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... will not work properly. If you are a musician, you know anything in this posting already. If you read it anyway, please feel free to correct me at points I might be wrong.
Why I started this thread: I find it interesting that harmony in music intervals can be expressed with very simple math. Understanding intervals gives a glimpse into the beauty of music. It cannot explain why a melody is nice, but it does reveal some parts of it.
The octave frequency ratio is 2:1. This is set in stone as we hear the same note regardless if we double or halve the frequency. C sounds like c or c', just in another octave, while for example E will sound as a different note.
The most simple ratio with a fraction is 3:2 = 1.5. This interval is also known as fifth. The fifth is fifth note from the base note, hence the name. The fifth for C is G. Try to play both notes together, it sounds good, now you know why.
The next-simplest interval after 3:2 is 4:3. The complementary interval of the fifth, which is needed to get to an octave, is 4:3, too, as 3:2 ("fifth") * 4:3 = 12:6 = 2:1. This interval is known as fourth, as it is the same like F, the fourth note of the base note C. We could use any other note as base but will stay with C for this posting.
We see a development for usable harmonic intervals: 2:1, 3:2, 4:3, ...
These are the ratios of the overtones to each previous overtone. Perfect overtones are generated by an idealized string. If it oscillates with 100 Hz (roughly G), it also generates vibrations at 200, 300, 400, 500, 600 Hz and so on. (Roughly notes g, d', g', b', d''.) These overtones are normally softer than the base tone, the loudness development of each overtone gives the whole tone its characteristics. So our ears and brains are trained to process overtone ratios. It is no wonder why we consider them harmonic.
![[image loading]](http://i.imgur.com/jl7jh.png) Approximation of the first five overtones for G
If we follow up, we get the 5:4 ratio next, which is also known as major third. The next interval would be 6:5, which is dubbed as minor third (all ratios considered for just intonation.) Major and minor thirds stacked are giving us the fifth again, as it should be. (5:4 * 6:5 = 30:20 = 3:2.) This sounds good so far.
The major triad, which is a major and then a minor third, gives us the known triad. For C major it is C-E-G. The frequency ratio is like 4:5:6, a part of the overtone spectrum with 1:2:3:4:5:6:... ratio. As we hear the same note in any octave as the same note, :6 is the same like :3, and :4 the same like :2 and :1, so in a way we get the sound of 1:2:3:4:5:6. This is why a triad sounds good. Or why it should sound harmonic, I will come to the issues soon.
The distance of the fourth to the fifth is of course a second, or to be more specific, the major second. It is 3:2 / 4:3 = 9:8. The second is the second "full" note above the base note. 9:8 hence is the frequency ratio of a full note. Now the trouble begins. Two seconds are slightly greater than a major third, and six of them are greater than an octave. Three major thirds are smaller than an octave.
With few exceptions, stacked intervals will not lead exactly to the target interval. It just roughly fits. If we tune any instrument for a perfect triad for a given key, we will have to compromise other intervals and also other keys.
The interval between minor and major third should give us the minor second. If we use the aforementioned ratios, it comes out to 25:24 which is of course too small. In just intonation, the minor second is 16:15. But even then two minor seconds are not giving us a major second as we would expect.
This means, even if you have an instrument on which you can play any pitch, you will not be able to play with perfect harmony as stacked intervals will not completely fit. To make it worse, we have to consider the overtone spectra. While slight differences in the base tones are bad enough, the interfering overtones will create many more unharmonic ratios.
![[image loading]](http://i.imgur.com/vRItL.png) The notes are clear, but what about the actual frequencies?
Interestingly enough, there is another relationship between the 12 notes, entirely based on the fifth and the octave. If we begin with any note, lets say with C, and then get the fifth, and then the next fifth and so on, we will have 12 fifths in 7 octaves where the circle completes. This way we hit any note exactly once, though in different octaves. A full-size keyboard with 88 keys does allow to play the entire circle beginning with C contra octave or some subcontra notes.
In reality of course it is not a circle, as 12 perfect fifths are slightly greater than 7 octaves. Since we must obey the octave, as only the octave ensures that we hear the same note, we can tune the fifth slighly smaller than the perfect 3:2 ratio to get exactly 7 octaves with 12 slightly-tuned fifths.
This is done for the equal temperament: We devide an octave to 12 intervals of the same size. Each minor second is 2^(1/12) times pitched higher than the previous key. With this approach, we however have no simple, or even any rational ratios beside the octave. Any ratio is irrational and therefore cannot be harmonic. The fifth (and as the complementary interval also the fourth) will almost fit. But the major third for example is really messed up. However we are used to it as any modern grand and many other instruments use equal temperament.
In equal temperament, the tritone is an interval which is its own complementary interval. Other intonations have other definitions for the tritone but it always lies between fourth and fifth. Equal temperament renders the tritone the most useless interval possible. It does not sound harmonic, nor disharmonic, it just sounds like nothing.
It does not matter if we use the fifth 12 times to create all notes from the two most simple ratios, 2:1 and 3:2, or if we use somewhat more complex ratios like we find in overtone spectra, we will never be able to create a throughout harmonic interval definition which allows to stack intervals properly. The most baffeling thing for me is that the current – and probably best – solution is equal temperament which provides only irrational ratios beside the octave.
If you have an instrument which can be tuned, do you tune it to equal temperament or do you prefer to have at least some intervals in just intonation?
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Yes, there are always compromises tuning instruments. Equal temperament is just the lesser of all evils probably, particularly since real music changes keys all the time. You could tune a piano to be more in tune for one key, but it's not like you can re-tune it in the middle of a piece.
My position is that, if equal temperament is really bothersome, hire some string and wind instrument players or vocalists. Those instruments are easily flexible enough in intonation that the correct ratios and tuning can be achieved on the fly with some listening and small adjustments. edit: though in practice, even world-class musicians get a little out of tune sometimes.
It's mostly just percussive (non-timpani) instruments that have this predicament.
edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
What to do?
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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
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I'm so used to equal temperament that other temperaments sound a little off to me. But that's because equal temperament is everywhere. Besides I play mostly with another instruments that can't be temperated so it would be a mess to play and to listen.
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I actually think this stuff is super cool.
I can remember a couple years ago when a theory and composition major friend of mine was explaining this to me I wasn't sure if he was serious but we sat down and did the math and a lot of stuff has made more musical sense since to me.
Like the second post said this isn't nearly as big of an issue for wind or string instruments as you can adjust your pitch with embrasure or in my case playing the trombone by having "sharp" and "flat" note positions depending on which harmonic you're in relative to the key of C (for which the instrument is correctly tuned).
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
Interestingly I do the same but tune B down from high E which I tune to the octave so my B string always feels a little out of tune cause I'm trying to force exact tuning even though the frets aren't set-up to do that.
Vocally I'm pretty sure most people (with a good sense of pitch) automatically hit the exact interval as the human voice is perfectly capable of "retuning" itself to a new key although I'll note that I know a few people with perfect pitch who get muddled by this stuff as they know exactly where say "g" is but because their singing in a particular key they have to consciously pull the note flat to adapt to the key. I think that would be super awkward.
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On June 11 2011 00:38 Myrmidon wrote:
edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
What to do? Live with the wrong intervals. Even if you could tune your instrument in real time, you cannot avoid wrong intervals unless you never play with sustain and too complex chords. The third is really off with equal temperament but as we have almost perfect fifths and fourth, which carry most of the harmonics at least for my ears, I can live with it.
We also know for example that Eb is actually not D#. We just play the same pitch for these two notes.
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On June 11 2011 01:19 EdSlyB wrote:
I'm so used to equal temperament that other temperaments sound a little off to me. But that's because equal temperament is everywhere. Besides I play mostly with another instruments that can't be temperated so it would be a mess to play and to listen. Same for me.
Interestingly, a real grand sounds better than a digital grand (at least someone I can afford.) An acoustic grand does not allow you to play a pure note because the vibrating strings will have an effect on surrounding strings of other notes. If I play a triad on my Yamaha DGX 630 keyboard, the three notes overlie themselves for the chord. If I play a triad on a real grand, the notes merge into the chord, I just hear the chord and not three notes. It sounds great, even though it is less accurate than on the keyboard.
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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 
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On June 11 2011 00:38 Myrmidon wrote:
Yes, there are always compromises tuning instruments. Equal temperament is just the lesser of all evils probably, particularly since real music changes keys all the time. You could tune a piano to be more in tune for one key, but it's not like you can re-tune it in the middle of a piece.
My position is that, if equal temperament is really bothersome, hire some string and wind instrument players or vocalists. Those instruments are easily flexible enough in intonation that the correct ratios and tuning can be achieved on the fly with some listening and small adjustments. edit: though in practice, even world-class musicians get a little out of tune sometimes.
It's mostly just percussive (non-timpani) instruments that have this predicament.
edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
What to do?
Play a good Instrument e.g Violin
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On June 11 2011 06:55 Kernkraft wrote:
Play a good Instrument e.g Violin 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.
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On June 11 2011 06:43 [F_]aths wrote:Show nested quote +On June 11 2011 00:38 Myrmidon wrote:
edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
What to do? Live with the wrong intervals. Even if you could tune your instrument in real time, you cannot avoid wrong intervals unless you never play with sustain and too complex chords. The third is really off with equal temperament but as we have almost perfect fifths and fourth, which carry most of the harmonics at least for my ears, I can live with it. We also know for example that E b is actually not D#. We just play the same pitch for these two notes.
Sorry if I misworded this, but I think you misinterpreted my question. With many instruments it's practical to be adjusting the pitch in real time and that is what's done in practice (e.g. violin as pointed below above, or pretty much any string or wind instrument), which was covered earlier in my post.
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.
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/head-asplode
So does this mean our biological sense of pitch/tuning is inaccurate?
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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 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.
Lets say you play the major scale with five major seconds and two minor seconds. When you are finished, you are slightly above the octave.
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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.
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On June 11 2011 07:05 ToxNub wrote:
/head-asplode
So does this mean our biological sense of pitch/tuning is inaccurate? Actually it means that there is no rational number between 1 and 2, for which the n-th exponentiation (where n is a positive integer) could be expressed with 2^m (where m is another positive integer.)
In other words, you cannot stack x equal, but rational intervals to an octave. The crux is that we need both the octave as well as rational – and if possible, quite simple – ratios to have truly harmonic sound. But we only can approximate it.
The current common solution uses equal, but irrational intervals (save for the octave, of course.) This is problematic but has its advantages.
There exist a number of five-tone-systems. (For me, this is a mystery for itself. Pentatonic scales can be found in any culture. I did not have yet discovered why they are so common.) Our western music often relies on a system with 12 notes per octave. Other systems use 19 or even more notes per octave but all systems are just approximations for exact harmonic ratios.
Our western music which is based on 12 minor seconds, knows a number of scales. There are enharmonic scales possible (C# major = Db major) but depending on how you would construct the scales, they could actually be different. Only equal temperament allows perfect enharmonics. We trick our senses to be able to switch scales within a piece of music.
If you listen carefully while you play a grand you actually can hear that some pitches are slightly off. It is not a biological, but mathematical issue, though an interesting one as you will have to sacrifice some properties to gain others.
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I expected to see some mention of logarithmic scaling of frequencies in your discussion of harmonics.
I was disappointed.
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On June 11 2011 09:26 MozzarellaL wrote:
I expected to see some mention of logarithmic scaling of frequencies in your discussion of harmonics.
I was disappointed.
He essentially did when referring to equal temperament tuning. i.e. exponential tuning.
F_relative = F_base*2^(n/12) where n is the number of semitones between F_relative and F_base.
Put this equation on a log base 2 scaled graph and it will look linear.
Also, it is implicit in his discussion of ratios. i.e. ratios are relative to a fundamental frequency, therefore as the fundamental frequency increases, the rate at which the fundamental some number of semitones relative to it changes at a logarithmic rate.
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On June 11 2011 09:26 MozzarellaL wrote:
I expected to see some mention of logarithmic scaling of frequencies in your discussion of harmonics.
I was disappointed.
I assumed this also. My band director has always told us these things but I never really understood the theory behind it until now, quite interesting.
This is the reason only instruments where pitch can be altered whilst playing (aka Wind instruments) can play perfectly in tune together.
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Interesting, but probably not too informative for musicians.
I can't tune my instrument, but if I was a singer I would totally use just intonation where ever possible. It just sounds so amazing.
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EPT
