In Reality, %99.9 of western culture doesn't hear just intonation as 'in tune', nor any other system other than equal temperament. The reason is that we use, as stated above, an irrational system of minutely adjusting tones so that all are equal. This was a long process beginning with the harmonic series and just intonation by way of the Pythagorean system, and subsequently modified to accommodate the use of more than one key (and then many) as history progressed (Pythagoran, Meantone, Werckmeister, Kirnberger, etc..) until we reached perfect equal temperament.
As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
Anyway, so I feel it's important that I should go and dispel some of the erroneous 'facts' in this thread, or clarify them further.
On June 11 2011 07:00 [F_]aths wrote: You still cannot play with perfect harmony as some strings are still oscillating when you play a new note. You cannot have perfect intervals as they just don't fit.
This is possible, but it requires a trained ear to be able to hear them. You're also not talking about pitch differences when talking about sympathetic vibrations. These timbral changes aren't going to affect the pitch nearly as much as you suggest (as in they won't) as the sympathetic vibrations only consist of overtones as part of that specific pitch's or pitch cluster's harmonic series. This only acts as a perceptual change, not an actual one. It's like adding reverb to a track that didn't have it before. The pitch hasn't actually changed at all, but the perception of that pitch may have based on the acoustical properties of the type of space used.
On June 11 2011 00:41 storm8ring3r wrote: when I tune my guitar I usually tune it using tuner for A440 and fifths for E, D G, tuner for B and harmony for E. so... equal temperament I guess
Depends, do you only tune at A440 or do you have a tuner that plays the pitch of all the other strings? If it's the former, then you actually tune in just intonation (or more accurately, you tune to perfect intervals), but the guitar's metal frets take care of the pitch adjustments into equal temperament.
On June 11 2011 06:43 [F_]aths wrote: We also know for example that Eb is actually not D#. We just play the same pitch for these two notes.
Only in 12-tet (equal temperament) is that the case, as you mentioned. Especially in just intonation because of the meandering pitch ratio's, enharmonics are critically important to distinguish. A really interesting thing a professor did for us in my undergrad was we took a small Palestrina, or Dufay work and plotted the pitch deviation from beginning to end (using JI). We started on A440 and depending on the piece the eventual A landed anywhere from 429 to 445. This assumes that you are horizontally making each pitch relationship a perfect one from one note to the next.
Tuning and Temperament is a fascinating subject, however I should state that simply grafting westernized music theory ideas onto another cultures music system is a really, really bad idea. Many times you will find that there is absolutely no correlation, or a very poor one, from one system to another especially when dealing with western culture vs. African, Indonesian or Asian cultures. In the African and Indian culture for example, there is no such thing as a time signature. Though each have very different functioning rhythmic systems they fundamentally have no concept that lines up with our system. Sure there are Tala's and cross-rhythmic structures where content converges, but those should not be considered time signatures in the way that western culture identifies them. Heterophony is a great example as it is integral to African musical culture, but until very recently (post 1900), it did not exist in western culture.
On June 11 2011 06:52 Z3kk wrote: I've always had perfect pitch but have been way too lazy to identify intervals versus frequency, etc. People who know I have perfect pitch will ask what the frequency is, but I can only tell the individual notes or notes in a chord and cannot identify frequency (though supposedly one is supposed to be able to do that?). This is also part of the reason I sort of shy away from composition/theory, etc. It's sort of childish, but I just play to enjoy the music instead of knowing and appreciating the theory and the like behind it
The average ability for one to discern a difference in pitch I think was ~3 Hz, which is less than 1/8th of a semitone in 12-tet. This really isn't distinguishable by the general populous, and even to many musicians. Only if you have perfect pitch, or have extensive experience with different tuning systems are these differences usually noticable.
On June 11 2011 07:06 [F_]aths wrote: As I tried to explain in the OP, one cannot have perfect pitch for all intervals. Even if you restrict yourself to one key, lets say C major (which is a quite tough limitation) the major second is not always the same between any given note in the scale. You could tune the second from the base note (C) perfectly, but the distance between some other notes would be different.
Let's say you play the major scale with five major seconds and two minor seconds. When you are finished, you are slightly above the octave.
Assuming you are restricting yourself to a fixed pitch system or instruments that are in 12-tet (there are many tuning systems and instruments that go beyond 12 tones/octave). With fixed pitch instruments (such as piano) this logic would hold true, but once you bring in a moving system this is no longer a hard and fast rule. That's not to say that the focal pitch in question isn't going to fluctuate a bit, but it's not like that's an impossibility and doesn't happen (or didn't).
So Just for emphasis, much of what's in the OP is right, but only conditionally, as if we are to assume that either the instrument or tuning system cannot move through linear space and is fixed in its position (and does't have microtonal capabilities). Here's a great example of a piece played in JI (it was definitely not supposed to be played in JI, but it's a good example of the tuning)
I'm not a musician and don't even know how to read music but I found this really informative; I study physics and it's nice to have a physical explanation that I can apply to music. I have a good ear and I've always wondered why certain instruments (when played live) sounded slightly "funny", I could never put my finger on it but I think this may explain it!
^^Bach Prelude and Fugue sounds really strange because of the sustains being too long...they are really disruptive to listening to the piece. Better example of just intonation is to listen to the Tallis Scholars. They are so awesome, and they sing Palestrina too.
On June 11 2011 15:44 Fontong wrote: ^^Bach Prelude and Fugue sounds really strange because of the sustains being too long...they are really disruptive to listening to the piece. Better example of just intonation is to listen to the Tallis Scholars. They are so awesome, and they sing Palestrina too.
It has nothing to do with the soundfont or the envelope being used, but the actual interaction of the pitches/overtones with one another that makes it sound so odd/jarring, although I would agree that the envelope is a little long, but that isn't why you perceive this as so disruptive. To be specific, what's making the timbres so odd to you are the interactions of the various beats between pitches, in that they are completely unfamiliar to your ear.
In 12-tet every semitone has the same ratio, 1.059. Here, beats should interact in an evenly out of tune manner. But in JI a semitone is closer to 1.066 so beats will differ depending on if the interval is a pure one or an inharmonic one (right, forgot to tell the OP that its not unharmonic, but inharmonic). To go a little further, you couldn't just add two semitones to get a M2 as this doesn't follow the harmonic series. In other words in JI, every interval class has its own unique ratio and therefore have a unique way of interacting with other pitch or pitches, this is why the beating or interaction of tones is so disruptive to the non trained ear, because these relationships aren't uniform whereas in 12-tet they would, but the point of JI isn't uniformity, its purity of intervals. This also reinforces what the OP was saying in that if you actually stack pure intervals on top of one another, your octave no longer becomes a pure interval, or 2:1 ratio, but is in fact a bit wider.
I can tell you that I hear this just fine, but that's because I have quite a bit of experience with JI and other tuning systems. Also the Tallis Scholars may predominantly sing early music, but they sing in equal temperament. The thirds are far to high, and their tuning is much too even to be JI. The reason it may sound more pure is that in choral singing many choirs actively do not use vibrato in order to strengthen pitch/harmonic relationships within a work. It also creates a more even uniformity in the overall sound and an makes it much easier to tune properly because you don't have to worry as much about an individuals voice interacting or fighting with another as much as if you were to add vibrato in there. That is another interesting discussion, but slightly off topic.
On June 11 2011 07:05 Myrmidon wrote: Here I'm asking what the role of the leading tone is, for a dominant triad. Taken out of context, a leading tone should be tuned upwards (compared to equal temperament) to get the correct interval/tuning. Also taken out of context, a third in a major triad should be tuned down (compared to equal temperament) to get the correct interval/tuning.
If you have a dominant triad resolving to tonic, aren't the two objectives conflicting? This question isn't directly about tuning systems.
I am still not 100% sure if I got you right.
I have to think this through:
If we want to play a major triad, we could tune it perfectly because in this case, the intervals do stack properly. (Major + minor third = perfect fifth.)
The interval for the leading tone is the major seventh (15:8). As you already wrote, the leading tone also is the fifth plus the major third (3:2 * 5:4 = 15:8.) This also fits if we use octave minus minor seconds (2:1 / 16:15 = 15:8).
But you want to play the dominant triad, meaning is the fifth, major seventh (which is the leading tone of the tonic), and (octave plus) major second, so we get 3:2, 15:8 and 18:8 (or 9:8 if we take out the octave.)
The intervals within the dominant triads are the same as for the major triad. If we use just intonation, we could play it right.
The distinction is between looking at music vertically (all the notes at any given time, like looking up and down a score) or horizontally (the progression of one line over time on a score).
I'm pretty sure that leading tones are generally not tuned 15:16, and in many cases looking at it in terms of a major 7th is incorrect, if that's not its role.
edit: I'm pretty sure this is a common example of contradictions in tuning, but it's been ages since I've taken any kind of theory and I never took anything remotely advanced anyway.
On June 11 2011 07:05 Myrmidon wrote: Here I'm asking what the role of the leading tone is, for a dominant triad. Taken out of context, a leading tone should be tuned upwards (compared to equal temperament) to get the correct interval/tuning. Also taken out of context, a third in a major triad should be tuned down (compared to equal temperament) to get the correct interval/tuning.
If you have a dominant triad resolving to tonic, aren't the two objectives conflicting? This question isn't directly about tuning systems.
I am still not 100% sure if I got you right.
I have to think this through:
If we want to play a major triad, we could tune it perfectly because in this case, the intervals do stack properly. (Major + minor third = perfect fifth.)
The interval for the leading tone is the major seventh (15:8). As you already wrote, the leading tone also is the fifth plus the major third (3:2 * 5:4 = 15:8.) This also fits if we use octave minus minor seconds (2:1 / 16:15 = 15:8).
But you want to play the dominant triad, meaning is the fifth, major seventh (which is the leading tone of the tonic), and (octave plus) major second, so we get 3:2, 15:8 and 18:8 (or 9:8 if we take out the octave.)
The intervals within the dominant triads are the same as for the major triad. If we use just intonation, we could play it right.
The distinction is between looking at music vertically (all the notes at any given time, like looking up and down a score) or horizontally (the progression of one line over time on a score).
I'm pretty sure that leading tones are generally not tuned 15:16, and in many cases looking at it in terms of a major 7th is incorrect, if that's not its role.
edit: I'm pretty sure this is a common example of contradictions in tuning, but it's been ages since I've taken any kind of theory and I never took anything remotely advanced anyway.
It's not a contradiction per se, but a matter of making compromises because of the theory involved. So let's look at your example again;
Example 1 (in a fixed pitch system and in strict JI):
The Dominant 7th chord will be built on the 5th scale degree which is the 3:2 ratio. From there the M3 = 15:8, then the m3 = 9:8, and finally the 7th = 4:3. So say we tuned to A JI and you want the E7 chord, then the notes are E[3:2] G#[15:8] B[9:8] D[4:3]. However this is a problem because these ratios do not represent perfect intervallic content. As we know, a true M3 is at 5:4 ratio, and 15:8 is != 5:4, so in this fixed pitch system, if you are using a chord that is not part of the root harmonic series (in this case A), your intervals will not be pure, ever (though in this case they are extremely close). The problem arises because you cannot tune on the fly to make perfect intervals, so this creates an odd color to your temperament, and irregular ratios.
Example 2 (with an instrument with a movable pitch system):
Same thing, you want E7 in AM. Well we have a fundamental difference now. In order to get a E7 in A all we have to do is take the 3:2 ratio, and build a dominant 7th chord on that note. What does that mean? Basically you will end up with a perfectly tuned dominant 7th, E[1:1] G#[5:4] B[6:5] D[16]. However, as you probably guessed, now we have another problem. The notes of the E7 will not resolve properly as now we have different (but perfect) ratios to deal with. The question then becomes, how do you resolve a problem that has no perfect solution?
Let's just focus on the m7th of the chord, in this case and Its easier to show this in Hz so let's do that. Say you start on A 220, the 5th, E, will be 330 and the 7th of EG#BD will be D 293.33. Now we resolve the D to C# like we are supposed to using 16:15 and we get 274.95. This is resolved perfectly, but we need to show how it relates back to the root. A perfectly tuned D in A JI is 293.26, if we move to the C# that relates to A we get 275. Notice how the C# is now just a little lower than it used to be?
E[330] G#[418.77] B[247.5] D[293.33] resolves to E[330] A[223.34] A[220] C#[274.95]
After looking at this you're probably thinking to yourself, but these are basically the same! Yea, they are, but notice how we now have two A's? Also note that the G# in JI is much higher than if it was fixed. Because of both of these instances, the G# pushes the A up about 3 Hz if we want to use perfect intervals (horizontally), but of course we can't have two different A's so one has to conform. The obvious choice is that in order to retain the best tonal structure the leading tone resolution has to be compromised in order to preserve the harmony, else we have two roots.
And that is the crux of every major temperament that had to deal with traditional western harmony, at some point you had to make a compromise in order to preserve the more important structure, whatever that might have been (horizontal or vertical).
equal temperament and just temperament are both inferior. Everything has been done with them, there is no room for doing anything new any more. The true scale of the elite is the Bohlen-Pierce 5-7-9 frequency ratio based scale.
E[330] G#[418.77] B[247.5] D[293.33] resolves to E[330] A[223.34] A[220] C#[274.95]
After looking at this you're probably thinking to yourself, but these are basically the same! Yea, they are, but notice how we now have two A's? Also note that the G# in JI is much higher than if it was fixed. Because of both of these instances, the G# pushes the A up about 3 Hz if we want to use perfect intervals (horizontally), but of course we can't have two different A's so one has to conform. The obvious choice is that in order to retain the best tonal structure the leading tone resolution has to be compromised in order to preserve the harmony, else we have two roots.
And that is the crux of every major temperament that had to deal with traditional western harmony, at some point you had to make a compromise in order to preserve the more important structure, whatever that might have been (horizontal or vertical).
Thank you, I think you made the issue clear: We cannot have a fixed tuned just intonated instrument. While we could play in just intonation with other instrument, we probably cannot play in an orchestra if other instruments play other melodies.
With computer technology you can adjust the tonality and introduce more just intonation even if the music modulates very quickly.
It will sound different but that's the point. Will it really be for the worse when actually in equal temperament every tone is off? And it will still not be 100% 'perfect' but it's the best we can do.
When I tune my violin viola the fifths are closer towards the A than they are in a tempered instrument.
I still remember my teacher telling me about the tunings and my then conductor talking about the circle of 5ths.
Another small caveat my accompanist told me is to tune the violin slightly higher than the piano. I've heard upper instruments trying to tune slightly lower or equal to the lower strings and I can safely say imho being a flat upper part is hell on the ears.
The problem with things other than equal temperment is that music is everywhere using that system. Using any other system (like Baroque systems) will commonly just sound horribly out-of-tune to the basic listener. We are used to equal temperment.
Further than that, with the worldwide popularity of pop music, equal temperment is becoming the standard across the globe. Other temperments will probably only be used when aiming for more archaic voices in music (like when people are mimicking the Gamelan or something).
As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
Those moods for keys were not universal. Different composers used different keys depending on what they associated with the key. The tuning system led to discrepancies, but moods were certainly not unique.
I created synthesized recordings (using some C code) of a Bach chorale in three different tunings:
--equitempered --"exact" just temperament, in which every note has its frequency adjusted based on its harmonic function so that these whole-number ratios are always present --"wrong" just temperament, in which an instrument tuned to one key is used to play in another key (which sounds like shit)
On June 11 2011 12:52 wo1fwood wrote:As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
I have to say, this is the biggest epiphany for me. I've been doing both mathematics and music for so long, but taken for granted (what I had been told) about all nth intervals being equal on a keyboard. I knew from just playing around on a keyboard that I liked certain chords more than others (differing only in their tonic), but I never had any idea why.
I am curious, however. How does this all apply to, say, brass instruments, who are necessarily built off harmonic intervals? I play the trumpet, and just for example, if I play everything with open fingers, these notes are theoretically all following an overtone series of the fundamental frequency of the horn. (right?). Does this mean then, that as my range gets higher and higher, I would end up being out of tune compared to a piano? And then that my (for example) C to E interval changes depending which octave I am at?
The crazy sorts of things you learn at Team Liquid.... gracias OP and all contributing smart peoples.
the physics in the sound is always good to know, but most professional musicians know that what is correctly in tune frequency-wise is not what is always correctly in tune to the human ear (some common examples include pushing the upper note in an octave slightly down to bring it "in tune," pushing the third slightly down in a major chord or slightly up in a minor chord, etc).
...which is the tempered tuning you guys are talking about, i guess. just small tricks of the trade that you learn from playing in serious orchestras. many touring solo pianists have the upper and lower ranges of their instruments tuned off from the machine -- i.e. the center of the keyboard is almost always in tune with whatever the machine is set to, then the upper range might be tuned lower than the machine and the lower range pushed up closer to the center etc.
the point is that what the machine says is correct is not always what will sound correct.
Both systems of tuning, and all other additional systems of tuning simply offer more possibilities for expression. Chopin and his generation would be amazed that our society plays almost entirely in Equal temperament. It really just comes down to what has been accepted as cultural norm, neither is right or wrong. But I think the worst approach is assuming musical validity in something based in physics, if in physics the scales seems irrational it does not make it musically unacceptable. Instead musicians should choose the system that offers them the correct expressive devices ex. 19th century tuning causes dissonance to be that must more effective and gripping.
I'm glad that knowledge of this is getting out there. Our culture seems to have simply locked down into an equally tempered mindset when that is not always the only option that is available.
I am a harpist and I prefer to avoid equal temperament for expressive reasons, however, it is sometimes necessity to use it, ie. orchestra.
All of the just-tempered scales stem from ratios of small prime numbers. The main question is how many prime numbers to include.
If you only use 2, then you get a series of octaves (2/1, 4/1, 8/1, etc.), which is a boring scale (no specific reason why it should be boring, but it's boring nonetheless).
If you use 2 and 3, this enables you to fill in the 'space' between the octaves, creating a more interesting scale with ratios like 3/2, 4/3, 9/8, 16/9, 27/16, 32/27, etc. This produces the Pythagorean scale, with its notable comma (a frequency ratio of 3^12 / 2^19 = 531441 / 524288 = 1.013643) after scaling by a fifth (3/2) twelve times. The Pythagorean scale of course works much better in some keys than in others.
The just-tempered scale most people refer to uses the prime numbers 2, 3, and 5. This enables the space between the octaves to be filled in with eleven very convenient ratios: minor 2nd = 16/15 major 2nd = 9/8 minor 3rd = 6/5 major 3rd = 5/4 perfect 4th = 4/3 diminished 5th = 45/32 perfect 5th = 3/2 minor 6th = 5/3 major 6th = 8/5 minor 7th = 9/5 major 7th = 15/8 This kind of scale sounds better in more keys than the Pythagorean scale.
Some versions even include the prime number 7, and use a frequency ratio of 7/4 for the minor 7th interval.
To much bla and no music ! .... first couple of sentences i was like ermmm .. just overtones ? cba to read the rest ... i always hated it when music gets mixed with TO much theory, and i still do till this day =( ... ofcourse there needs to be knowledge in theory but to much .. is to much ... comprende!?
On June 12 2011 23:47 DoubleReed wrote: Those moods for keys were not universal. Different composers used different keys depending on what they associated with the key. The tuning system led to discrepancies, but moods were certainly not unique.
Yeah, I probably should have been clearer on that. There was a general consensus so to speak, whether you were referencing Galeazzi, or Christian Schubert's, or Helmholtz' writings on the subject on what each key inherently was supposed to convey, though it is true as you stated, sometimes that could be highly subjective considering the composer writing the piece, whether it was following that consensus or more of a personal affectation supplanting what the former was (I'm going to try to ask an musicology prof. about this how much of a consensus and get back to you).
EPT
]. However, as you probably guessed, now we have another problem. The notes of the E7 will not resolve properly as now we have different (but perfect) ratios to deal with. The question then becomes, how do you resolve a problem that has no perfect solution?
