I find combinatorial categorization almost pathogically compelling, so I couldn't resist pushing a little further about scales and tetrachords. I made a table with every tetrachord I've run across, picked abbreviations for them, and included synonyms from Arabic (Maqam) and Turkish (Makam) traditions, then tried to lay out as many scales as I could in terms of those tetrachords. For the scales, I looked at the seven modes of each of several source scales (major, melodic minor, harmonic minor, harmonic major, and double harmonic), including the fancy mode names and synonyms where I could find them (mostly from Wikipedia, especially the modes article).
Finally, I made tables of what scales get produced by combining tetrachords. Here we can see the names that get attached to all the combinations, and also all the blank spots on the map, scales we can play that have no name.
I had to draw some limits. I ignore the microtonal tetrachords, limiting myself to scales that can be played on an ordinary Western instrument. I had to include several tetrachords that don't give a perfect fourth: the Lydian tetrachords, which fill a tritone, and the diminished tetrachord, but I didn't write down every permutation or rotation of intervals to invent new tetrachords, I just included the ones I found reference to. I'll leave it to someone else to figure out why Lydian #2, m3-s-T, is in the canon, but the just-as-reasonable-looking m3-T-s is not. The perfect fourth tetrachords can be combined arbitrarily into scales making an octave, separated by a tone, but we need to do something different for the non-perfect ones. Either we need to combine a Lydian (tritone) tetra with a diminished, separated by a tone; or we need to put a Lydian with a perfect and separate by a semitone. Similarly, there's a couple places where an augmented second joins the two.
I also skipped some important scales. The major and minor pentatonic scales, and the related blues scale, an elaboration of the minor pentatonic, don't fit easily into a tetrachord concept, both because they have the wrong number of notes (all the above scales are heptatonic, ie have 7 notes) and because they don't have a fourth. I'll skip the bebop scales, despite their awesome names (A Doriolian ♭5? B altered quintal?) for a similar reason: they are octatonic, basically inserting a passing note into another scale. You can of course outline four note patterns in these, but it's hard to see the scale as being constructed of them. Or maybe I'm just lazy.
At some point, of course, this kind of exercise becomes silly: mapping out and analyzing parts of harmonic space which are unused, not because they are great new territories for invention and creativity which you and you alone have realized, but rather because they suck. The world's cultures have been making music for a long time, and if a scale is undiscovered, there's likely to be a good reason for that. Still, it's kind of fun to try and get a global picture of the possibilities, patterned after the scales we are used to.
Oh, yeah, the table is here, done up as a spreadsheet so it's easy for me to fix.
Refs: Wikipedia, http://www.tonalcentre.org/, various jazz harmony books, http://docs.solfege.org:81/3.9/C/scales/modes.html
Added: Another mammoth list of modes is here, 1200 of them, part of the astonishing microtonal scale analysis and construction program Scala. It's written in Ada, and has a 9/11 Truth icon at the end, just in case you need additional evidence that the author is looney, but really, I think that list of scales should be sufficient.
Tuesday, August 10, 2010
Monday, August 9, 2010
Tetrachords
So I sat down at the keyboard, deciding that I would figure out major and minor scales, and what triads they contained at each position, and got lost just in the major scale. See, the whole TTsTTTs thing always bothered me -- so asymmetric. Why two tones, then a semitone, followed by three... never made sense. Why not the other way around, or some other pattern. And the whole layout of the piano keyboard contains this pattern. The structure of the major scale is built into the interface. It's mathematically unpleasing, which is one of the reasons why, when extending my interval practice into scales and arpeggios, I often stuck to the totally symmetric ones, built entirely from a single interval: chromatic, whole tone, dim7, augmented.
While diddling, I remembered reading a claim that the scale is in fact symmetric, being built from two identical four-note tetrachords, each consisting of a perfect fourth divided as TTs. That claim confused me at the time, too, since it's clearly not symmetric. Still, it kinda is, since you've got the same note pattern in each half (approximately half, anyway) of the octave, separated by a whole step.
So what happens if you start minor, eg C-D-Eb-F, ie TsT? Does repeating that in the upper half give you one of the minor scales? Well no. You get a Dorian mode, which has a minor feel, from the flatted third, but is not one of the classical ones (natural, harmonic, or melodic). To get melodic minor (going up melodic, TsTTTTs, sometimes called jazz minor if you're planning on playing the same notes going down) you put the major tetrachord on the top, TTs, since those notes are the same as the major scale. Fine, so different scales can get built by combining different tetrachords. That's kind of interesting, since we can therefore use the tetrachords as an organizing concept to relate different scales which have parts which sound the same. To build a natural minor, we use the last tetrachord, sTT, on the top. If you build a scale from two of those, you get the Phyrgian mode, C-Db-Eb-F G-Ab-Bb-C. I guess it's almost obvious that building a scale from two of the same of any these tetrachords (what wikipedia calls diatonic tetrachords, and names the three Lydian, Dorian, and Phrygian) has to give you a mode of the major scale, since the semitones are the same distance apart. But there are only three of them anyway. I suspect pianists learn those shapes, and use those patterns in playing.
The asymmetry of the major scale comes from placing the upper tetrachord a whole step above the lower tetrachord, but not vice versa. So we can distinguish upper and lower. This is not the only choice, and other choices give us other scales. Each tetrachord covers a fourth, so to make a complete octave, we need to add a total of two semitones. We can put them both between lower and upper, giving the major scale, so that the last note of the upper tetrachord, C, is the same as the first note of the lower tetrachord; or both after the upper tetrachord, so that the last note of the lower tetrachord, F, is the first note of the upper tetrachord, giving us the Lydian mode. Conceivably we could divide them equally, with a semitone between lower and upper, and another between upper and lower. This gives the eight note scale, C-D-E-F-F#-G#-A#-B-C, a scale for which I don't know a name. It sounds kinda like a whole tone scale, but with passing tones inserted. Like the whole tone scale, it is symmetrical, or at least, more symmetrical than the major scale. Might be worth looking at what scales are generated by those choices for the other tetrachords.
We could also symmetrize by not demanding we end up with an octave. You lose the idea of the scale as a closed circle, but maybe it could work anyway. Without adding any gaps, just extending the TTs (Lydian) tetrachord, two fourths make a m7, and you start moving through the circle of fifths as you go up the scale, every part sounding kind of in a major key, but mutating as you go: C-D-E-F, F-G-A-Bb, Bb-C-D-Eb, and so on. Or if you put a whole step between both upper and lower, you get something similar, but moving up by fifths instead of by fourths: C-D-E-F, G-A-B-C, D-E-F#-G, A-B-C#-D, ...
You might have noticed I skipped harmonic minor. It's that minor third, which doesn't occur in any permutation of TTs. Wikipedia puts this into a separate class of tetrachords, calling it chromatic, with two semitones and a minor 3rd. Lots more scales can come out of here no doubt, and almost certainly as well from another class of tetrachord that I wouldn't have thought of, because you can't play it on a piano: major 3rd, quartertone, quartertone.
Finally, why a fourth? This, at least, I have a good answer for. It's the smallest perfect interval, and the perfect intervals are unique, not because they have simple integer ratios of frequencies (though they do), and not because we call them a special name (though we do), but rather because it is for perfect intervals alone that we have hardware in our brains to measure. Perfect intervals are innate, all other intervals are learned and cultural. And the fourth is the smallest one.
So I didn't figure out major and minor, but I did manage to clear up a longstanding confusion of mine over the symmetries, and lack thereof, of the basic major scale. Seems like enough for an evening.
While diddling, I remembered reading a claim that the scale is in fact symmetric, being built from two identical four-note tetrachords, each consisting of a perfect fourth divided as TTs. That claim confused me at the time, too, since it's clearly not symmetric. Still, it kinda is, since you've got the same note pattern in each half (approximately half, anyway) of the octave, separated by a whole step.
So what happens if you start minor, eg C-D-Eb-F, ie TsT? Does repeating that in the upper half give you one of the minor scales? Well no. You get a Dorian mode, which has a minor feel, from the flatted third, but is not one of the classical ones (natural, harmonic, or melodic). To get melodic minor (going up melodic, TsTTTTs, sometimes called jazz minor if you're planning on playing the same notes going down) you put the major tetrachord on the top, TTs, since those notes are the same as the major scale. Fine, so different scales can get built by combining different tetrachords. That's kind of interesting, since we can therefore use the tetrachords as an organizing concept to relate different scales which have parts which sound the same. To build a natural minor, we use the last tetrachord, sTT, on the top. If you build a scale from two of those, you get the Phyrgian mode, C-Db-Eb-F G-Ab-Bb-C. I guess it's almost obvious that building a scale from two of the same of any these tetrachords (what wikipedia calls diatonic tetrachords, and names the three Lydian, Dorian, and Phrygian) has to give you a mode of the major scale, since the semitones are the same distance apart. But there are only three of them anyway. I suspect pianists learn those shapes, and use those patterns in playing.
The asymmetry of the major scale comes from placing the upper tetrachord a whole step above the lower tetrachord, but not vice versa. So we can distinguish upper and lower. This is not the only choice, and other choices give us other scales. Each tetrachord covers a fourth, so to make a complete octave, we need to add a total of two semitones. We can put them both between lower and upper, giving the major scale, so that the last note of the upper tetrachord, C, is the same as the first note of the lower tetrachord; or both after the upper tetrachord, so that the last note of the lower tetrachord, F, is the first note of the upper tetrachord, giving us the Lydian mode. Conceivably we could divide them equally, with a semitone between lower and upper, and another between upper and lower. This gives the eight note scale, C-D-E-F-F#-G#-A#-B-C, a scale for which I don't know a name. It sounds kinda like a whole tone scale, but with passing tones inserted. Like the whole tone scale, it is symmetrical, or at least, more symmetrical than the major scale. Might be worth looking at what scales are generated by those choices for the other tetrachords.
We could also symmetrize by not demanding we end up with an octave. You lose the idea of the scale as a closed circle, but maybe it could work anyway. Without adding any gaps, just extending the TTs (Lydian) tetrachord, two fourths make a m7, and you start moving through the circle of fifths as you go up the scale, every part sounding kind of in a major key, but mutating as you go: C-D-E-F, F-G-A-Bb, Bb-C-D-Eb, and so on. Or if you put a whole step between both upper and lower, you get something similar, but moving up by fifths instead of by fourths: C-D-E-F, G-A-B-C, D-E-F#-G, A-B-C#-D, ...
You might have noticed I skipped harmonic minor. It's that minor third, which doesn't occur in any permutation of TTs. Wikipedia puts this into a separate class of tetrachords, calling it chromatic, with two semitones and a minor 3rd. Lots more scales can come out of here no doubt, and almost certainly as well from another class of tetrachord that I wouldn't have thought of, because you can't play it on a piano: major 3rd, quartertone, quartertone.
Finally, why a fourth? This, at least, I have a good answer for. It's the smallest perfect interval, and the perfect intervals are unique, not because they have simple integer ratios of frequencies (though they do), and not because we call them a special name (though we do), but rather because it is for perfect intervals alone that we have hardware in our brains to measure. Perfect intervals are innate, all other intervals are learned and cultural. And the fourth is the smallest one.
So I didn't figure out major and minor, but I did manage to clear up a longstanding confusion of mine over the symmetries, and lack thereof, of the basic major scale. Seems like enough for an evening.
Saturday, July 31, 2010
Little Jake pickup test
So I picked up one of Trent Jacob's Little Jake pickups. After some trouble with the installation (a DIY job, and let me tell you, I didn't try it on one of my good bocals), I'm now playing with it. You're supposed to use it with a preamp, either a custom or something like the LR Baggs Gigpro. I'm putting it straight into the input of my audio interface, an M-Audio FastTrack Pro. It needs a fair bit of gain to get an acceptable signal level, but the FastTrack preamp is kinda noisy at very high gain, so I'm thinking about getting a proper preamp. Here's a short sample, first recorded with the Little Jake, next recorded in parallel with a decent quality (but rather old) dynamic mic, an AT802.
Little Jake Test by TFox17
Little Jake Test Control by TFox17
A couple of things to point out: the Little Jake does a very good job of rejecting outside noise. It won't feedback, if you're using it to drive a loud amplifier in the same room. When I've played against a delay, or self-recordings with a mic, it was always a struggle to get the signal through amp loud enough that I could hear it while playing while also avoiding feedback. Even without feedback, there's still effects, such as some uncontrolled reverb from the signal leakage. So that's the primary purpose of using a pickup, is to avoid stuff like that. You get side benefits too. Key clacks are gone, for instance, and ambient noise in the room is gone too. The only thing you're getting is the sound actually inside the bassoon. Of course, external sounds can propagate to the inside of the bassoon too, but the bassoon sound is so much louder there, that they are swamped out. It's a signal to noise thing, kind of the ultimate version of close miking. It's not perfect: I was recording while playing a metronome through headphones, adjusted rather too loud (don't ask), and I could end up making out a bit of the metronome sound in the Little Jake recording.
The tone color, however, seems quite different from the far field mike recording. Not a big deal, I think, since mostly people use this as a starting point for effects, as opposed to looking for a true bassoon sound. Still, interesting.
Updated: For reference, as a comparison of microphone differences, here's my other microphone, an AT2020 condenser mic, against the 802. They've been normalized, to try and get their volumes equal, but are otherwise unprocessed. Mic location is bell height, about 5 ft in front of the bassoon, a few feet in front of the wall, with the mics both facing forward, and within a few inches of each other. When I did this yesterday I was surprised at how different they sounded, but today I'm surprised at how similar they are. Certainly there's a lot of things the AT2020 hears that seem to be gone in the 802, but I seem to need the nice headphones to hear them (the laptop speakers aren't enough), and they all seem to be things I'd rather not hear anyway (breathy noises, that kind of thing).
AT2020 Test by TFox17
AT802-Test by TFox17
Little Jake Test by TFox17
Little Jake Test Control by TFox17
A couple of things to point out: the Little Jake does a very good job of rejecting outside noise. It won't feedback, if you're using it to drive a loud amplifier in the same room. When I've played against a delay, or self-recordings with a mic, it was always a struggle to get the signal through amp loud enough that I could hear it while playing while also avoiding feedback. Even without feedback, there's still effects, such as some uncontrolled reverb from the signal leakage. So that's the primary purpose of using a pickup, is to avoid stuff like that. You get side benefits too. Key clacks are gone, for instance, and ambient noise in the room is gone too. The only thing you're getting is the sound actually inside the bassoon. Of course, external sounds can propagate to the inside of the bassoon too, but the bassoon sound is so much louder there, that they are swamped out. It's a signal to noise thing, kind of the ultimate version of close miking. It's not perfect: I was recording while playing a metronome through headphones, adjusted rather too loud (don't ask), and I could end up making out a bit of the metronome sound in the Little Jake recording.
The tone color, however, seems quite different from the far field mike recording. Not a big deal, I think, since mostly people use this as a starting point for effects, as opposed to looking for a true bassoon sound. Still, interesting.
Updated: For reference, as a comparison of microphone differences, here's my other microphone, an AT2020 condenser mic, against the 802. They've been normalized, to try and get their volumes equal, but are otherwise unprocessed. Mic location is bell height, about 5 ft in front of the bassoon, a few feet in front of the wall, with the mics both facing forward, and within a few inches of each other. When I did this yesterday I was surprised at how different they sounded, but today I'm surprised at how similar they are. Certainly there's a lot of things the AT2020 hears that seem to be gone in the 802, but I seem to need the nice headphones to hear them (the laptop speakers aren't enough), and they all seem to be things I'd rather not hear anyway (breathy noises, that kind of thing).
AT2020 Test by TFox17
AT802-Test by TFox17
Saturday, July 24, 2010
Scales inside of scales
When I started playing again, I didn't want to be locked into my existing limited models, notes off the page, all major scales, plus the occasional foray into the three minors they teach you in school. If you expand your horizons, however, there are a very large number of possible scales to play with, even if you limit yourself to the ones common used in some musical tradition and ignore the infinities of mathematical possibilities. So I grabbed some of my wife's jazz theory books, typed in the names of some scales, and picked one at random to play from time to time when I was in the mood. I soon got to be able to distinguish a melodic from a jazz minor, minor and major pentatonic, remember the names of all the modes, and so on. I couldn't play them fast or fluently, but I could puzzle through them one at a time, and I felt like I was learning something.
Still, there's always stuff you miss. For instance, it seems reasonable that a minor pentatonic fits inside a minor scale. But there's more. The inimitable Paul Hanson explains something I wouldn't have guessed: in fact, you can fit *three* different pentatonics inside. For a Dorian, you have a minor pentatonic available starting on the 1st, 2nd, and 5th scale tones, all without leaving the scale. (Likewise, you could do major pentatonics starting on 3, 4, or 7, with similar embeddings for any other mode.) In this video, he demonstrates with bass player Craig Harris.
Still, there's always stuff you miss. For instance, it seems reasonable that a minor pentatonic fits inside a minor scale. But there's more. The inimitable Paul Hanson explains something I wouldn't have guessed: in fact, you can fit *three* different pentatonics inside. For a Dorian, you have a minor pentatonic available starting on the 1st, 2nd, and 5th scale tones, all without leaving the scale. (Likewise, you could do major pentatonics starting on 3, 4, or 7, with similar embeddings for any other mode.) In this video, he demonstrates with bass player Craig Harris.
Thursday, July 22, 2010
Another week, another lesson
Milde Scale 1 went well this time. Really well in fact, I think better than I've ever played it in practice. It got so I was nervous about ruining the streak of smooth notes, and messed up from losing focus. I'd done lots of practicing, missing only one day, and I'd played twice yesterday and also twice today, before my lesson. It doesn't seem like things improve, but I guess they do. I was going to ask about the finger movement thing, but it didn't happen, so
Still complaints about tone in upper register, despite my work in that direction. Especially A4 and up. Lips firm against teeth but as loose as possible against the reed mute the reed less apparently, if that doesn't sound too contradictory. All support from the air, very fast air. No biting or anything like that of course. G4 is a good note to start on, since it's easy to get a good tone there, and then work up. Thin and dead is what I'm trying to avoid.
Elgar: very long lines, look for the dissonant downbeats->resolve figures (appogiatura). It's all very exposed, so every aspect of every note counts. I think this is the first piece I've played which has been hard due to phrasing, rather than just notes.
Scales: just keep the ones I'm working on, C F G D, their relative melodic minors, and arpeggios, maybe slowly work up speed. I played them at 60 in the lesson, which felt comfortable, though I've also been working them slower in practice, eg 53.
Bocals: I played on a new-to-me bocal, a CD1 I got off of eBay. It feels different than my other one, thinner and more metallic somehow. Probably no audible difference to an external listener, but it feels quite different anyway. For reed, I played on G5. I formed a bunch of tubes before I went on vacation, and really need to start finishing some of those reeds.
Still complaints about tone in upper register, despite my work in that direction. Especially A4 and up. Lips firm against teeth but as loose as possible against the reed mute the reed less apparently, if that doesn't sound too contradictory. All support from the air, very fast air. No biting or anything like that of course. G4 is a good note to start on, since it's easy to get a good tone there, and then work up. Thin and dead is what I'm trying to avoid.
Elgar: very long lines, look for the dissonant downbeats->resolve figures (appogiatura). It's all very exposed, so every aspect of every note counts. I think this is the first piece I've played which has been hard due to phrasing, rather than just notes.
Scales: just keep the ones I'm working on, C F G D, their relative melodic minors, and arpeggios, maybe slowly work up speed. I played them at 60 in the lesson, which felt comfortable, though I've also been working them slower in practice, eg 53.
Bocals: I played on a new-to-me bocal, a CD1 I got off of eBay. It feels different than my other one, thinner and more metallic somehow. Probably no audible difference to an external listener, but it feels quite different anyway. For reed, I played on G5. I formed a bunch of tubes before I went on vacation, and really need to start finishing some of those reeds.
Monday, July 19, 2010
Struggling
Had another lesson. Terrible, in part. The fingering changes is one reason, another is the time off I took, and I'm still getting back up to speed. But Milde 1, the first scale study, was both slower and more uneven than I'd played it at the last lesson. Parts of it he just said, there's nothing I can say to help you there. And the lesson on Elgar Romance covered nearly identical ground as the previous one. Which is to say, I've made precisely zero progress.
Still, I learned things. For instance, apparently my embrochure is wrong too: not enough wrapped around the teeth. I spent a year playing long tones with an incorrect embrochure, I guess, not to mention the decade of playing when I was younger. Anyway, he said that a thinner embrochure, ie less lip on the reed, would damp less, and let the high A4 ring more, and match the lower A's. So yeah, yet another thing to change.
And while practicing tonight, trying to figure out why I can play the second bar of Milde 1 okay by itself, but never after the first bar, I realized that I have two positions for my left hand. If I start in the bottom register, my hand is wrapped a little more, and my fingers extended more, but in the tenor register I put the ball of my first finger a little more in front to support the weight. This affects how I move the fingers, and whether I can only bend from the knuckle or from the ball (not sure the right anatomical descriptions of these). Anyway, I'm trying to learn to be consistent, so that if I play a scale up, I don't need to shift. So that's yet another thing that I didn't know existed, not to mention the things I knew were problems, but hadn't figured out how to fix yet. So much to learn.
Still, I learned things. For instance, apparently my embrochure is wrong too: not enough wrapped around the teeth. I spent a year playing long tones with an incorrect embrochure, I guess, not to mention the decade of playing when I was younger. Anyway, he said that a thinner embrochure, ie less lip on the reed, would damp less, and let the high A4 ring more, and match the lower A's. So yeah, yet another thing to change.
And while practicing tonight, trying to figure out why I can play the second bar of Milde 1 okay by itself, but never after the first bar, I realized that I have two positions for my left hand. If I start in the bottom register, my hand is wrapped a little more, and my fingers extended more, but in the tenor register I put the ball of my first finger a little more in front to support the weight. This affects how I move the fingers, and whether I can only bend from the knuckle or from the ball (not sure the right anatomical descriptions of these). Anyway, I'm trying to learn to be consistent, so that if I play a scale up, I don't need to shift. So that's yet another thing that I didn't know existed, not to mention the things I knew were problems, but hadn't figured out how to fix yet. So much to learn.
Wednesday, July 14, 2010
Restarting
So I'm back from traveling, and starting to practice again. First couple of days were a bear, it felt totally alien to have this buzzing thing in my mouth. And not just me, but the reeds, too, felt like they weren't used to working. I'm over that now, but my teacher changed some fingerings on me, and I'm having great trouble adjusting. I'm to add the Eb vent on D3 (and I'm guessing Eb3 as well), put down the whisper key on G3 (and probably F#3 and G#3), and remove the vent on B3 and C4. D3 is the one that kills me, because it basically adds a pinky finger motion opposite to fourth finger, when getting the fourth finger to behave is bad enough. Still, it helps the intonation, on what was previously perhaps my worst note. Still, at this point, I feel like getting back into the practice of practicing is the most important step. It doesn't matter if I sound worse than I did at my last lesson, the point is to practice practicing.
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