitous. It was in the later 20th century that musicians interested in historic performing practices began to rediscover the old temperaments and use them for older music.

Nowadays their use is common enough that when designing an organ, serious consideration needs to be given to what temperament will be used. When playing a harpsichord, one has the luxury of being able to retune for different kinds of music.

This is a basic outline of what temperament is about. It can be a complicated subject, and I welcome both basic questions about it and more advanced discussions.

Colorful Mage
Jan-06-2006, 05:06
Sorry I've been inactive the past few days; I've been very busy.

This is a great explanation of temperament. I'm left with no questions. I should be in chorus tomorrow, so I'll ask my chorus teacher if I can play around with the different temperaments on his electric piano.

Great summary of the topic, though. Temperaments are, I guess, an extremely important part of music that I had been mostly overlooking.

Thomas Dressler
Jan-20-2006, 19:56
Did you get to try them? If so, what did you think?

yury habrus
Mar-10-2006, 15:22
Thank you for this reach and clear information! And could you tell something more about modern and old tuning techniques?

Thomas Dressler
Mar-14-2006, 08:22
Do you have specific questions? If you do, don't be shy to ask, and I or someone will try to answer. https://www.magle.dk/ubbthreads/images/graemlins/smile.gif

Are you asking specifically about how to set temperaments? That's actually a bit complicated to explain, but how complicated depends on which temperament you're tuning. The easiest of all is Pythagorean (the one with the pure fifths.) You just tune through the circle of fifths (C-G, G-D, D-A, A-E, etc) with pure fifths. Well, actually, you have to decide where you're going to put the "wolf" (the out of tune fifth that "howls.") Start with the top note of that fifth and tune through the circle of fifths, and just allow the last one to be really out of tune.

The most difficult temperament of all is equal. I believe that until the invention of the electronic tuner, equal temperament was often not tuned really equal. I do know numerous stories of tuners who would modify the temperament enough to make some keys sound better than others. You do have to remember that when setting temperaments, there is always give and take. Whenever you make one key sound more in tune, there is always another key that sounds worse. I cannot explain exactly how to tune equal temperament by ear because I never use it on my harpsichord, but I do know that it's pretty complicated to do well.

Meantone can be complicated, but it's a sort of fortunate accident that the version called 1/4 comma is pretty easy, once you practice it a bit. Here's how I do it: Start at Middle C and tune the third, C-E perfectly in tune. Now, you have to temper the fifths C-G, G-D, D-A, A-E. You need to spread the amount of tempering among these fifths so that by the time you end up at E, the C-E third is pure. With practice, you get used to how they should sound, but you need to realize that what I mean by tempering is to tune them perfectly in tune, but then put them slightly out of tune. I won't go into the details on how to do this unless you specifically ask, because it's complicated to explain. But once you temper those fifths, then you just go through the rest of the notes and tune all the thirds purely in tune, such as G-B, D-F#, A-C#, etc.

The one thing you need to realize about Meantone is that you cannot use a black note for two different notes. In other words, C sharp and D flat ARE NOT the same note. If you tune that black key as C sharp, it will sound horrible if you try to use it as a D flat. Some old organs and harpsichords actually had extra black notes so you could have more notes. There would be, for instance, another black key so you could have both G sharp and A flat.

Kirnberger's temperament is not too difficult, as you temper the fifths the same way as you do for 1/4 comma Meantone, but then you tune in pure fifths rather than pure thirds.

I don't know if this is helpful or more confusing. Please ask anything specific that you want to know and I'll try to help.

Mar-14-2006, 08:29
Hey, just a question, this is going a bit off topic, have you learnt prelude and Furgue in G major BWV 860? That has to be one of my favourite harpsichord pieces, I am learning it at the moment, it is way beyond my level, but will get there one day.


Thomas Dressler
Mar-14-2006, 08:40
I've learned the Prelude, but haven't worked on the Fugue. I do like it though, and I do plan to work on it one of these days!

I like BWV 861 (G minor) very much and I play both the Prelude and Fugue for this one. The fugue has a couple really tricky spots though. Measures 17 and 18 have some difficult stretches in them.

Enjoy working on the G major! The Well Tempered Klavier has such interesting, varied, and sometimes really romantic sounding pieces in it. I always go back to working on these pieces, and never get tired of them.

Mar-14-2006, 10:20
I love how he put it down, the rhythm and the melody and the tempo are just lovely.

Mar-14-2006, 11:22
Hi Tom,

It's pure joy to see you posting again. I always learn so much from you since you make such great sense in what you write.

Lately, I have been trying to adapt the *dis-tempered klavier* https://www.magle.dk/ubbthreads/images/graemlins/grin.gif https://www.magle.dk/ubbthreads/images/graemlins/nut.gif https://www.magle.dk/ubbthreads/images/graemlins/whistling2.gif https://www.magle.dk/ubbthreads/images/graemlins/banana.gif p& f 's to the organ - it ain't easy but it's fun nonetheless.


Giovanni https://www.magle.dk/ubbthreads/images/graemlins/tiphat.gif

ps: I trust your Mom is feeling better these days https://www.magle.dk/ubbthreads/images/graemlins/pray.gif

Thomas Dressler
Mar-14-2006, 20:46
Thanks Giovanni!

Yes, my mom is doing better, but recovery from shoulder operations can be long--she's had her right arm immobilized for about 6 weeks--a very long time not to be using your arm at all!

I play some of the WTC Preludes and Fugues on the organ, too! Some of them adapt better than others, but I play my favorite, the G minor one from Book 1, on the organ, as well as a few others.

Best to you,


Thomas Dressler
Mar-17-2006, 08:15
Gareth asked in another thread about meantone temperament. I had some deadlines to meet so I couldn't answer right away, and it also gave me some time to think about this question. Temperament can be a really complicated subject, and I don't pretend to know everything about it. If you search on the internet, you'll find that many sites talking about it immediately launch into mathematics so complex my eyes start to glaze over.

Perhaps someday I'll feel like I have a handle on the math and theory part, but at this point I'd say that practical musicians don't need to know all the complex stuff. But there are some basic ideas concerning temperament that are important and might help with understanding.

Let me explain it in as simple a way as I can (of course, making generalizations that may at times not be true, but they help with understanding this from the practical point of view.)

The first thing to think about is the whole concept of pitch. We don't usually think about this. . .C is C, and D is D, right??? There are 12 notes in an octave and that's just the way it is, right??? Not necessarily.

Think of a siren, going gradually up in pitch, and gradually back down. There are no sharply defined pitches, just a gradual rise and fall. Eventually, as you go up this gradual wail, you will reach a point where the sound is "the same but higher" as where you started. You've gone up the distance of an octave. So you see, there is no "God given" rule as to how many notes there should be in an octave. There could be an infinite number. You could divide an octave up into 2 notes, 12 notes, 24 notes, or however many you want.

Think of the color spectrum, and narrow it down to gradations from red to yellow. In between there are all kinds gradations from red to orange to yellow. How do you know where red ends and orange begins? It's difficult. But just as a painter needs defined colors to paint, in order to make music as we know it, we need defined pitches.

The actual way these pitches were defined was not done, as one might imagine, by arbitrarily stopping at 12 different places in the gradual wail up the octave. It was done by ratios, or in simple terms, by the relationships of one pitch to another, and another, etc. It is difficult to explain this to modern people because we are so accustomed to the equal tempered scale which has 12 equally spaced pitches in an octave. But probably the main reason some musicians dislike equal temperament is that it is the supreme compromise. It takes the least account of pitches' relationships to one another in order to create an artificial, equal stepped scale. But in the world of western music based on polyphony, we listen to pitches TOGETHER, in relation to one another. So the point of this is to explain (without going into much detail) that pitches were and are defined in relation to one another, not as an absolute ("C is C and that's it.)

So depending on how the pitches are being related to one another, they may fall in different places. In other words, take the note D for example. In the key of C, with everything being played in pure intervals, the note D will be different than it would be in the key of D. Ultimately, for perfect tuning, you could tune a keyboard only to play in one key, and then your music could not modulate--it would sound good only in that key, period.

The problem is that musicians don't like to stay in one key, and not modulate. But as soon as you move out of your one in tune key, things start to sound bad, as the relationships of notes are wrong. You could add extra keys to allow this, but you'd have a jillion keys just to play in a few different tonalities. People liked the simplicity of 12 keys for an octave.

Well, if you're going to stick with only 12 keys (or sometimes a few more) to an octave, then you're going to have to figure out how to have the notes do double duty, or triple duty, or whatever. This method of compromising, or putting some notes purposely out of tune so they can be used in more than one way, is called tempering.

Over the years, there have been many ways of compromising. At first, musicians wanted to stay as close to pure as possible, so they accepted the fact that if they didn't want to play in EVERY key, they could still keep some keys relatively pure. Understand that the fewer keys you want to play in, the more pure the tuning can be. You have to compromise the purity more and more in order to use more keys.

One commonly accepted way of doing this was Meantone. This temperament allowed you to play in keys with up to about 4 sharps or flats, but anything beyond that was badly out of tune. Another thing to understand is that the more pure the thirds are in a given temperament, the worse the fifths are. In Meantone, the thirds are good but the fifths are kind of off. Composers who wrote for Meantone tuning used thirds liberally because they sound so good, but avoided plain fifths. Meantone was used up until about 1700 or so in Germany, and even later in other places. In England and parts of the US, Meantone was used until the mid-19th century in some places.

Bach, however, was one composer who didn't like those limitations. He wanted to play in ALL possible keys. Werckmeister had invented a temperament that compromised a little more than Meantone and allowed tolerable use of all the keys. Of course, this means less purity in all of them, even the good keys. While Werckmeister's temperaments, and others of the time, allowed one to play in all the keys, it was still not a complete compromise. Some keys still sounded better than others, but the worst ones were tolerable, not like the bad ones in Meantone that sound awful. But the compromise is that the good ones do not sound as good as in Meantone. There's always give and take when tempering.

Equal temperament is the complete compromise. All the keys sound the same, and all are somewhat out of tune. The fifths are ok, not wonderful, not bad. But if you're used to listening, the thirds in equal are REALLY out of tune. On the other hand, this is what most modern musicians are used to. It actually happens that when they hear a REALLY pure third as in Meantone, they think it's OUT of tune.

In my opinion this is a shame for several reasons. One is that we rarely hear old music written for Meantone, and I believe it's partly because it sounds terrible in equal temperament. The older composers made liberal use of thirds in exposed, strategic spots because they sound so good in meantone. In equal, thirds are one of the worst sounds, and it makes that music sound bad.

My personal opinion is that historic music should be played in a temperament that is sympathetic to the way the music is composed. Bach sounds alright but not as he intended it in equal temperament. He used the different degrees of in-tune-ness of different keys on purpose, sometimes deliberately creating a jangly effect by playing in a jangly key. In equal, you completely lose this use of color. But the worst problem is playing Meantone music in equal. It simply sounds bad to those who have become accustomed to hearing REALLY in tune thirds.

There is a more involved explanation that is not too technical here (http://www.users.dircon.co.uk/~oneskull/3.6.04.htm).

I hope this helps. Let me know if this does not make it clear.

Mar-17-2006, 19:52
Hi Tom,

Good show on posting more about temperament and including a link to Bicknell's treatise.

Somewhat off-topic: Have you had your choir sing any works by Carlo Gesualdo? There's some real chromaticism in his works and he being an early Italian composer.

Giovanni https://www.magle.dk/ubbthreads/images/graemlins/tiphat.gif

Mar-18-2006, 04:16
Hi Tom.

That stuff sounds really complicated, but I sort of get where you are coming from.


Thomas Dressler
Mar-20-2006, 07:15
Hi Giovanni and Gareth,

Yes, the temperament stuff can get really, really, REALLY complicated, but at its simplest it can just boil down to knowing about a few different temperaments and when to use them. And just knowing a little about it will make it so you have an idea what to do if you encounter an instrument in a different temperament.

I have to admit, Giovanni, that I have not done any Gesualdo with my choirs, and mostly just because it's too difficult for my volunteers. But I do find it to be extremely interesting. Now there's a colorful person who wrote colorful music!! https://www.magle.dk/ubbthreads/images/graemlins/smile.gif