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Some final edits Getty Ritter 7 years ago
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22 "meantones and microtonal tuning"
33 ("music") 1485060589)}
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5 So, in my \link{http://what.happens.when.computer/2016-10-18/pythagorean-tuning-and-equal-temperament/|previous post on the topic of music theory}, I described the basic system of building a tuning by repeatedly applying a musical interval to a root note: when this process is performed with a perfect fifth, we get twelve distinct notes before producing a note which is \em{almost} identical to the original root note, which is the \em{original} genesis of our twelve-note system. I also described how we could perform a different transformation—dividing the space of the octave into equally-spaced chunks—and get a similar but non-identical tuning that nonetheless suffices for most modern music.
5 So, in my \link{http://what.happens.when.computer/2016-10-18/pythagorean-tuning-and-equal-temperament/|previous post on the topic of music theory}, I described the basic system of building a tuning by repeatedly applying a musical interval to a root note: when this process is performed with a perfect fifth, we get twelve distinct notes before producing a note which is \em{almost} identical to the original root note. This tuning—called Pythagorean tuning—forms the \em{original} genesis of our twelve-note system. I also described how we could perform a different transformation—dividing the space of the octave into equally-spaced chunks—and get a similar but non-identical tuning that trades perfect harmony for flexibility, and how that system is the predominantly used system today.
66
7 At one point, I mentioned and glossed over a stage in musical history, which I want to go into more detail about now. In particular, I want to describe the family of tunings that were popular after Pythagorean tuning fell out of favor in the 16th century until Equal Temperament became popular in the 19th century: the \em{meantone temperaments}.
7 At one point, I mentioned and glossed over another stage in musical history, which is I want to go into more detail about now. In particular, I want to describe the family of tunings that were popular after Pythagorean tuning fell out of favor in the 16th century, and before Equal Temperament became popular in the 19th century: the family of \em{meantone temperaments}.
88
99 \h1{Quarter-Comma Meantone}
1010
11 One big problem with Pythagorean tuning, aside from \em{the wolf} that I demonstrated in the last post, is that there are several other pleasing ratios which are very poorly approximated by the notes available. One of those ratios is the one we call a \em{major third}: that is, the interval that corresponds to a \tt{5:4} ratio. In Pythagorean tuning, the closest approximation we have is \tt{81:64}, which is out-of-tune enough that many musicians who composed for Pythagorean tuning would avoid using it entirely!
11 One big problem with Pythagorean tuning, aside from \em{the wolf} that I demonstrated in the last post, is that there are several other pleasing ratios which are very poorly approximated by the notes available. One of those ratios is the one we call a \em{major third}: that is, the interval that corresponds to a \tt{5:4} ratio of frequencies. In Pythagorean tuning, the closest approximation we have is \tt{81:64}, which is out-of-tune enough that many musicians who composed for Pythagorean tuning would avoid using it entirely, or at least treat it as a dissonant interval when it was used.
1212
13 Beginning in the sixteenth century, musicians began to adopt alternate tuning systems where the \tt{5:4} ratio—also called the \em{major third}—was better approximated and more usable. One way of doing this was to build a system of tuning using the same techniques as in Pythagorean tuning—that is, generating a set of tones by repeatedly increasing frequencies by the \tt{3:2} ratio, also called a \em{fifth}—except unlike in Pythagorean tuning, they decided to choose a ratio that differed \em{slightly} from the exact \tt{3:2} ratio we used before. Systems that follow the same process as Pythagorean tuning but with a slightly different approximation of the fifth are called \em{meantone temperaments}.
13 Beginning in the sixteenth century, musicians began to adopt alternate tuning systems where the major third was better approximated and thus more generally usable. One way of doing this was to build a system of tuning using the same techniques as in Pythagorean tuning—that is, generating a set of tones by repeatedly increasing frequencies by the \tt{3:2} ratio, also called a \em{fifth}—except unlike in Pythagorean tuning, they decided to choose a ratio that differed \em{slightly} from the exact \tt{3:2} ratio we used before. Systems that follow the same process as Pythagorean tuning but with a slightly different approximation of the fifth are called \em{meantone temperaments}.
1414
15 But what ratio do we choose? We could just keep trying slightly different approximations of \tt{3:2}, seeing what works and what doesn't work, but that's not a very principled way of designing a system of tuning. Instead, let's start with our desired goals and then figure out what ratio we want by working backwards. Let's stipulate right up front that we want a system where major thirds are related to the root by \em{exactly} a \tt{5:4} ratio, and then find a frequency that's a close approximation of \tt{3:2}—let's call this unknown ratio \\(\\rho\\)\ref{rho}—that nevertheless accomodates our initial stipulations.
16
15 But what ratio do we choose? We could just keep trying slightly different approximations of \tt{3:2}, seeing what works and what doesn't work, but that's not a very principled way of designing a system of tuning. Instead, let's start with our desired goals and then figure out what approximation we want by working backwards. Let's stipulate right up front that we want a system where the relationship between the root and a major third is \em{exactly} a \tt{5:4} ratio, and then find a frequency that's a close approximation of \tt{3:2}—let's call this unknown ratio \\(\\rho\\)\ref{rho}—that nevertheless accomodates our initial stipulations.
1716 \sidenote{The variable \\(\\rho\\) is not a standard variable name, so don't use it around music theoreticians and expect them to know what you mean. I just wanted to have a shorter name than "the approximation of the perfect fifth ratio".}
1817
19 So let's start by asking: how \em{did} we get the value of a major third in Pythagorean tuning? If we run the process of generating Pythagorean tuning again, we can see that the frequency which most closely approximates the \tt{5:4} ratio was generated after four applications of the \tt{3:2} ratio. To demonstrate this, I'm going to reproduce the relevant parts of Pythagorean tuning in this chart—but in this chart, I'm \em{not} going to wrap the frequencies back around into the same octave:
18 So let's start by asking: how did we \em{originally} derive a frequency for the major third in Pythagorean tuning? If we run through the process of generating Pythagorean tuning again, we can see that the frequency which most closely approximates the \tt{5:4} ratio was generated after four applications of the \tt{3:2} ratio. To demonstrate this, I'm going to reproduce the relevant parts of Pythagorean tuning in this chart—but in this chart, I'm \em{not} going to wrap the frequencies back around into the same octave:
2019
2120 \table{
2221 \tr{ \th{Frequency} \th{ Formula } \th{Ratio w/r/t root} }
2928
3029 In the previous post, I had adjusted the frequencies back down an octave whenever they left the 400–800 Hz range so that they'd be easier to visualize together, but here I didn't: consequently, the frequencies 900 Hz and 1350 Hz are one octave higher than the ones we reported in the last post, and the frequency 2025 Hz is \em{two} octaves higher than the one we originally reported. That is to say, 2025 Hz is two octaves plus a major third above 400 Hz in Pythagorean tuning.
3130
32 But wait! We know the octave ratio is \tt{2:1}, and we're currently trying to build a system where the major third is \em{exactly} \tt{5:4}. Two octaves plus a major third in the system we're attempting to create should be \em{exactly the same} as the root multiplied by \\(2 \\times 2 \\times \\frac\{5\}\{4\} = 5\\) (that is, 2000 Hz), but as we already know, Pythagorean tuning gives us the approximation \\((\\frac\{3\}\{2\})^4 = 5.0625\\) (or 2025 Hz). In the new system we're building, we still expect to get to a major third by four applications of our approximated fifth, so let's replace the exact fraction \\(\\frac\{3\}\{2\}\\) by our variable \\(\\rho\\), and apply some basic algebra to the the resulting equation \\( \\rho^4 = 2 \\times 2 \\times \\frac\{5\}\{4\} \\): our mystery frequency turns out to be \\(\\rho = \\sqrt[4]\{5\}\\).
31 But wait! We know the octave ratio is \tt{2:1}, and we're currently trying to build a system where the major third is \em{exactly} \tt{5:4}. Two octaves plus a major third in the system we're attempting to create should be \em{exactly the same} as the root multiplied by \\(2 \\times 2 \\times \\frac\{5\}\{4\} = 5\\) (that is, 2000 Hz), but as we already know, Pythagorean tuning gives us the approximation \\((\\frac\{3\}\{2\})^4 = 5.0625\\) (or 2025 Hz). In the new system we're building, we'd still expect to get to a major third by four applications of our approximated fifth, so let's replace the exact fraction \\(\\frac\{3\}\{2\}\\) by our variable \\(\\rho\\), and apply some basic algebra to the the resulting equation \\( \\rho^4 = 2 \\times 2 \\times \\frac\{5\}\{4\} \\): our mystery frequency turns out to be \\(\\rho = \\sqrt[4]\{5\}\\).
3332
34 Now that we have that frequency, let's use it just like we used \tt{3:2} in Pythagorean tuning and look at the resulting scale. This time, I'm restricting the resulting notes to the 400 to 800 Hz range and then sorting the resulting frequencies. For comparison, I'm also going to include the values for Pythagorean tuning:
33 Now that we have that frequency, let's use it just like we used \tt{3:2} in Pythagorean tuning and look at the resulting scale. This time, I am moving the resulting notes back into the 400 to 800 Hz range and then sorting the resulting frequencies. For comparison, I'm also going to include the corresponding values from Pythagorean tuning:
3534
3635 \table{
3736 \tr{ \th{ Freq. (Meantone) } \th{ Cents (Meantone) } \th{ Freq. (Pythagorean) } \th{ Cents (Pythagorean) } }
6160
6261 \h1{One-Third-Comma Meantone}
6362
64 Quarter-comma meantone quickly became the most common exemplar of meantone temperament after it was first proposed in the 16th century, but it was also far from the only one proposed or used. One of the first rigorous mathematical descriptions of quarter-comma meantone comes from the 16th-century music theorist and organist \link{https://en.wikipedia.org/wiki/Francisco_de_Salinas|Francisco de Salinas}. In his writing, he \em{also} described a different meantone system with a similar motivation: instead of trying to build a meantone system that accomodates perfect major thirds, what if we built a meantone system that accomodates perfect \em{minor} thirds?
63 Quarter-comma meantone quickly became the most common exemplar of meantone temperament after it was first proposed in the 16th century, but it was also far from the only one proposed or used. One of the first rigorous mathematical descriptions of quarter-comma meantone comes from the 16th-century music theorist and organist \link{https://en.wikipedia.org/wiki/Francisco_de_Salinas|Francisco de Salinas}. In his writing, he also described a different meantone system with a similar motivation: instead of trying to build a meantone system that accomodates perfect major thirds, what if we built a meantone system that accomodates perfect \em{minor} thirds?
6564
6665 A perfect minor third corresponds to a ratio of \tt{6:5}, which has a slightly more dissonant, haunting sound than most of the ratios we've been working with. A perfect minor third sounds like this:
6766
6867 \audio{/static/microtonal/minor-third.mp3}
6968
70 I'll elide the math behind this tuning, but it proceeds very much like the process above, except that the Pythagorean approximation of a minor third is derived after \em{nine} applications of the \tt{3:2} ratio, which is also several octaves higher, so we end up having to solve a rather different equation for our new \\(\\rho\\) value, but the end result is that our fifth ends up being 694.786 cents: it ends up differing from a perfect major third by exactly \em{one-third} of the syntonic comma. Let's start doing the same generative process as before and build our scale:
69 I'll elide the math behind this tuning, but it proceeds very much like the process above, except that the Pythagorean approximation of a minor third is derived after several more applications of the \tt{3:2} ratio, which is ends up with a note several octaves higher, so we end up having to solve a rather different equation for our new \\(\\rho\\) value, but the end result is that our fifth ends up being 694.786 cents: it ends up differing from a perfect major third by exactly \em{one-third} of the syntonic comma. Let's start doing the same generative process as before and build our scale:
7170
7271 \svgimg{/static/microtonal/third-incomplete.svg}
7372
7574
7675 \svgimg{/static/microtonal/third-complete.svg}
7776
78 \em{Now} we've gotten close back to the root note: the red tone which I put in the bottom-right is the last tone we generated, and it's close enough to the root for our purposes (within twenty-five cents, anyway, which is closer than we got before.) But something here is different from before: by using a slightly different approximation of a fifth, we've ended up developed a scale that has not twelve but \em{nineteen notes}!
77 \em{Now} we've gotten back closer to the root note: the red tone which I put in the bottom-right is the last tone we generated, and it's close enough to the root for our purposes (within twenty-five cents, anyway, which is closer than we got before.) But something here is different from before: by using a slightly different approximation of a fifth, we've ended up developing a scale that has not twelve but \em{nineteen notes}!
7978
8079 \audio{/static/microtonal/19-scale.mp3}
8180
82 When Salinas proposed this system, he described it as sounding "languid" but not "offensive to the ear". Building an octave from nineteen notes has a different fun property, as well: recall that when using a twelve-note octave, if we keep moving up a fifth (that is, seven steps up in the scale), we'll eventually get back to the root, but only after having stepped through every single note in the scale. This isn't true for other intervals: if we go up two steps, for example, then we'll only hit every other note in the scale before returning to the root. But in a nineteen-notes scale, then every possible choice of step size (except nineteen, of course) will have this same property: moving up any consistent amount will cycle through the scale, hitting every possible note before returning to the root. That means we don't just have a circle of fifths, we can produce, say, a circle of major thirds
81 When Salinas proposed this system, he note that nineteen notes wasn't too unwieldy to use, and he described it as sounding "languid" but not "offensive to the ear". Building an octave from nineteen notes has some interesting new properties, as well. For example, recall that when using a twelve-note octave, if we keep moving up a fifth (that is, seven steps up in the scale), we'll eventually get back to the root, but only after having stepped through every single note in the scale. This isn't true for other intervals in a twelve-note system: if we go up two steps, for example, then we'll only hit every other note in the scale before returning to the root. But in a nineteen-notes scale, then every possible choice of step size (except nineteen, of course) will have this same property: moving up any consistent amount will cycle through the scale, hitting every possible note before returning to the root\ref{coprime}.
82 \sidenote{This is because, while 7 is coprime to 12, \em{every} number between 1 and 19 is coprime to 19. Because, well, 19 is, uh, prime.}
83 That means we don't just have a circle of fifths, we can also produce, say, a circle of major thirds
8384
8485 \audio{/static/microtonal/circle-of-major-thirds.mp3}
8586
9596
9697 \audio{/static/microtonal/19th-subminor-third.mp3}
9798
98 Now that we know we can reasonably split an octave into nineteen more-or-less equal chunks, what if we did the same thing we did in the last post, and just split the octave into nineteen equal chunks?
99 Now that we know we can reasonably split an octave into nineteen more-or-less equal chunks, what if we did the same thing we did in the last post, and just split the octave into nineteen chunks of exactly equal size?
99100
100 If we do that, it turns out to be \em{incredibly} similar to the previous tuning we developed, so close it's difficult to hear the difference. Here's a table of the same notes derived using the meantone approach and the equal temperament approach, as well as the difference in cents between the notes:
101 If we do that, it turns out to be \em{incredibly} similar to the previous tuning we developed, so close it's difficult to hear the difference. Here's a table of the same notes derived using the meantone approach and the equal temperament approach, as well as the absolute difference in cents between the notes:
101102
102103 \table{
103104 \tr{ \th{Freq. (Meantone)} \th{Cents (Meantone)} \th{Freq. (ET)} \th{Cents (ET)} \th{Difference}}
130131
131132 \audio{/static/microtonal/19th-third.mp3}
132133
133 All this is interesting! We built a scale using a very similar approach as other scales, but it turns out to have some radically different properties that allow for a very different kind of music to be created. And indeed, some composers were eager to work with alternate systems like this. Around the same time that Salinas first proposed this system, a French composer—\link{https://en.wikipedia.org/wiki/Guillaume_Costeley|Guillaume Costeley}—independently invented a nearly-identical system\ref{cost}
134 All this is interesting! We built a scale using a very similar approach as other scales, but it turns out to have some radically different properties that allow for a very different kind of music to be created. And indeed, some composers were eager to work with \em{microtonal} systems like this, which divide the octave into a larger number of smaller steps. Around the same time that Salinas first proposed this system, a French composer—\link{https://en.wikipedia.org/wiki/Guillaume_Costeley|Guillaume Costeley}—independently invented a nearly-identical system\ref{cost}
134135 \sidenote{Costeley's system was built up from a different mathematical basis, and consequently isn't \em{exactly} the same as the one-third-comma meantone that Salinas was describing, but it's remarkably close. I should also mention that, while I presented them in reverse order, Costeley's composition came first, in 1558, while Salinas' book \em{De musica libri septem} wouldn't be published until 1577.}
135 and wrote a \em{chanson} called \em{Seigneur Dieu ta pitié} with this system, designed to be played on a newly-invented instrument called the \link{https://en.wikipedia.org/wiki/Archicembalo|\em{Archicembalo}}:
136 and wrote a \em{chanson} called \em{Seigneur Dieu ta pitié} with this system, designed to be played on a newly-invented instrument called the \link{https://en.wikipedia.org/wiki/Archicembalo|\em{Archicembalo}}. Consequently, this piece can be considered the very first 19-note composition:
136137
137138 \youtube{wT6-Ndx1EbM|560|315}
138139
139 While never achieving \em{major} popularity among musicians at large, various nineteen-tone tunings have been revisited regularly throughout musical history, and have achieved a modest popularity among some composers, in part because nineteen is still a "workable" number of tones for an octave. Recall in the previous post that, when generating the frequencies for a Pythagorean scale, I decided to stop after having generated a frequency that was was "close enough" to the root, because we wouldn't generate a tone closer to the root until after another 53 steps. That was a motivated choice, but still a choice: I certainly \em{could} have kept generating new tones and adding them to the scale until I had generated all 53 notes. After all, we're just concerned with building up a set of frequencies we can use together, so why \em{not} compose music on a 53-note scale?
140 While never achieving \em{major} popularity among musicians at large, various nineteen-tone tunings have been revisited regularly throughout musical history, and have achieved a modest popularity among some composers, in part because nineteen is still a "workable" number of tones for an octave. There are many other microtonal systems, and not all of them are so convenient! For example: recall in the previous post that, when generating the frequencies for a Pythagorean scale, I decided to stop after having generated a frequency that was was "close enough" to the root, because we wouldn't generate a tone closer to the root until after another 53 steps. That was a motivated choice, but still a choice: I certainly \em{could} have kept generating new tones and adding them to the scale until I had generated all 53 notes. After all, we're just concerned with building up a set of frequencies we can use together, so why \em{not} compose music on a 53-note scale?
140141
141142 \svgimg{/static/microtonal/53tet.svg}
142143
143 The problem is, of course, that to write music for such a scale, I would need instruments that could \em{play} the notes on this scale. The nineteenth-century English \link{https://en.wikipedia.org/wiki/Robert_Holford_Macdowall_Bosanquet|music theorist Robert Holford Macdowall Bosanquet} actually \em{did} invent a device capable of playing music using this 53-note extended Pythagorean tuning! It looks like this\ref{attr1}:
144 The problem is, of course, that to write music for such a scale, I would need instruments that could \em{play} the notes on this scale. The nineteenth-century English music theorist \link{https://en.wikipedia.org/wiki/Robert_Holford_Macdowall_Bosanquet|Robert Holford Macdowall Bosanquet} actually \em{did} invent a device capable of playing music using this 53-note extended Pythagorean tuning! It looks like this\ref{attr1}:
144145 \sidenote{This image was borrowed from \link{http://tardis.dl.ac.uk/FreeReed/organ_book/node26.html|this page of photographs of enharmonic harmoniums}, and is copyrighted by the Science and Society Picture Library.}
145146
146147 \center{\img{/static/microtonal/bosanquet.jpg}}
147148
148 Actually playing this instrument is very difficult, and \em{certainly} more difficult than mastering a twelve-key piano keyboard! And that's just for a keyboard: attempting to build, say, a guitar fretboard that supports 53 tones per octave would be incredibly difficult! This is a major advantage of a twelve-note scale: it's easy to build physical instruments to accomodate it. Because nineteen is still \em{pretty} small, we can still build reasonable physical devices to accomodate it: for example, piano keyboards that include a handful of extra black keys:
149 Actually playing this instrument is very difficult, and \em{certainly} more difficult than mastering a twelve-key piano keyboard! And that's just for a keyboard: attempting to build, say, a guitar fretboard that supports 53 tones per octave would be incredibly difficult! This is a major advantage of a twelve-note scale: it's easy to build physical instruments to accomodate it. Because nineteen is still \em{pretty} small, we can still build reasonable physical devices to accomodate it: for example, people have build nineteen-note piano keyboards that just include a handful of extra black keys:
149150
150151 \svgimg{/static/microtonal/keyboard-layout.svg}
151152
152153 Of course, we now have a lot of digital instruments, so for modern music, we can be far more adventurous, and indeed composers like electronic music pioneer \link{https://en.wikipedia.org/wiki/Wendy_Carlos|Wendy Carlos}\ref{wendy}
153154 \sidenote{Wendy Carlos is a personal hero of mine! She's composed music you've certainly heard, including spectacular movie soundtracks for \em{A Clockwork Orange}, \em{The Shining}, and \em{Tron}, and she also pioneered a lot of early work in electronic music. Her 1968 album of Bach pieces played on a Moog—called \em{Switched-On Back}—was a major factor in bringing electronic instruments to a wider audience. Look up her music!}
154 have taken full advantage of the current obviated need for physical instruments. But still: it's possible to build a nineteen-tone flute in a way that's it's not for a fifty-three-tone flute, which is why nineteen-tone scales are often a first exploration for budding microtonal musicians.
155 have taken full advantage of the current obviated need for physical instruments and experimented with wilder and stranger systems of tuning. But still: it's possible to build a nineteen-tone flute in a way that's it's not for a fifty-three-tone flute, which is why nineteen-tone scales are often a first exploration for budding microtonal musicians.
155156
156157 \h1{And Even More Scales}
157158
158159 So far, I've demonstrated scales that include twelve, nineteen, and fifty-three tones: are there more? Why, of \em{course}!
159160
160 In addition to choosing equal-tempered scales by approximating just intonations, as I've done so far, there are numerous other ways we could construct scales: if we wanted to, we could even just choose equal-tempered scales by running through numbers and discovering whether those choices result in pleasing intervals or not.\ref{friends}
161
161 In addition to choosing equal-tempered scales by approximating just intonations, as I've done so far, there are numerous other ways we could construct scales. If nothing else, we could just keep choosing numbers, splitting the octave into that number of equal parts, and seeing whether the resulting scale is convenient for music or not: not the most efficient way of doing this kind of exploration, but certainly possible.\ref{friends}
162162 \sidenote{Several friends I talked to assumed that twelve-tone equal temperament had been discovered more or less this way, rather than by successive modifications of just intonation!}
163163
164 For example, if we want to keep the same set of intuitions we've built up for twelve-tone scales, but want a greater degree of tonal richness, we could just subdivide each 12-TET (that is, 12-Tone Equal Temperament) interval into two, and we'd get a 24-TET scale. The 24-TET scale also has a precedent in Middle Eastern music theory: while Middle Eastern music doesn't use a 24-tone scale directly (instead opting for a larger numbers of unusual scales built around certain small intervals), the Lebanese historican and music theorist \link{https://en.wikipedia.org/wiki/Mikhail_Mishaqa|Mikhail Mishaqa} described how many of the scales used in Arabic and Persian music can be described and analyzed as being (in effect) selected subsets of a 24-TET tuning.
164 Consider: if we want to keep the same set of intuitions for composing for twelve-tone scales, but also want a greater degree of tonal richness, we could just subdivide each 12-TET (that is, 12-Tone Equal Temperament) interval into two and get a 24-TET scale. The 24-TET scale also has a precedent in Middle Eastern music theory: while Middle Eastern music doesn't use a 24-tone scale directly (instead opting for a larger number of smaller scales built by repeated application of small intervals), the Lebanese historican and music theorist \link{https://en.wikipedia.org/wiki/Mikhail_Mishaqa|Mikhail Mishaqa} described how many of the scales used in Arabic and Persian music can be described and analyzed as being (in effect) selected subsets of a 24-TET tuning.
165165 Other popular choices have included 29-TET (which includes closer-to-perfect fifths than 12-TET), 31-TET (which includes reasonably approximated thirds) or even 72-TET, championed by microtonal musicians like \link{https://en.wikipedia.org/wiki/Joe_Maneri|Joe Maneri}.
166166
167167 We also can go \em{lower} than twelve tones per octave and still produce workable scales: in particular, 5-TET and 7-TET are both viable options. Both 5-TET and 7-TET in fact show up in various kinds of non-western music theory: for example, Thai musical instruments are regularly tuned to a close approximation of 7-TET, and some scholars contend that there's a 5-TET basis to one of the two major methods of tuning Javanese \em{gamelan} ensembles.
168168
169169 There's yet another wrinkle we can add to our tuning systems: we've been treating the octave as an omnipresent fact of life in all these systems, but we can throw that out the window, too! For example, the well-known \link{https://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale|Bohlen-Pierce scale} is a scale that deliberately ignores the \tt{2:1} octave in its structure, and instead uses a \tt{3:1} ratio, sometimes called a \em{tritave}, as a basic element in its tuning, which ends up giving it a very different auditory flavor, as well as making the system very different to compose for.
170 Yet another example: the aforementioned Wendy Carlos has an \em{absolutely wonderful} album called \em{Beauty In The Beast} where she experiments both with various non-Western tunings as well as some non-octave repeating scales of her own creation, the so-called \em{alpha} and \em{beta} scales. These scales are created by splitting a smaller interval, such as the perfect fifth or minor third, into equal parts that can then be repeated: these scales don't end up repeating on an octave level, but rather repeat only on the smaller intervals on which they are modeled.
170 Yet another example: the aforementioned Wendy Carlos has an absolutely wonderful album called \em{Beauty In The Beast} where she experiments both with various non-Western tunings as well as some non-octave repeating scales of her own creation, the so-called \em{alpha} and \em{beta} scales. These scales are created by splitting a smaller interval, such as the perfect fifth or minor third, into equal parts that can then be repeated: these scales don't end up repeating on an octave level, but rather repeat only on the smaller intervals on which they are modeled.
171171
172172 \h1{Well, How Do They Sound?}
173173