In 1962, Thomas Kuhn's The Structure of Scientific Revolutions introduced the concept of the paradigm shift. Since then, the term has been so widely abused that the venerable Wikipedia says that it "bears no meaning whatsoever". Well, now I'm joining the pack, and with the help of some quotes stolen from some Wikipedia articles that I didn't even read on Wikipedia, I'm going to describe the way I look at tuning theory as a new paradigm.
Put simply, the regular mapping paradigm is a new way of thinking about microtonal scales according to their relationship to just intonation. So far, I believe this paradigm is only shared (or even understood) by a handful of Yahoos on The Tuning-Math List None of us are academic music theorists or famous musicians. There are no peer reviewed articles describing our paradigm. There have been no conferences devoted to the new paradigm. There have been no articles in hip magazines outlining the new movement in music theory. The rest of the world hasn't really noticed that anything's happened yet.
Still, despite the momentous waves that haven't been made, the rediscovery of miracle temperament did change the way I think about the music I'd like to write. The fact that the temperament had to be rediscovered for us to get excited about it is already a clue that a paradigm shift is required to assimilate it. It isn't even that the theory was lacking back when George Secor originally discovered it. He managed to describe it, after all. In fact, none of the concepts I'll be describing here is really new.
I'll outline here some other paradigms used in microtonal circles, and show how miracle can be understood in them. In each case the understanding is incomplete, because miracle requires concepts that don't really belong in any of them. It's a sign of the diversity of microtonal thinking that the new paradigm doesn't replace a single old one, but unifies three different ones. I wonder what Kuhn would make of that.
If you've seen The Sound of Music you'll know that, as reading begins with ABC, so music begins with do re mi. The dominant musical paradigm in the western world involves 7 note diatonic subsets of a 12 note chromatic scale. It's been dominant for hundreds of years in Europe, and also holds in India and China. You probably already know about it (indeed share it) so I won't talk too much about it here.
Under this paradigm, keyboard instruments have 7 white and 5 black keys to the octave. Temperaments are different tunings of the 12 note octaves on keyboard instruments. Guitars have 12 frets to the octave. Instruments are generally built to play 12 notes to the octave. Some instruments may only handle 7, or wind instruments may have 6 holes (to give a 7 step octave) with clever fingerings to get the other 5 notes. Notation is either based on a 7 note scale with accidentals (like staff notation) or a closed 12 note scale (like guitar tablature).
You could argue that there are really two paradigms at work here: 7 note diatonicism and 12 note chromaticism. Well, yes, but the two get mixed up so much you can't tell which is operating when. Besides, I'm not here to talk about music history (of which I'm about as ignorant as paradigm shifts) so for the sake of simplicity let's call it one paradigm.
When a paradigm has held sway for as long as the do re mi of the diatonic scale it's only to be expected that some people have found it inadequate to contain their ideas. Such is the strength of a paradigm that they'll usually try to bend the prevailing one to accomodate their new thinking. Perhaps you'd rather say that some of these extensions should constitute new paradigms. It doesn't really matter.
Meantone is the do re mi with consonant thirds and perfect fifths. It's been tuned different ways throughout history, and most tunings of the meantone era were sufficiently different from equal temperament that you could get microtonal intervals by using more than 12 notes. In the 16th and 17th centuries, people really did build instruments with extra keys to handle these extra notes. But the idea didn't catch on, and most music didn't make use of the microtonality anyway, so the paradigm survived.
I've got a page on this temperament. It's a common first step from extended meantone because you keep the same notes but interpret them differently. As it still involves 7 from 12 scales I call it an extension of the do re mi paradigm, rather than a new one.
The atonal movement was really about discarding the do re mi itself but keeping the 12 note scale it was embedded in.
When your keyboard doesn't have the note you want to play, it's natural to wish another key between the nearest two you have. Quartertones have arisen in different guises throughout history. Ancient Greek music had them, although it really predates the do re mi paradigm. Arabic notation uses quartertone symbols. Vicentino split some intervals of extended meantone into nominally equal parts. More recently, composers have played with quartertones as an extension of atonality.
If two equal divisions aren't enough, why not try more? Composers who want to use all possible notes as the fancy takes them, but can't escape the pull of the chromatic scale, can take it as high as you like. It's also a convenient way of getting musicians to play new notes.
I hesitate to pin down the magical shape-shifting sruti. I'm sure some of you will quibble with my very equation of the sa ri ga with the do re mi. Still, one way of thinking about srutis is as small deviations from a 12 note chromatic.
How do you describe it within the do re mi paradigm? Well, basically, you can't. However 72-equal is an almost optimal miracle tuning, and it's also a multiple of 12. So by dividing the semitones of the chromatic scale into 6 equal parts you can notate music written in the temperament.
There are two problems with this approach: it's horribly complex and you lose flexibility of tuning. With the sagittal system you need 13 accidentals to notate the 21 notes of the blackjack scale. The notation doesn't reflect the structure of the temperament but at least you can get musicians to play it with minimum effort.
This shows that the do re mi paradigm is incapable of dealing with any tuning system in a simple way if it has a radically different structure to the diatonic scale. Many composers are still happy with this because they don't want to think of such structures. For many microtonalists, it's painfully inadequate.
Just intonation (JI) begins as a way of tuning the do re mi. It doesn't really work within that paradigm because you find that the same note requires different tunings at different times. When you start looking at consonances beyond the 5-limit of major and minor chords, the whole thing starts to break apart. The scales start finding their own structure that doesn't include the do re mi and you find yourself looking at a new paradigm.
Harry Partch is the key figure here. His instruments aren't based on 7 or 12. The notation is based around the instruments. The notes are named as ratios. The harmonies are taken from simple ratio relationships. He made compelling music with no trace of the do re mi.
The principle behind JI as a universal principle is that the intervals of JI are the ones the ear most likes to hear. Everything else is a better or worse imitation. To describe the reality you describe JI.
Historically speaking, miracle temperament did arise as an extension of JI. George Secor's Xenharmonikôn article is about a new keyboard layout for Partch's infamous 43 note scale. The business of tempering is presented as an afterthought. Of course, as soon as you start tempering you leave JI behind. Consonances in a miracle scale don't always look like consonances after you map them to Partch's scale. They might still sound like consonances but when you start thinking like that you're already stretching the JI paradigm.
Not surprisingly, the JI paradigm is inadequate when dealing with temperaments. You can approximate any tuning to arbitrary precision with a sufficiently complex JI system. But you gain complexity and lose flexibility. JI is only one part of the microtonal world.
When you're bored of the same old 12 notes, it's natural to try some other equal temperaments. They're very easy to describe in theory, and lend themselves to electronic production. Ivor Darreg found he could get a whole range of them by abusing a knob on his synthesizer. So he wrote music in all of them, and others have followed his lead, notably Easley Blackwood.
The paradigm breaking feature of equal temperaments is that you can easily hit upon something weird like 11 or 13 that twists the do re mi into something grotesque. The easiest thing is to give up your familiar landmarks and think about step sizes instead of more or less out of tune versions of familiar intervals.
You don't need to keep the octave. You can choose any interval and repeat it to get an equal temperament, whether it adds up to something like an octave or not.
There's also a good theory behind equal temperaments. As theorists got hold of computers, they started producing tables showing how well each temperament approximated different intervals from just intonation. From these, you can get some idea of how weird a temperament's going to sound before you tune it up.
The idea behind equal temperament as a universal paradigm is that equal steps are an easy way of thinking. You can program a synthesizer or dream up a notation that works with any equal temperament. Whatever scale you really want, you can find an equal temperament that gets arbitrarily close to them. So when you start thinking in terms of equal temperaments it's tempting to see everything as progressively complex equal temperaments.
Miracle scales can be described as subsets of either 31, 41, or 72 note equal temperament. For example, the 21 note "blackjack" scale looks like this:
In fact, you can describe any regular temperament class as a set of equal temperaments. A regular temperament is an approximation to JI where a given target ratio is always approximated to give the same tempered interval. A regular temperament class is a set of regular temperaments that have the same pattern of different sized intervals, but different specific sizes. You can even think of them as being families of equal temperaments with increasingly large numbers of notes. But that's a needlessly complex way of thinking about them, which is a sure sign that you're in the wrong paradigm.
The description of blackjack as a subset of 72-equal was successful enough that most of the early criticisms revolved around it being an equal temperament. This view doesn't capture the flexibility of the miracle tuning. You can vary the relative interval sizes over a continuous range, and they needn't form a closed scale. It's also needlessly complicating to use 72 notes to define a scale with only 21 notes and 2 step sizes.
Not only is the equal temperament paradigm incapable of describing JI scales in a simple fashion, it doesn't work well with all temperaments. There are other tunings, including most of those empirically deduced from folk practice, that aren't close to simple equal temperaments and so aren't well described in terms of equal temperaments. Equal temperaments are only one part of the microtonal world.
Mervyn King memorably described a paradigm as "a word too often used by those who would like to have a new idea but cannot think of one." Well, that cerainly applies here. I've said it before and I'll say it again: none of the ideas here is really new. Back to Wikipedia: "There are anomalies for all paradigms, Kuhn maintained, that are brushed away as acceptable levels of error, or simply ignored and not dealt with." So, why did we stop ignoring the anomalies in the above paradigms and start to deal with them?
People have been talking about linear temperaments for a long time now. If the microtonal world really consisted of three mutually exclusive paradigms like I described above, this would have caused our brains to explode. As it is, we knew there were regular temperaments that weren't equal temperaments but didn't think much about it.
It helped that there weren't many unequal regular temperaments that we had to worry about. I had meantone, schismatic and diaschismatic explained on my website along with a page about neutral third scales. As it happens, these can all be viewed as extensions of other paradigms.
Meantone and schismatic are both listed above as extensions of do re mi. Meantone is one of the main ideas behind traditional harmony. Schismatic is different, in that you have to spell chords differently, but it still works with a spiral of fifths, a 7 note diatonic, and a 12 note chromatic.
Diaschismatic is a bit more of a challenge. It requires the octave to be divided into two equal parts, and so has two spirals of fifths. That means it isn't a linear temperament at all with the current terminology. It's notable that Paul Erlich's landmark paper on a variety of diaschismatic (now known as pajara) is titled Tuning, Tonality, and Twenty-Two-Tone Temperament. So it appears to be about an equal temperament, and many people think it is an advocacy for 22-equal in general rather than one particular regular temperament class that includes it.
As for neutral third scales, a neutral third is the quartertone between a major and minor third. Hence these are really about (flexibly tuned) quartertones and so an extension of the do re mi. They even have 7 notes!
There are now hundreds of unequal regular temperaments that have been listed, pondered, discussed, and some have even been used for making music. It's much harder to think of them as anomalies in a microtonal world dominated by just intonation and equal temperaments.
Once we started looking through all these regular temperaments, people also started giving them names. When you have scores of things with names, they're a big part of your world view. So scores of named unequal regular temperaments means that some of us take them pretty seriously.
One thing about the temperaments we're looking at now is that they can be extremely accurate. The price for this is that they're also frighteningly complex. For example, dear old Hemiennealimmal gets the 11-limit to within about 0.2 cents of JI, but needs a 72 note scale to get it done properly. Before, we thought 29 notes of schismatic temperament was worryingly complex.
An obvious problem with such temperaments is that you can't fit them onto a normal keyboard. The equal temperament paradigm described above could be thought of as part of a "tuning table paradigm". The idea is, a tuning is something you put on your keyboard and then make music with. Equal temperaments happen to make the simplest tuning tables. Hairy beasties like hemiennealimmal won't work as a tuning table. You need to think in a more abstract way, and so really consider the nature of the temperament.
One reason for looking at these super-accurate temperaments was to get arbitrarily close to JI. This caused an enormous amount of consternation on the tuning lists. You can understand the strength of this if you think about the new temperaments as challenging the paradigm of just intonation as being that which is not tempered.
Consistency is a concept Paul Erlich came up with to ensure that the best approximations to different JI intervals add up the right way in an equal temperament. Some previous theorists ignored inconsistency, and so gave high marks to inconsistent temperaments for different intervals that couldn't work together in the same chord. Once you can brand them inconsistent you can also consider them defective and so ignore them.
A more recent development is that people are looking at inconsistent temperaments and facing up to their inconsistency. To do this, you have to state how you interpret each of the inconsistent intervals you're using and when. That means the interpretation becomes an important part of the temperament, and so even with equal temperaments you're looking at a regular mapping paradigm. The pure equal temperament paradigm assumes that you know everything about an ET from how many notes there are to the octave.
If you've been paying attention, by now I've hopefully convinced you that either we need a new paradigm here or that I'm a raving lunatic. Well, now you know what it isn't, you may well wonder what this new paradigm looks like. It's really too early (and maybe I'm too closely involved) to say what the indispensible features are. The following things are notably unoriginal.
The simplest tunings under this paradigm are built up from a small number of intervals called generators. For just intonation, there's a generator for each prime number in the ratio. For equal temperaments, the step size is the generator. The do re mi scale can use generators of an octave and fifth, or a semitone and tone, or two different tones, and so on. All intervals in such a tuning come about by adding and subtracting the generators.
For a strict linear temperament, you can always choose one generator to be the octave. The other generator is then simply called "the generator". For any regular temperament that approximates the octave, you can still choose one of the generators so that it equally divides the octave. This is then known as "the period". Unequal temperaments where the period doesn't equal the octave are a hallmark of the regular mapping paradigm.
The mapping tells you which intervals in your scale correspond to which ratios in just intonation. For an equal temperament, the mapping can be written as a list of numbers showing how many steps there are to each prime-number interval. For higher ranked regular temperaments there's a different list for each generator.
In tuning-math circles, it's conventional to write the lists of numbers in the mapping in a notation similar to Dirac's bra vectors. For example, the mapping for 12 note equal temperament in the 5-limit is
<12, 19, 28]
That tells you that the mapping of an interval of 2:1, commonly known as an octave, is 12 steps. The mapping of an interval of 3:1 is 19 steps. A perfect fifth is a mapping of 3:2, and so the difference between the mapping of 3:1 and the mapping of 2:1. Here that means 19-12=7, so a perfect fifths is 7 steps from 12-equal. The mapping of an interval of 5:1 is 28 steps. A major third (assuming meantone temperament) is the mapping of 5:4, and so the difference between the mapping of 5:1 and twice the mapping of 2:1. Here that means 28-2*12=4. So a major third is 4 steps from 12-equal.
The mapping for a 7 note diatonic scale in meantone temperament is
< 7, 11, 16]
Follow the sums through, and you'll see that a perfect fifth maps to 4 steps. A fifth is 4 diatonic steps, yes. That's diatonic numbering for you. Similarly, a major third maps to 2 diatonic steps.
The specific list of numbers you get depends on which generators you choose. The mapping from tempered to just intervals doesn't depend on the choice of generators. As such, the mapping is more fundamental than the generators but it's easier to explain the generators first. Choosing one generator to be the period reduces the number of different ways of writing the mapping, and so this is the first step towards a standardized representation of the mapping.
The fact that I'm calling all this "the regular mapping paradigm" obviously means that I think the mapping is pretty important. It may even be the most original of these key ideas. I can't think of anybody showing mappings as lists like this before the paradigm started to take shape.
Scales with the same mapping have many things in common. They'll work with the same generalized keyboard mappings. You can use the same system of notation for them. You can move melodies and chord sequences between them. If they're regular temperaments, they'll have similar patterns of large and small intervals.
Perhaps following the regular mapping paradigm means feeling that the things that the mapping tells you are the most important features of a scale.
If a given generator always means an interval of the same size then you have a regular tuning. When you apply the mapping you have a regular temperament. The rank of a regular temperament is the number of generators it has. So an equal temperament is rank 1, a linear temperament is rank 2, and so on.
It's tempting to think that this paradigm is all about regular temperaments. However, you can also apply the mapping to irregular tunings. So I'd like to point out that, although regular temperaments are extremely important they aren't all that the paradigm's about.
Knowing the optimal accuracy of a regular temperament can even be useful if you don't plan to use the temperament. The closer a regular temperament gets to just intonation, the closer a JI rendition will come to having regular scale steps. Also, if you want to play some chords that would be consonances in the temperament but map to the wrong ratios in the JI rendition, the better the temperament the closer those JI intervals will be to the ideal JI intervals.
Unison vectors were originally described by Fokker without reference to mappings or unequal regular temperaments. That's a shame, because it's easy to define the one in terms of the other: a unison vector is an interval that maps to a unison.
In musical terms, unison vectors tell you which intervals become equivalent to each other. If you write a chord sequence that won't work in JI, you can inspect it to see what unison vectors it depends on. You can then translate it to any scale whose mapping shares those unison vectors.
A suitable set of unison vectors contains the same information as the mapping. There will usually be different unison vectors you could choose, the same way that there are different lists of numbers that could denote the mapping. We have mathematical algorithms to convert between mappings and sets of unison vectors so the two concepts are nearly equivalent.
Some people don't like the term "unison vector" and we haven't agreed on an alternative. Still, the name lives on in my heart.
Miracle temperament was the first regular temperament I looked at seriously that doesn't fit any of the above paradigms. I've done my best to bend it into shape in the descriptions above, but also shown you why each description is inadequate.
In the regular mapping paradigm, miracle temperament is a rank 2 temperament with the mapping
[< 1, 1, 3, 3, 2], < 0, 6, -7, -2, 15]>
And that's all there is to it! Its properties all follow from it being a regular temperament and having that mapping.
The linear mapping paradigm ties together a lot of different threads of microtonal thinking. While some of these don't require all of the key ideas I listed above, you can think of them easily without leaving the paradigm.
In regular mapping terms, the do re mi scale is usually related to meantone temperament. I've already given you the mappings for the 12 note chromatic and 7 note diatonic in the mapping section.
You can also write the mapping for the meantone (and so typical do re mi) according to it's period (the octave) and generator (here a perfect fourth)
[<1, 2, 4], <0,-1,-4]>
Using this mapping, you can show that 5, 7, and 12 note scales are likely to be important. You can show that using dry mathematics. You can't derive the whole edifice of common practice harmony from it, or jazz or bossa nova. But it's a perfectly good foundation.
Looked at using unison vectors, that foundation becomes even more embarrasingly simple. The whole of the Western tradition (and some Eastern ones as well) requires 81:80 to be a unison vector. That's it.
None of this should come as a surprise because the whole idea of regular temperaments started out as a generalization of meantone.
Just intonation is a regular tuning with a mapping onto itself. That's all there is to it. Regular mappings are somewhat wasted on it, but it's entirely consistent with them.
So far so trivial, but using regular mappings you can convert interesting scales, chords and melodies into whatever other tuning system you happen to be interested. That means regular mapping composers have a lot to learn from JI composers, even if those JI composers would prefer the influence only flowed one way.
Scales, chords and melodies that work with any given mapping will not always work with JI. Otherwise we wouldn't need temperaments, would we? Because regular temperaments can get arbitrarily close to JI the results may sometimes be indistinguishable -- but for a few impossible chord sequences. The errors may even be so small that JI itself works as a valid tuning.
Of course, you can find a JI tuning arbitrarily close to any other scale as well. But that's a different matter because the JI logic might not mean anything.
Equal temperaments are a special case of regular temperaments. Everything from the regular mapping paradigm applies to them directly. The more advanced concepts like mappings and unison vectors can be useful. There's no redundant baggage when thinking about equal temperaments in these terms.
You can also think of equal temperaments as special cases of regular temperaments. There's more to this than finding an equal tuning with arbitrary close approximations to any given interval. Regular temperaments classes have a range of acceptable tunings, some of which will be equally tuned. For rank 2 temperaments in particular, there's a simple procedure to find equal temperaments that get closer and closer to the tuning you're interested in. Then you can write your regularly tempered scale as a subset of an equal temperament.
Whether you think of equal temperaments as a special case of regular temperaments, or regular temperament scales as ways of taking subsets of equal temperaments, is a good test of what paradigm you're working in. If you are happy with equal tunings, the scales themselves are the same however you look at them.
A Moment of Symmetry (MOS) scale has a period and one other generator, and some other properties. Because it isn't clear if the period is allowed to be different from the octave in an MOS, some people talk about Distributionally Even (DE) scales instead. Some other terms are "well formed" and "myhill's property". They all come to the same thing.
When you're working with rank 2 temperaments, a lot of the tunings will naturally be MOS/DE. But the MOS concept itself doesn't say anything about the mapping. So MOS is fully compatible with the regular mapping concept, but only a small part of it.
Periodicity blocks are a concept Fokker came up with that relate to unison vectors. He may have thought of them as being either equal temperaments or just intonation, but in English he always described them as just intonation scales.
A periodicity block is defined by a set of unison vectors. It contains the maximum number of notes within the octave and a given prime limit such that no interval between them is entirely made up of unison vectors. Fokker also seems to have an algorithm for choosing a particular scale for a given set of unison vectors.
In regular mapping terms, a periodicity block works with two different mappings. One describes an equal temperament in which the unison vectors map to unisons and the other is the trivial mapping for just intonation. The number of notes in the abstracted equal temperament matches the number of notes in the periodicity block. In these terms, it's easy to see how a periodicity block can be either an equal temperament or a JI scale.
Periodicity blocks are an example of how the regular mapping paradigm needn't only apply to temperaments. Periodicity blocks do have enough desirable properties in JI that they're considered valuable in their own right.
Generalized keyboards date back at least as far as the 19th Century. They're designed so that all keys have the same fingering. Or, a particular chord or melody can be changed to start on a different note without changing the fingering. This is called "transpositional invariance". Button accordions work a bit like this. Generalized keyboards can be used with any systems of two generators. A three dimensional keyboard is theoretically possible, but would be difficult to play. With electronic protocols such as MIDI the keyboard needn't have a fixed tuning, and so the generalized nature reaches its full potential.
Naturally, generalized keyboards will tend to work best with rank 2 temperaments. But Erv Wilson has also applied them successfully to just intonation. In a similar way, you can use them for higher rank regular temperaments, or any scale that shares its mapping with a rank 2 temperament.
Unison vectors help you fit a higher rank scale to a two dimensional keyboard. You can find possible unison vectors by looking at intervals within the true scale of a similar size. The difference between them could be a good unison vector. The smaller the unison vectors you choose, the less violence the transpositional properties of the keyboard will do to the scale. So long as you don't cheat and use unison vectors that are so complex they have no musical meaning.
Adaptive tuning is any keyboard/score and synthesizer type system where the tuning is automatically changed to reflect the notes played. Usually that means you play as if in a temperament and the chords are magically tuned to just intonation. This field is still in its infancy, and working systems usually assume a 12 note octave.
For adaptive tuning to work with arbitary systems, you need to define the mapping from the scale used by the keyboard or score into the intervals you want to hear. A regular temperament mapping is a good way to do this. So adaptive tuning becomes regular temperament without the fixed temperament. The result is that, like with flexible pitch instruments, the composer can write music for a particular mapping rather than a particular tuning.
There are other trends related to the new paradigm that should make adaptive tuning more practical. If you use a more accurate temperament, there's less bending for the adaptive tuner to do, and so less inconsistencies in what the listener hears. If you use a keyboard or notation based on an unequal regular temperament (without the temperament) it means you can be more specific about what intervals you intend than if you used a one dimensional input scale. Finally, the output may be a different regular temperament to the input, rather than just intonation, so that the adaptive tuning becomes more like a way of choosing the key center.
Another interesting idea is to use different mappings at different times. If a certain chord sequence requires certain unison vectors to be tempered out, then you apply the corresponding mapping. If a different sequence requires different unison vectors then you need a different mapping. In between you use the mapping determined by the unison vectors that both sequences have in common. This is tricky to think about and notate. You could write in JI but with explicit approximations, for example. But it's even harder to think about if you leave the regular mapping paradigm.
So far, all the talk's been about mappings from just intonation. That assumes we're only interested in harmonic timbres, or at least we want to write harmony as if all timbres were harmonic. In fact, the paradigm doesn't depend on any specific properties of the harmonic series.
Take any instrument with an inharmonic timbre, or a group of instruments that you want to sound good together. By ear or egg-headed theory you can find some intervals that are consonances within this system. Then, construct an artificial "inharmonic series" that contains the intervals you're interested in. You can define mappings from this series the same as from the original harmonic series for JI.
William Sethares has already done work with equal temperaments and inharmonic timbres. There's nothing special that goes wrong when you increase the rank of the temperament.
You can also find timbres that work with a higher-rank regular tuning. I tried this with my piece for the first MakeMicroMusic day and nobody complained. You could even try to get the timbres to follow the actual tuning of the regular temperament in use.
As with other JI intervals, you can replace the octaves if you like. You can also stretch and shrink octaves to taste.
In terms of group theory, the mapping is a homomorphism from one group of intervals to another. Sets of intervals are free abelian groups. The mapping for a regular temperament is strictly an epimorphism. The mapping from JI to itself is an isomorphism. Unison vectors make up the kernel of the homomorphism. Periodicity blocks are like transversals, although strictly speaking periodicity blocks are sets of notes, but group theory deals with intervals.
This group theory formulation doesn't account for human limits to pitch perception. For sets of intervals to be groups, you have to assume they can get arbitrarily large or small and still be perceived correctly. These aren't details we usually worry about. We use the mathematics as if the intervals were perfect and apply common sense limits later on if we need to.
If you don't know anything about group theory it doesn't matter. You don't gain much by thinking this way. But if you do understand group theory these concepts should all be familiar.
I'm not sure exactly what diatonic set theory is, but from the page on Wikipedia it all seems to be consistent with regular mappings. Probably musical set theory in general works as well. Also, Rothenberg's work on propriety, efficiency, etc. Like the MOS, these properties don't depend on the mapping but aren't broken by it either.
The Sagittal notation system was developed by George Secor and Dave Keenan, both of whom are identified with the regular mapping paradigm. George originally discovered miracle temperament and Dave did the first systematic search for linear temperaments. Sagittal symbols stand for small intervals, defined by either a range of sizes or an archetypal JI ratio. Finding a notation for an equal temperament is therefore related to finding a mapping, and temperaments with similar mappings may have similar notations. You are, in a sense, notating the mapping rather than the tuning.
The description from Xenharmonikôn only mentions JI and ETs, but you can be sure that anything that works with both of them is also fine with the regular temperaments in between. There are some standard symbols for rank 2 temperaments ... somewhere ...
Ideally, you'd use custom staves to house your Sagittal notation. The standard system uses a usual staff with 7 notes to the octave. This isn't a good match for mappings that don't work naturally with 7 note scales. Herman Miller has looked at pseudo-nominals to adapt such tunings to a normal staff. But it would be nice to set the real nominals to be the notes you want.
For now, you'll have to use a standard system to get music performed. If the regular mapping paradigm takes off, so that composers write lots of music in unequal regular temperaments and the like, and musicians are interested in performing it, I think it's plausible that those musicians might learn to read other staves. I reckon 7, 9, 10, and 12 nominals to the octave should do for most purposes. That only means 4 times as much training as for a single system and, considering that professional musicians are really very good these days, certainly isn't impossible. Say you need to learn 10 pieces that use a given set of nominals to be really comfortable with them. That means a repertoire of 40 pieces for those 4 nominals. Perhaps élite musicians could also handle 11 or 13 notes, and some non-octave systems.
It would be nice to claim that the regular mapping paradigm provides a cure for all the ills of the microtonal world. There are, however, some ideas that don't naturally fit within it. While adopting the paradigm doesn't mean you suddenly become unable to think about these things (any more than you were incapable of grasping miracle temperament before) they do have to be treated as ugly extensions.
Some microtonalists at some times abandon the idea that the harmonic series, or a related concept for inharmonic timbres, is something they have to think about for their music. If you don't recognize some kind of JI ideal then the mapping doesn't really tell you anything. As the regular mapping paradigm puts that mapping first, it's useless in this context.
Resultant tones are sum and difference tones. When two sine waves sound together, the ear can also hear notes derived from the sum and difference of their frequencies. Because regular mappings work with (some abstraction of) generators rather than frequencies, these calculations are harder to do.
You can always define functions to return the resultant tones, but it's easier to stick with the frequency ratios. Scales defined around resultant tones are unlikely to work with regular mappings.
Some composers think of just intonation in terms of a static, harmonic series. When you do that the limit stops mattering, and defining your scale as a regular tuning can be extremely cumbersome. Also, this way of thinking usually ties in with a desire for extreme purity of harmony that renders temperament irrelevant. In such a context it's easier to stick with the frequency ratios and not worry about generators or mappings.
In addition, high prime JI often ties in with control over resultant tones. LaMonte Young's Dream House does this, and also the chorus effect from Greg Schiemer's Tempered Dekanies.
In terms of regular mappings, well temperaments look like equal temperaments. They share a mapping from JI to intervals defined as a number of scale steps. But they aren't regular tunings. So there's no mapping from intervals defined using scale steps to the specific intervals you hear. You have to map directly from notes to pitches.
You could define a generator for each independent note. Usually that means each note within the octave. But that means an interval from JI can end up being represented differently by the generators. Really, it's better to say that the regular mapping paradigm doesn't tell you anything about the irregularity of the temperament.
You can still use well temperaments within the paradigm. You can think of them as being like equal temperaments but with good and bad keys. More importantly, you can write a piece ostensibly in an equal temperament, but only really care about the mapping. So the performers are free to use any irregular tuning they like.
An inconsistent temperament has more than one viable mapping. That gives it an ambiguous position in a paradigm that's all about the mappings. On the one hand, it's difficult to describe the inconsistency without using mappings. On the other hand, it may break the principle that we choose a mapping rather than a tuning before writing a piece of music.
I described inconsistent equal temperaments above as being a paradigm breaking development. So it's worth pointing out that they aren't properly covered by the new paradigm either. You can use an inconsistent temperament with only one mapping, but if the listener hears intervals that work with a different mapping you can't really tell them otherwise.
It isn't only equal temperaments that can be used in an inconsistent manner. For example, in quarter comma meantone an augmented sixth (e.g. C-A#) is a good approximation to the just interval 7:4. However, conventional music makes considerable use of dominant seventh chords, where a minor seventh (e.g. C-Bb) may also be identified with 7:4. (This mapping even has a different name -- "dominant temperament".) Although it's a much poorer approximation, if the ear can hear it, the theory should account for it. The more complex a temperament gets, the more likely a simpler interval might get close enough to an interval it isn't intended to be close to that the listeners will notice. This more general form of inconsistency looks like a big problem that we're overlooking because it doesn't fit within our paradigm.
I can't think of a better way of dealing with inconsistent temperaments. For now, we'll have to talk about them having more than one mapping, and so treat them as an ugly extension of the regular mapping paradigm. If they become popular enough, perhaps somebody will think of a theory that can handle them in a more natural manner. For now it's only speculation, but I can see this anomaly prompting the next paradigm shift.
The regular mapping paradigm is an interesting new way of looking at the pitch structure of music. Although it doesn't cover all microtonal ideas, it does tie up some areas that a few of us have been looking at. At this stage it's too early to say if it will become a dominant paradigm in musical thought, or remain an obscure school of thought outside any influential artistic or academic community.
One unavoidable problem with the paradigm is that it makes things more complex than they were before. That's because it unifies existing ideas, and so it has to be more complicated than any one of them. That means a lot of musicians will never have a use for it. But perhaps they'll pick up on scales developed by people who do think within the paradigm. At least, the musical world's diverse enough to support more than one approach, and there are always going to be a few people pushing for complexity.
Maybe the paradigm will also prove useful for non-Western musics. It's inspired by things like harmony and modulation that aren't so common in other traditions. But then, they are successful ideas that are being adapted to other musics. Perhaps cultures that aren't rooted in diatonic scales will find ways to harmonize their own melodies. There's not really much more I can say about it because, if this kind of development is to have true value, it has to come from within the culture.
Here's a list of specialist or non-standard terms. These descriptions aren't intended to be definitive, and in some cases they're inconsistent with general usage, but they should be enough to help you understand them in context here. For more depth, see the Tonalsoft Encyclopedia.
Thanks to Yahya Abdal-Aziz, Paul Hjelmstad, Carl Lumma, Herman Miller, Joe Monzo, and Gene Smith for suggesting improvements on the earlier drafts.
As soon as I finished polishing this web page, I was pointed to an article called X_System by Andrew Milne, William A. Sethares, and Jim Plamondon. It covers the basic ideas behind this paradigm, but I only recognize William Sethares from the tuning lists.
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The Chromelodeon may look like it's based around 7 and 12 notes, but this is misleading. From Herman Miller on tuning-math:
Harry Partch's chromelodeon was tuned to the 43-note scale, not anything related to the 7-and-12 paradigm. He just retuned all the reeds and left the keyboard as it was (retuning the reeds alone would have been a pretty big task).
Both the original chromelodeon and the "New Chromelodeon II" have piano-style keyboards with labeled keys. The "Old Chromelodeon II" had extra keys added to the piano-style keyboard, but an octave of the 43-note scale still took up the space of two piano octaves.
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Mark Lindley and Ronald Turner-Smith did publish a book including a variation on group theory to take account of the limitations of human hearing. See the associated article at Music Theory Online and review
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From Paul Hjelmstad on tuning-math:
[Enumerative combinatorics] combines Group Theory and Combinatorics. It shows, for example, the number of chords (0-, 1-, 2- triads, tetrachords, etc) in 12t-ET in polynomial form. You can also look at different symmetries like D4 X S3 for example. (The "M5" symmetry).