Whenever I read harmony books I get ticked off at the sophomoric explanations of harmonic phenomena that are ever present. I should correct that and use the past tense, actually, because the harmony books in which I find these tend to be older, apparently before the concept of observation was developed. Hindemith, in particular, talks about combination tones and their overtones to discuss consonance, and I find that rather silly. He also says that intervals have roots.
What is the root of a given interval? For the major third, for instance, it's the bottom member; for the minor sixth, its inversion, the top member. How does he know? See, I disagree with him, because one would have to consider the interval in a vacuum to identify a root, and an interval is never in a vacuum! I hold that context is critical, and that the tonal center forms a third note around which the two notes of the interval gravitate. When trying to figure out the "meaning" of an interval, the (local) key is an inseparable part of the problem. Hence we have a three-body problem, which complicates the simplicity of a two-note interval with necessary ambiguity.
My favorite example is the major sixth, because either note can be the "root". What's worse, if the major sixth is G E and the key of C is well-understood, the root is the C, a note not even in the interval! If an E minor harmony is implied, E is the root; if a dominant function in the key of C is implied, G is the root. If the interval is present by itself, without any context -- or, rather, with itself as its context -- a trick of the ear can change the root, like the spinning dancer or the cube that looks like it's either coming out of the page or into the page. The beautiful simplicity of the analysis of only two tones is, sadly, not to be.
I hope that this helps dispel the notion that the overtone series has much to do with harmony beyond providing the fifth. That notion needs dispelling, and any little bit counts.
Showing posts with label harmony. Show all posts
Showing posts with label harmony. Show all posts
Tuesday, January 29, 2008
Monday, December 10, 2007
In Which I Rant About The Overtone Series
GAH. I'm reading through Hindemith's "The Craft of Musical Composition", and like every other book written about music, no introduction is complete without a discussion of the overtone series. And somehow, maybe because they're musicians rather than physicists, it's ALWAYS unsatisfactory. I've NEVER seen the overtone series explained correctly except in the Griffiths Intro to Quantum Mechanics book. Yeah, QM, not music theory. It's ridiculous. The overtone series isn't what it claims to be, and explanations of other phenomena involving it are just wrong. I don't know why. So here we go.
There is no such thing as the overtone series. There is AN overtone series, but not THE overtone series, except in mathematics. That description makes a lot of sense to someone who has studied Fourier analysis and less sense to someone who hasn't, but it's worth a try. Pretend that there is a perfect string with constant density and infinitesimal width and that it has some finite length L and is held taut and fast at both ends. This is an idealization, remember. Now, pull the string in some shape with very small deviations from the resting state. Regardless of what that shape is, and this is the magic of it, you can write it as a sum of sine functions. This is called a Fourier decomposition, or a Fourier series. Those sine functions will look like your normal sine wave, but only those sine waves that have value 0 at the endpoints will be in the series -- so sin(pi*L), sin(2pi*L), sin(3pi*L), ..., sin(n*pi*L), for any integer n. For your perfect idealized string, the relationship between wavelength and frequency is such that if the whole string (sin(pi*L)) vibrates at a particular frequency (which depends only on the string's length, tension, and density), half the string (sin(2pi*L)) will vibrate at double the frequency, a third of the string (sin(3pi*L)) will vibrate at triple the frequency, and so on. So WHATEVER you do to the string at the start (assuming your stretching is very small), it will be vibrating in some weird pattern, but that weird pattern will be a sum of the frequencies of the whole string, half, one third, one fourth, and so on. If the whole string vibrates at a frequency of 64 Hz, after you make your weird initial condition, you will hear the 64 Hz, twice that (128 Hz), three times that (192 Hz), and so on. You'll actually hear all those notes. That is THE overtone series. It's described as C', C, G, c, e, g, bb, c', ..., with some exact pitches, where the G's are a little tiny bit sharp, the E's are somewhat flat, the Bb's are very flat, and so on, and authors make various claims as to its usefulness in various situations and as various explanations.
Of course, what I described is the perfect string. Real strings don't work like that, though they come close. Piano strings especially don't work like that, and if you look at them, you'll see that they aren't just plain strings, since piano makers try to correct those effects. The perfect string, where the overtones are integer multiples of the fundamental, is a linear medium. Real instruments aren't. The overtone series of a crotale, for instance, which is a round metallic pitched plate, is nothing like a string's -- since it's a circle, you can't break down a perturbation into sines; you have to use a different set of functions at different frequencies. Non-pitched instruments are even more nonlinear. Strings come close to being linear, but aren't exactly. Not even the human ear is exactly linear in its response. Vibrating air columns aren't. Reeds certainly aren't, and they respond differently at different frequencies.
This perfect string is used to describe the sounds that you hear from an instrument. An instrument produces, along with the fundamental that you hear clearly, a series of overtones at varying strengths. An oboe has strong overtones, for instance, which explains its rich sound, while the clarinet has weak ones, which explains it's dark quality. But since no instrument is really linear, this overtone series isn't exactly the one described. The set of frequencies sounded is called a spectrum, and it doesn't necessarily look like the overtone series described. It's just not perfectly in tune.
The overtone series is considered the basis of all everything. It's annoying, because, well, it isn't. Only the octave and fifth are actually important. The overtone third isn't the same third as the one in our triads, and this is obvious when we consider the minor triad -- it's not more dissonant than the major triad, but the minor third is much farther away from the "natural" third of the harmonic series. The third in our triads is, I believe, psychologically conditioned in us rather than an aural phenomenon like the octave and fifth. When those are in tune, mathematical cancellations in the sound waves (and the sound waves in the air themselves ARE linear, unlike the ones on the string) are audible, and the interval is perceived as "open". If the third is tuned correctly, it can also sound "open", and an in-tune major ninth can also sound "open". That is a DIFFERENT characteristic from tonal character. We can see from the scales of non-Western cultures that thirds can actually vary widely in pitch, and that major doesn't necessarily mean happy, and so on. The third isn't really a function of the overtone scale. The other notes aren't, either! At least not the way we do it. We take a fifth -- actually, something very close to the fifth -- and stack it repeatedly until we get to the original note (assuming everything is octave-reduced, of course), and THAT is what our system of harmony is based on, not the overtone series -- or rather, only indirectly the overtone series. Luckily, 3^12 = 531441 is very close to 2^19 = 524288, a difference of 1.36% (or 1.35%, depending on how you count it), so the perfect fifth needs to be fudged only a tiny bit to make it fit the octaves. There are twelve of them, so we get 12 evenly spaced tones in an octave, each therefore having a frequency of 2^(1/12) the previous pitch. This is NOT the overtone series! The overtone series does affect consonance and dissonance, to some extent, but only in the "beats" between notes of similar pitch -- play a C and a C#, and the waves will interfere and sound somewhat ugly. Play a C and a B, and it won't be so bad, but there will be beats between the B and the first overtone of C to create the dissonance. The dissonance can be hidden, say by playing C E G B, in which case the ear will hear a C major chord and an E minor chord rather than a strident major seventh interval. However, only the first few overtones (essentially up to the perfect fifth, not even so far as the "perfect" third) actually have this aural property to a reasonable strength. The higher terms in the overtone series are meaningless in this respect.
Finally, people try to say that the most stable voicings for chords have a spacing like that of the overtone series, wide at the bottom and tight at the top. THIS IS A CROCK OF BULLSHIT. The reason for this has nothing to do with the overtone series, except perhaps that some of the basic principles are the same. Two notes in the lower register are MUCH closer in frequency than the same two notes in a higher register due to the logarithmic nature of the scale. Two low C's are 32 Hz apart, from 32 Hz to 64 Hz, whereas two high C's are, say, 4096 Hz apart, from 4096 Hz to 8192 Hz. It's 8 octaves from the 32 Hz to the 8192 Hz, by the way. Part of the ear's pitch resolution is based on linear frequency, not the logarithm of it, which is what musical pitch is based on. This changes for the very high notes, of course, at the threshhold of hearing, but it's still very clear in, say, the piccolo range. So at the upper end of things, close pitches can be resolved by the ear much more easily than at the lower end of things. Furthermore, the overtone series DOES play a role, but it's only that part of the reason for the dissonance of low pitches is that the overtone series clash. They shouldn't clash much, but remember that the instruments are nonlinear, so they end up clashing quite a bit, especially on the piano. On a string or electric bass, which produces rather pure pitches, the clash is due to the bad resolution rather than the overtones, since the overtones are mostly absent.
OK, I think I've exhausted my rant. Feel free to disagree, of course.
There is no such thing as the overtone series. There is AN overtone series, but not THE overtone series, except in mathematics. That description makes a lot of sense to someone who has studied Fourier analysis and less sense to someone who hasn't, but it's worth a try. Pretend that there is a perfect string with constant density and infinitesimal width and that it has some finite length L and is held taut and fast at both ends. This is an idealization, remember. Now, pull the string in some shape with very small deviations from the resting state. Regardless of what that shape is, and this is the magic of it, you can write it as a sum of sine functions. This is called a Fourier decomposition, or a Fourier series. Those sine functions will look like your normal sine wave, but only those sine waves that have value 0 at the endpoints will be in the series -- so sin(pi*L), sin(2pi*L), sin(3pi*L), ..., sin(n*pi*L), for any integer n. For your perfect idealized string, the relationship between wavelength and frequency is such that if the whole string (sin(pi*L)) vibrates at a particular frequency (which depends only on the string's length, tension, and density), half the string (sin(2pi*L)) will vibrate at double the frequency, a third of the string (sin(3pi*L)) will vibrate at triple the frequency, and so on. So WHATEVER you do to the string at the start (assuming your stretching is very small), it will be vibrating in some weird pattern, but that weird pattern will be a sum of the frequencies of the whole string, half, one third, one fourth, and so on. If the whole string vibrates at a frequency of 64 Hz, after you make your weird initial condition, you will hear the 64 Hz, twice that (128 Hz), three times that (192 Hz), and so on. You'll actually hear all those notes. That is THE overtone series. It's described as C', C, G, c, e, g, bb, c', ..., with some exact pitches, where the G's are a little tiny bit sharp, the E's are somewhat flat, the Bb's are very flat, and so on, and authors make various claims as to its usefulness in various situations and as various explanations.
Of course, what I described is the perfect string. Real strings don't work like that, though they come close. Piano strings especially don't work like that, and if you look at them, you'll see that they aren't just plain strings, since piano makers try to correct those effects. The perfect string, where the overtones are integer multiples of the fundamental, is a linear medium. Real instruments aren't. The overtone series of a crotale, for instance, which is a round metallic pitched plate, is nothing like a string's -- since it's a circle, you can't break down a perturbation into sines; you have to use a different set of functions at different frequencies. Non-pitched instruments are even more nonlinear. Strings come close to being linear, but aren't exactly. Not even the human ear is exactly linear in its response. Vibrating air columns aren't. Reeds certainly aren't, and they respond differently at different frequencies.
This perfect string is used to describe the sounds that you hear from an instrument. An instrument produces, along with the fundamental that you hear clearly, a series of overtones at varying strengths. An oboe has strong overtones, for instance, which explains its rich sound, while the clarinet has weak ones, which explains it's dark quality. But since no instrument is really linear, this overtone series isn't exactly the one described. The set of frequencies sounded is called a spectrum, and it doesn't necessarily look like the overtone series described. It's just not perfectly in tune.
The overtone series is considered the basis of all everything. It's annoying, because, well, it isn't. Only the octave and fifth are actually important. The overtone third isn't the same third as the one in our triads, and this is obvious when we consider the minor triad -- it's not more dissonant than the major triad, but the minor third is much farther away from the "natural" third of the harmonic series. The third in our triads is, I believe, psychologically conditioned in us rather than an aural phenomenon like the octave and fifth. When those are in tune, mathematical cancellations in the sound waves (and the sound waves in the air themselves ARE linear, unlike the ones on the string) are audible, and the interval is perceived as "open". If the third is tuned correctly, it can also sound "open", and an in-tune major ninth can also sound "open". That is a DIFFERENT characteristic from tonal character. We can see from the scales of non-Western cultures that thirds can actually vary widely in pitch, and that major doesn't necessarily mean happy, and so on. The third isn't really a function of the overtone scale. The other notes aren't, either! At least not the way we do it. We take a fifth -- actually, something very close to the fifth -- and stack it repeatedly until we get to the original note (assuming everything is octave-reduced, of course), and THAT is what our system of harmony is based on, not the overtone series -- or rather, only indirectly the overtone series. Luckily, 3^12 = 531441 is very close to 2^19 = 524288, a difference of 1.36% (or 1.35%, depending on how you count it), so the perfect fifth needs to be fudged only a tiny bit to make it fit the octaves. There are twelve of them, so we get 12 evenly spaced tones in an octave, each therefore having a frequency of 2^(1/12) the previous pitch. This is NOT the overtone series! The overtone series does affect consonance and dissonance, to some extent, but only in the "beats" between notes of similar pitch -- play a C and a C#, and the waves will interfere and sound somewhat ugly. Play a C and a B, and it won't be so bad, but there will be beats between the B and the first overtone of C to create the dissonance. The dissonance can be hidden, say by playing C E G B, in which case the ear will hear a C major chord and an E minor chord rather than a strident major seventh interval. However, only the first few overtones (essentially up to the perfect fifth, not even so far as the "perfect" third) actually have this aural property to a reasonable strength. The higher terms in the overtone series are meaningless in this respect.
Finally, people try to say that the most stable voicings for chords have a spacing like that of the overtone series, wide at the bottom and tight at the top. THIS IS A CROCK OF BULLSHIT. The reason for this has nothing to do with the overtone series, except perhaps that some of the basic principles are the same. Two notes in the lower register are MUCH closer in frequency than the same two notes in a higher register due to the logarithmic nature of the scale. Two low C's are 32 Hz apart, from 32 Hz to 64 Hz, whereas two high C's are, say, 4096 Hz apart, from 4096 Hz to 8192 Hz. It's 8 octaves from the 32 Hz to the 8192 Hz, by the way. Part of the ear's pitch resolution is based on linear frequency, not the logarithm of it, which is what musical pitch is based on. This changes for the very high notes, of course, at the threshhold of hearing, but it's still very clear in, say, the piccolo range. So at the upper end of things, close pitches can be resolved by the ear much more easily than at the lower end of things. Furthermore, the overtone series DOES play a role, but it's only that part of the reason for the dissonance of low pitches is that the overtone series clash. They shouldn't clash much, but remember that the instruments are nonlinear, so they end up clashing quite a bit, especially on the piano. On a string or electric bass, which produces rather pure pitches, the clash is due to the bad resolution rather than the overtones, since the overtones are mostly absent.
OK, I think I've exhausted my rant. Feel free to disagree, of course.
Thursday, November 29, 2007
A Scale! Ahava Rabbah!
http://en.wikipedia.org/wiki/Ahava_Rabbah
I'm a Jewish atheist, or Humanist, or Pastafarian, or whatever. I don't believe in gods, but I'm Jewish. Figure it out if you feel so inclined (Wikipedia can help). But the Ahava Rabbah scale is one of my favorite scales (mentioned in the last post as well), also known, according to the article linked above, as the Spanish phrygian, Jewish scale, Hungarian scale, or phrygian dominant. I'll stick with Ahava Rabbah, at least for now, but it really is a phrygian with raised 3rd.
I love this scale. I suppose the augmented second sounds "mystical", which in today's postpostmodern age is quite the cliche, but to me it really does. Listening to liturgical music that uses it makes me feel, well, elevated, which I'm sure the religious would interpret as "closer to Hashem". (This is because the religious can't write "Adonai", as that is using the name of God in vain -- instead, they use "Hashem", meaning "the name", in writing and singing, and they use "Adonai", "Lord", in actual prayer. They will sometimes use "Adoshem" when emphasis on pronunciation is necessary, and "Elokeinu" for "Eloheinu". Now you know.) The point is that I love this scale, and modulating to it is an interesting touch, much like that major chord in a minor piece that is like sunlight.
The typical cadences in Ahava Rabbah are not V-I as might be expected, since the 5 chord is actually vo rather than the v that you can change into a V. Instead, the cadences are bII-I and bvii-I (the convention is confusing and I can't be bothered to follow it; hopefully this is clearly referring to the chords on the 2 and 7 of this scale). bvii is especially nice, because since the important half step occurs between 3 and 4, an improvisatory-style melody would tend to hover around there, and the 7th degree would be a refreshing note to signal the cadence. In some chanting schemes this is actually done. A nice melodic cadence using bvii-I would be, in C: Bb' F E Db C; the Shabbat prayer Shalom Aleichem uses this, though the position of the tonal center makes this a half cadence instead. In fact, this is just the conventional phrygian cadence of Western tonal music, iv6-V, but with the function of an authentic cadence (and since this is used in mostly monophonic liturgical music anyway, with no attention to inversion numbers).
But yeah, Ahava Rabah, awesome scale.
I'm a Jewish atheist, or Humanist, or Pastafarian, or whatever. I don't believe in gods, but I'm Jewish. Figure it out if you feel so inclined (Wikipedia can help). But the Ahava Rabbah scale is one of my favorite scales (mentioned in the last post as well), also known, according to the article linked above, as the Spanish phrygian, Jewish scale, Hungarian scale, or phrygian dominant. I'll stick with Ahava Rabbah, at least for now, but it really is a phrygian with raised 3rd.
I love this scale. I suppose the augmented second sounds "mystical", which in today's postpostmodern age is quite the cliche, but to me it really does. Listening to liturgical music that uses it makes me feel, well, elevated, which I'm sure the religious would interpret as "closer to Hashem". (This is because the religious can't write "Adonai", as that is using the name of God in vain -- instead, they use "Hashem", meaning "the name", in writing and singing, and they use "Adonai", "Lord", in actual prayer. They will sometimes use "Adoshem" when emphasis on pronunciation is necessary, and "Elokeinu" for "Eloheinu". Now you know.) The point is that I love this scale, and modulating to it is an interesting touch, much like that major chord in a minor piece that is like sunlight.
The typical cadences in Ahava Rabbah are not V-I as might be expected, since the 5 chord is actually vo rather than the v that you can change into a V. Instead, the cadences are bII-I and bvii-I (the convention is confusing and I can't be bothered to follow it; hopefully this is clearly referring to the chords on the 2 and 7 of this scale). bvii is especially nice, because since the important half step occurs between 3 and 4, an improvisatory-style melody would tend to hover around there, and the 7th degree would be a refreshing note to signal the cadence. In some chanting schemes this is actually done. A nice melodic cadence using bvii-I would be, in C: Bb' F E Db C; the Shabbat prayer Shalom Aleichem uses this, though the position of the tonal center makes this a half cadence instead. In fact, this is just the conventional phrygian cadence of Western tonal music, iv6-V, but with the function of an authentic cadence (and since this is used in mostly monophonic liturgical music anyway, with no attention to inversion numbers).
But yeah, Ahava Rabah, awesome scale.
Sunday, November 18, 2007
Scales, Harmonic and Melodic
There are two similar but somewhat different things we call "scales", and sometimes those two things are different and sometimes the same. There is the harmonic scale and there is the melodic scale, and they are each sets of pitches from which one draws harmony and melody, respectively. The melodic scale is an ORDERED set of pitches, and the harmonic scale is not -- this is one of the main distinctions between them. This is important when we consider what the silly "common practice" people call the minor key.
It took me a while to realize that the three minor "scales", natural, harmonic, and melodic (and I don't mean the jazz melodic scale, I mean the CP one), were not all meant to be played. My first encounter with scales, as such, was in beginning band in sixth grade, or maybe in my piano computer program in fifth grade, and they were major. I knew what the minor keys were, but I never had to play minor scales. When I saw "harmonic minor" and "melodic minor" in a book from the school media center in sixth grade, or maybe someplace else, I saw them as curiosities, and I liked the sound of the harmonic minor scale, and I wondered at this melodic scale thing that goes up and down differently.
Eventually, in college, while taking a music theory course, I realized several things. First, the augmented second in the harmonic minor is essentially wrong. Second, the leading tone HAS to be a half step below the tonic, which is why the 7th is raised. Therefore, the 6th needs to be raised as well if you're moving to the raised 7th from below. So the minor key, with a flatted 7th and 6th, needs to be modified. The V chord has a leading tone when it goes to i, so it has to be V rather than v. Everything else still works, so the scale with this modification is the harmonic scale. The b6 and b7 still want to be used, but they can't be used if the tonic is approached by step from below. So if you're going to PLAY the scale, you have to raise the 6th and 7th on the way up, but going down you can keep the minor key feeling.
Personally, I find this all very silly, and enough other people find it silly, so we play just one minor scale: the melodic minor. When we play scales, which are a useful exercise for instrumental technique, the melodic minor should be played (if we are practicing for CP music), since those are the note patterns that we are likely to see. There is no reason to practice playing the harmonic minor scale -- this is because the harmonic minor scale was never meant to be played! The harmonic minor is the set of notes from which we draw the harmony. There are really only five chords, i, ii0, iv, V, VI, in the minor key (though with melodic minor melodic materials, you can include also ii and IV, but these are essentially chromatic alterations). III and VII don't really have functions so much, and even VI is a little iffy, so we can essentially limit ourselves to using the notes from the harmonic scale. The melody, conceived entirely separately and heard entirely separately (this is not true, but let's consider it as such), uses a different set of notes, and given the rules of CP, they are different going up from going down.
The melody and harmony, let me stress, use DIFFERENT SETS OF NOTES. There are plenty of examples, of course.
First, the blues scale. Mark Levine's Jazz Theory Book claims that nobody understands how it manages to work, but I claim that I do. (: In the key of C, the blues scale is C Eb F F#/Gb G Bb C. The twelve bar blues progression is C7 C7 C7 C7 F7 F7 C7 C7 G7 F7 C7 C7. Now, let's consider ripping the nonfuctional dominant 7ths out first, so we have C C C C F F C C G7 F C C. This uses the harmonic scale C D E F G A B, so we have that as a harmonic scale and C Eb F F#/Gb G Bb C as the melodic scale. The only common notes here are the tonic, subdominant, and dominant, the roots of the chords! We have here an extremely inflexible system, since this blues scale depends in a way on the fact that we're not using it freely but over a progression that essentially works like the traditional 12-bar blues. We can't really modulate in the scale; the scale has no symmetries! Of course, that's also true of the normal major scale; we have to introduce new notes to modulate. In this case, though, the tendency to the tonic is very strong from the Eb and the Bb. The Gb also tends very strongly down in a way that leads to the tonic. NONE OF THOSE THREE NOTES NEEDS TO BE IN TUNE. They aren't in the harmonic scale, and they form no harmonies. They can be wildly flat and they'll work just as well! (If C7 and F7 are used, though, the 7ths in those chords should probably be the same as the ones in the melody, since those are essentially melodic tones added to the chords rather than harmonic/functional ones).
The blues scale is essentially a harmony that works with bitonality over the C major harmonies in the progression. Let's see how the notes match up, ignoring common tones. Over a C chord, we have the Eb clashing with the E. The Eb has a strong melodic pull to the tonic, so you can hear it as an appoggiatura, essentially. The F clashes with the E as well, but that pulls to the Eb which pulls to the tonic. The Gb clashes with the G, but that pulls to the F which pulls to the Eb which pulls to the tonic. The F# pulls to the G, which is stable. The Bb pulls to the tonic or the fifth, but it doesn't actually clash with anything. The tonic essentially remains a pedal through the other chords, and the blues scale is essentially an elaboration of the tonic pedal. You can use the same logic to consider the notes over the other chords. Over F, then, you have C being stable as the fifth, Eb pulling down to it, F being stable as the tonic, Gb pulling down to it, F# not really working, G being the fifth of the pedal C blues scale, and Bb pulling the same as over the C chord in the tonic pedal scale. Over G7, C is the pedal, Eb tends to C in the pedal, F if the seventh, G is the root, F# tends to it, Gb tends down to the F, and the Bb tends either way in the pedal. So the clashes aren't really clashes because the harmony and the melody are heard separately.
Another example are the modes of the harmonic minor scale, taken melodically. The interesting ones in Jewish music are phrygian #3, dorian #4, and aeolian #7 (the original harmonic minor); they have names in Hebrew that I don't remember. The others (locrian #6, ionian #5, lydian #2, and, uh, mixolydian #1 which is really weird) don't have very useful corresponding harmonic scales, that is, the melody notes can't really be used to form harmonies very easily. You could make a doubly "harmonic" scale as well by taking a phrygian #3#7 or an aeolian #4#7. In these cases, the augmented second is a characteristic interval, though it doesn't have to be -- it might well be treated as a leap rather than a step.
The phrygian #3 scale in particular is a MAJOR scale. The tonic chord is major! The harmonic scale for it is certainly phrygian #3, but the melodic scale isn't, if you don't like augmented seconds. It's just as well to raise the 2nd going up and lower the 3rd coming down if it makes sense in the music when stepwise motion is called for. Of course, a solution is to avoid stepwise motion in that direction, never going lower than the raised 3rd in a scalar passage or higher than the 2nd, and leaping across the gap when necessary. An example from Fiddler on the Roof, "If I were a rich man", in C: G F G F E, C. E F G F G F E F G A Bb A Bb A G. This is a somewhat bad example because it isn't really in this mode, but see how it just avoids the Db or D altogether? This is actually mixolydian (which is another Jewish mode as well), of course. A better example, still from Fiddler on the Roof (it's really the easiest source for this, even though it's not particularly authentic), is the fathers' theme in "Tradition", also here in C: C CDE F GF#GAbG E FEFGF Db ED#EFE G C CDE F GF#GAbG EFEFGF Db C. In this case the majorness and the Db are very clear. There is one occasion of Db going to E, but it's certainly as a leap rather than a scale. When E has to be approached by step from below, the passing tone is D, not Db, but when the harmony is a bII Db chord (or a bvii Bbm chord), the Db is necessary. This way you can have music in the mode that still has smooth voice leading.
And so it is that harmonic and melodic scales are not only different notes but different things altogether. Comments?
It took me a while to realize that the three minor "scales", natural, harmonic, and melodic (and I don't mean the jazz melodic scale, I mean the CP one), were not all meant to be played. My first encounter with scales, as such, was in beginning band in sixth grade, or maybe in my piano computer program in fifth grade, and they were major. I knew what the minor keys were, but I never had to play minor scales. When I saw "harmonic minor" and "melodic minor" in a book from the school media center in sixth grade, or maybe someplace else, I saw them as curiosities, and I liked the sound of the harmonic minor scale, and I wondered at this melodic scale thing that goes up and down differently.
Eventually, in college, while taking a music theory course, I realized several things. First, the augmented second in the harmonic minor is essentially wrong. Second, the leading tone HAS to be a half step below the tonic, which is why the 7th is raised. Therefore, the 6th needs to be raised as well if you're moving to the raised 7th from below. So the minor key, with a flatted 7th and 6th, needs to be modified. The V chord has a leading tone when it goes to i, so it has to be V rather than v. Everything else still works, so the scale with this modification is the harmonic scale. The b6 and b7 still want to be used, but they can't be used if the tonic is approached by step from below. So if you're going to PLAY the scale, you have to raise the 6th and 7th on the way up, but going down you can keep the minor key feeling.
Personally, I find this all very silly, and enough other people find it silly, so we play just one minor scale: the melodic minor. When we play scales, which are a useful exercise for instrumental technique, the melodic minor should be played (if we are practicing for CP music), since those are the note patterns that we are likely to see. There is no reason to practice playing the harmonic minor scale -- this is because the harmonic minor scale was never meant to be played! The harmonic minor is the set of notes from which we draw the harmony. There are really only five chords, i, ii0, iv, V, VI, in the minor key (though with melodic minor melodic materials, you can include also ii and IV, but these are essentially chromatic alterations). III and VII don't really have functions so much, and even VI is a little iffy, so we can essentially limit ourselves to using the notes from the harmonic scale. The melody, conceived entirely separately and heard entirely separately (this is not true, but let's consider it as such), uses a different set of notes, and given the rules of CP, they are different going up from going down.
The melody and harmony, let me stress, use DIFFERENT SETS OF NOTES. There are plenty of examples, of course.
First, the blues scale. Mark Levine's Jazz Theory Book claims that nobody understands how it manages to work, but I claim that I do. (: In the key of C, the blues scale is C Eb F F#/Gb G Bb C. The twelve bar blues progression is C7 C7 C7 C7 F7 F7 C7 C7 G7 F7 C7 C7. Now, let's consider ripping the nonfuctional dominant 7ths out first, so we have C C C C F F C C G7 F C C. This uses the harmonic scale C D E F G A B, so we have that as a harmonic scale and C Eb F F#/Gb G Bb C as the melodic scale. The only common notes here are the tonic, subdominant, and dominant, the roots of the chords! We have here an extremely inflexible system, since this blues scale depends in a way on the fact that we're not using it freely but over a progression that essentially works like the traditional 12-bar blues. We can't really modulate in the scale; the scale has no symmetries! Of course, that's also true of the normal major scale; we have to introduce new notes to modulate. In this case, though, the tendency to the tonic is very strong from the Eb and the Bb. The Gb also tends very strongly down in a way that leads to the tonic. NONE OF THOSE THREE NOTES NEEDS TO BE IN TUNE. They aren't in the harmonic scale, and they form no harmonies. They can be wildly flat and they'll work just as well! (If C7 and F7 are used, though, the 7ths in those chords should probably be the same as the ones in the melody, since those are essentially melodic tones added to the chords rather than harmonic/functional ones).
The blues scale is essentially a harmony that works with bitonality over the C major harmonies in the progression. Let's see how the notes match up, ignoring common tones. Over a C chord, we have the Eb clashing with the E. The Eb has a strong melodic pull to the tonic, so you can hear it as an appoggiatura, essentially. The F clashes with the E as well, but that pulls to the Eb which pulls to the tonic. The Gb clashes with the G, but that pulls to the F which pulls to the Eb which pulls to the tonic. The F# pulls to the G, which is stable. The Bb pulls to the tonic or the fifth, but it doesn't actually clash with anything. The tonic essentially remains a pedal through the other chords, and the blues scale is essentially an elaboration of the tonic pedal. You can use the same logic to consider the notes over the other chords. Over F, then, you have C being stable as the fifth, Eb pulling down to it, F being stable as the tonic, Gb pulling down to it, F# not really working, G being the fifth of the pedal C blues scale, and Bb pulling the same as over the C chord in the tonic pedal scale. Over G7, C is the pedal, Eb tends to C in the pedal, F if the seventh, G is the root, F# tends to it, Gb tends down to the F, and the Bb tends either way in the pedal. So the clashes aren't really clashes because the harmony and the melody are heard separately.
Another example are the modes of the harmonic minor scale, taken melodically. The interesting ones in Jewish music are phrygian #3, dorian #4, and aeolian #7 (the original harmonic minor); they have names in Hebrew that I don't remember. The others (locrian #6, ionian #5, lydian #2, and, uh, mixolydian #1 which is really weird) don't have very useful corresponding harmonic scales, that is, the melody notes can't really be used to form harmonies very easily. You could make a doubly "harmonic" scale as well by taking a phrygian #3#7 or an aeolian #4#7. In these cases, the augmented second is a characteristic interval, though it doesn't have to be -- it might well be treated as a leap rather than a step.
The phrygian #3 scale in particular is a MAJOR scale. The tonic chord is major! The harmonic scale for it is certainly phrygian #3, but the melodic scale isn't, if you don't like augmented seconds. It's just as well to raise the 2nd going up and lower the 3rd coming down if it makes sense in the music when stepwise motion is called for. Of course, a solution is to avoid stepwise motion in that direction, never going lower than the raised 3rd in a scalar passage or higher than the 2nd, and leaping across the gap when necessary. An example from Fiddler on the Roof, "If I were a rich man", in C: G F G F E, C. E F G F G F E F G A Bb A Bb A G. This is a somewhat bad example because it isn't really in this mode, but see how it just avoids the Db or D altogether? This is actually mixolydian (which is another Jewish mode as well), of course. A better example, still from Fiddler on the Roof (it's really the easiest source for this, even though it's not particularly authentic), is the fathers' theme in "Tradition", also here in C: C CDE F GF#GAbG E FEFGF Db ED#EFE G C CDE F GF#GAbG EFEFGF Db C. In this case the majorness and the Db are very clear. There is one occasion of Db going to E, but it's certainly as a leap rather than a scale. When E has to be approached by step from below, the passing tone is D, not Db, but when the harmony is a bII Db chord (or a bvii Bbm chord), the Db is necessary. This way you can have music in the mode that still has smooth voice leading.
And so it is that harmonic and melodic scales are not only different notes but different things altogether. Comments?
Thursday, October 18, 2007
A Chord
When I was listening to my old college singing group (the Harvard Noteables) rehearsing last year, one thing struck me quite a bit. The seniors were singing "Hear My Song", from Jason Robert Brown's Songs for a New World, and towards the end, from 5:03 to 5:08 if you have the recording, they sang a wild chord! I keep thinking now that, if I were writing a marching show of Songs for a New World, this chord would be the point in the closer where the entire audience starts screaming. So what is it?
That section of the song is in the key of B. I'm doing this by ear, by the way, so feel free to correct me. This is Jason Robert Brown, so his chords are all 2's and 4's -- in the key of C, these would be the tonic CDG, the subdominant CFG (though the bass player would most likely play the F), and the dominant GCD (which actually makes it equal to a 64 chord). While this looks rather drab, with a C present in the important chords, it's important to note that his music's harmonic interest comes from having other chords besides those three. My point is really about the character of the primary chords he uses.
Back to the music. Starting at 4:50, we have repeated melodic precadential material over a dominant F#4 chord (or B2/F# if you prefer, but there is ambiguity between the V and I64 feelings). The bass alternates the F# with an E, so this could also be a quartal dominant 7th in whatever inversion it is that has the 7th in the bass (3rd in CP theory -- CP being Common Practice -- but that isn't very meaningful here). After three bars, this is followed by a chord with an A in the bass guitar and soprano, a B in the man's voice, and possibly an E somewhere, for an A2 chord (though I hear it as an A chord with an added 9, since that B is important). The next bar features a B2 chord, with mostly parallel motion (though the male voice goes to a different note to jump instead of moving in parallel), and the next bar, that wild chord, has a G in the bass, an F# in the man, D in the woman. I think. And possibly an A someplace else. If I can get a hold of the score, I'll check. But at least on the piano, G A F# D sounds about right, with a ninth between the G and the A. This looks like a D/G chord, which would be -- ahem -- bIII/b6^, if that makes any sense. However, it's clearly a G9 chord of some sort, in this case a Gmaj9 with no third. It's hard to fit non-CP chords into CP conventions! In that case, this is a bVI chord, but there is no third because that would make a minor chord with the F# and D and that would change the energy of that chord. But it's ambiguous enough that it could be both a D/G and a Gmaj9.
Where does the chord get this energy? Partly from that beautiful major ninth, I think. The major ninth, where the top note is tonicized, is a very harmonic interval that does not resolve as might be expected, since the bass and not the soprano is the source of tension. As I hear it, it has no directional pull, but it clearly can't end a piece. Another example is in Meadowlark. This major ninth could actually be considered a diminished tenth, possibly. I don't know. One possible resolution is for both notes to resolve a half step in, to an octave; an example would be (in diminished 10th form) D# f to E e, but the idea can be expanded to resolve to E g instead, which is the same harmony in this case. Consider the following progression of two-note chords: F e, Bb d, F e, D# f, E g. The D# f chord is pretty much an Eb9, but since it's so foreign to the key, it might be better to interpret it as D# dim10, or however you want to write it. It's the inverse of the augmented 6th, but since it's a major 9th it sounds much more open (since the 9th is made up of two open fifths stacked) and therefore much more powerful. IF IT'S IN TUNE. I'm also hoping that the 9th really is in the chord, or I'll look stupid. (: The chord two chords before the chord in question is a 9th chord with this property.
Another part of the power of that chord, I think, is that the (human) voices all leap up to the note. Persichetti uses this as well, for great effect, in his Psalm, for a similar I - bVI progression that could be written C c e g, C c e g, C c e g, Ab eb ab c'. The bVI is a very powerful chord, since it's new to the scale in major and just a major chord in minor, but if the upper voices leap up to it, it's a gigantic swell of excitement. In our example, the chord is a bVI in the bass, giving it this powerful function, even though the other notes aren't those of a bVI major chord.
Finally, that chord is powerful because it's in tune. I can't pick out the notes of the singers at all, yet they're so crunchy!
Thoughts?
That section of the song is in the key of B. I'm doing this by ear, by the way, so feel free to correct me. This is Jason Robert Brown, so his chords are all 2's and 4's -- in the key of C, these would be the tonic CDG, the subdominant CFG (though the bass player would most likely play the F), and the dominant GCD (which actually makes it equal to a 64 chord). While this looks rather drab, with a C present in the important chords, it's important to note that his music's harmonic interest comes from having other chords besides those three. My point is really about the character of the primary chords he uses.
Back to the music. Starting at 4:50, we have repeated melodic precadential material over a dominant F#4 chord (or B2/F# if you prefer, but there is ambiguity between the V and I64 feelings). The bass alternates the F# with an E, so this could also be a quartal dominant 7th in whatever inversion it is that has the 7th in the bass (3rd in CP theory -- CP being Common Practice -- but that isn't very meaningful here). After three bars, this is followed by a chord with an A in the bass guitar and soprano, a B in the man's voice, and possibly an E somewhere, for an A2 chord (though I hear it as an A chord with an added 9, since that B is important). The next bar features a B2 chord, with mostly parallel motion (though the male voice goes to a different note to jump instead of moving in parallel), and the next bar, that wild chord, has a G in the bass, an F# in the man, D in the woman. I think. And possibly an A someplace else. If I can get a hold of the score, I'll check. But at least on the piano, G A F# D sounds about right, with a ninth between the G and the A. This looks like a D/G chord, which would be -- ahem -- bIII/b6^, if that makes any sense. However, it's clearly a G9 chord of some sort, in this case a Gmaj9 with no third. It's hard to fit non-CP chords into CP conventions! In that case, this is a bVI chord, but there is no third because that would make a minor chord with the F# and D and that would change the energy of that chord. But it's ambiguous enough that it could be both a D/G and a Gmaj9.
Where does the chord get this energy? Partly from that beautiful major ninth, I think. The major ninth, where the top note is tonicized, is a very harmonic interval that does not resolve as might be expected, since the bass and not the soprano is the source of tension. As I hear it, it has no directional pull, but it clearly can't end a piece. Another example is in Meadowlark. This major ninth could actually be considered a diminished tenth, possibly. I don't know. One possible resolution is for both notes to resolve a half step in, to an octave; an example would be (in diminished 10th form) D# f to E e, but the idea can be expanded to resolve to E g instead, which is the same harmony in this case. Consider the following progression of two-note chords: F e, Bb d, F e, D# f, E g. The D# f chord is pretty much an Eb9, but since it's so foreign to the key, it might be better to interpret it as D# dim10, or however you want to write it. It's the inverse of the augmented 6th, but since it's a major 9th it sounds much more open (since the 9th is made up of two open fifths stacked) and therefore much more powerful. IF IT'S IN TUNE. I'm also hoping that the 9th really is in the chord, or I'll look stupid. (: The chord two chords before the chord in question is a 9th chord with this property.
Another part of the power of that chord, I think, is that the (human) voices all leap up to the note. Persichetti uses this as well, for great effect, in his Psalm, for a similar I - bVI progression that could be written C c e g, C c e g, C c e g, Ab eb ab c'. The bVI is a very powerful chord, since it's new to the scale in major and just a major chord in minor, but if the upper voices leap up to it, it's a gigantic swell of excitement. In our example, the chord is a bVI in the bass, giving it this powerful function, even though the other notes aren't those of a bVI major chord.
Finally, that chord is powerful because it's in tune. I can't pick out the notes of the singers at all, yet they're so crunchy!
Thoughts?
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