gdritter repos when-computer / b0db12a
Merge branch 'master' of rosencrantz:/srv/git/when-computer Getty Ritter 7 years ago
1 changed file(s) with 5 addition(s) and 5 deletion(s). Collapse all Expand all
+5
-5
drafts/temperament-and-tuning.telml less more
4242
4343 In music and music theory, notes which "go together" are said to be \em{consonant} and notes which "don't go together" are called \em{dissonant}. When talking about two notes, we can refer to their difference as an \em{interval}, so consonance and dissonance are generally a property of intervals.
4444
45 There isn't a strictly-defined separation between consonance and dissonance: some pairs of sounds are notes consonant (such as notes related by the the \tt{2:1} ratio) and some are clearly dissonant, but there is no well-defined cutoff point where an interval stops being consonant and start being dissonant. It's best to think of them as relative to one another: a pair of sounds can be \em{more consonant} or \em{more dissonant} than another pair, rather than being \em{consonant} or \em{dissonant} on an absolute scale.\ref{condis} \sidenote{Also note that the terms \em{consonant} and \em{dissonant} are old, and have been informally and sometimes contradictorily defined for centuries: some people define them in terms of frequencies, some in terms of perception, some in terms of both. Defining them as \em{pleasant} and \em{unpleasant} is reductive, but not necessarily a bad intuition.}
45 There isn't a strictly-defined separation between consonance and dissonance: some pairs of sounds are clearly consonant (such as notes related by the the \tt{2:1} ratio) and some are clearly dissonant, but there is no well-defined cutoff point where an interval stops being consonant and start being dissonant. It's best to think of them as relative to one another: a pair of sounds can be \em{more consonant} or \em{more dissonant} than another pair, rather than being \em{consonant} or \em{dissonant} on an absolute scale.\ref{condis} \sidenote{Also note that the terms \em{consonant} and \em{dissonant} are old, and have been informally and sometimes contradictorily defined for centuries: some people define them in terms of frequencies, some in terms of perception, some in terms of both. Defining them as \em{pleasant} and \em{unpleasant} is reductive, but not necessarily a bad intuition.}
4646
4747 \h1{Picking Points in Sound-Space}
4848
144144
145145 As an interesting aside: the existence of Pythagorean comma is also the reason for one of the odd vestigial features of musical notation: namely, the existence of \em{enharmonically equivalent notes}. When learning modern music notation, you quickly learn that there are some odd redundancies in staff notation, including the fact that some notes can be expressed more than one way: for example, A♭ and G♯ are two ways of writing the same note. That's true with modern tunings and notation, but one reason for having both notations is that they \em{did} used to connote different notes in tunings like the Pythagorean tuning: if we use A♭ as our root note, then we'd write our close-to-the-root-but-not-quite note as G♯, and the two would differ from each other by the Pythagorean comma. This distinction is no longer made, but we're still stuck with the notation.
146146
147 Back to the scale we're generating, where we have have to decide what to do with the not-quite-the-root note: we could include our it in the scale we're creating, and maybe even continue generating new frequencies from our \tt{3:2} generator. This would build a scale that includes more and more notes, but in addition to requiring some a physically unwieldy keyboards, our scales are already getting diminishing returns from including these new notes: our goal was to choose a set of notes that sounded good together. Already, we've confused would-be composers to including two very-nearly-the-same notes when all the other notes are nicely distinct, and that new root isn't going to sound nearly as nice when played with some of the notes we generated earlier. Anyway, since we our new note is \em{almost} just the original root note, we could just leave it off, stop the scale with only twelve notes. Sure, that means that one of our intervals is gonna be a \em{teeny} bit less consonant than the others, but most of them sound great, right?
147 Back to the scale we're generating, where we have have to decide what to do with the not-quite-the-root note: we could include our it in the scale we're creating, and maybe even continue generating new frequencies from our \tt{3:2} generator. This would build a scale that includes more and more notes, but in addition to requiring some a physically unwieldy keyboards, our scales are already getting diminishing returns from including these new notes: our goal was to choose a set of notes that sounded good together. Already, we've confused would-be composers to including two very-nearly-the-same notes when all the other notes are nicely distinct, and that new root isn't going to sound nearly as nice when played with some of the notes we generated earlier. Anyway, since our new note is \em{almost} just the original root note, we could just leave it off, stop the scale with only twelve notes. Sure, that means that one of our intervals is gonna be a \em{teeny} bit less consonant than the others, but most of them sound great, right?
148148
149149 As it turns out, that's exactly what Western music did for about two thousand years.
150150
154154 \sidenote{Surprising, I know.}
155155 in the 6th century BCE, and was historically quite popular in Western music up until about the 16th century. It has a nice basis in mathematics, and features the \tt{2:3} ratio throughout, which means it has a lot of nice consonances between its various notes. However, the West has stopped using this tuning nearly as much as it once did. What are the problems with it? Why would we want something else?
156156
157 Well, for one, there are a lot of ratios other than \tt{2:3} that \em{also} sound nice: for example, the \tt{4:5} ratio, being small and simple, is also a consonant interval. If we apply that ratio to our root note from before of 400 Hz, then we get \\(400 * \\frac\{5\}\{4\} = 500\\). This frequency isn't in the scale we created. The closest we have is 506.25 Hz, which corresponds a ratio of \tt{81:64}—close, but not really as pleasant as a real \tt{4:5} ratio. Take a listen: the following clip alternates back and forth between a perfect \tt{3:4} interval and the less-pleasing \tt{81:64} interval we get from Pythagorean tuning:
157 Well, for one, there are a lot of ratios other than \tt{2:3} that \em{also} sound nice: for example, the \tt{5:4} ratio, being small and simple, is also a consonant interval. If we apply that ratio to our root note from before of 400 Hz, then we get \\(400 * \\frac\{5\}\{4\} = 500\\). This frequency isn't in the scale we created. The closest we have is 506.25 Hz, which corresponds a ratio of \tt{81:64}—close, but not really as pleasant as a real \tt{5:4} ratio. Take a listen: the following clip alternates back and forth between a perfect \tt{3:4} interval and the less-pleasing \tt{81:64} interval we get from Pythagorean tuning:
158158
159159 \audio{/static/tuning/ditone.mp3}
160160
237237
238238 \table{
239239 \tr{ \th{ frequency } \th{ cents } \th { change in cents } }
240 \tr{ \td{ 400.00 Hz } \td{ 0.00 cents } \td{ } }
240 \tr{ \td{ 400.00 Hz } \td{ 0.00 cents } \td{ +100 cents } }
241241 \tr{ \td{ 423.78 Hz } \td{ 100.00 cents } \td{ +100 cents } }
242242 \tr{ \td{ 448.98 Hz } \td{ 200.00 cents } \td{ +100 cents } }
243243 \tr{ \td{ 475.68 Hz } \td{ 300.00 cents } \td{ +100 cents } }
257257
258258 It's a pretty convenient system, and it's in pretty wide use today: at the very least, it's how a lot of people think about tuning. In practice, a number of instruments can't use equal temperament \em{exactly} because of physical limitations on the instrument itself, and singers tend to use something closer to the Pythagorean tuning (because it's easier to make your voice go up a perfect fifth than an ever-so-slightly-imperfect fifth) but we tend to treat equal temperament as the ideal tuning for most modern instruments.
259259
260 I have more things to say, but this post is already a bit long, so I'm going to split the rest of this material into a separate post. Next time: \em{microtonal music}!
260 That's the high-level overview of where twelve came from: it's has its roots in Pythagorean tuning, where it made sense to only include twelve tones, and our current system of equal temperament came from a reasonable compromise that ironed out some of the rough edges in Pythagorean tuning. Of course, there are still more tunings and temperaments to discuss—including some very interesting systems that result in more than twelve tones!—but this post is already a bit long, so I'm going to split the rest of this material into a separate post. Next time: \em{microtonal music}!