User:VectorGraphics/Intro to xenharmony
1. 12edo
Most modern music is made with the "12edo" tuning system, which divides to octave into 12 equally spaced notes. This standard (for composition, production of instruments, etc) is because it is the simplest possible equal tuning system that includes a diatonic scale, and because it includes all the intervals that were used in historical Western traditions. However, 12edo is far from the only system out there.
First, let's give a primer on how tuning systems work.
2. L + Ratio
Because pitch perception is logarithmic, any interval can be characterized as a ratio between pitches. The easiest such ratios to tune are simple ratios of whole numbers: 2/1, 3/2, 4/3, 5/4, 5/3, etc, so many historical instruments were tuned to these intervals. Using whole number ratios also has another advantage in that due to the fundamental theorem of arithmetic, which says that any rational number can be written in terms of powers of primes in exactly one way, there is an inherent mathematical structure to any set of whole number ratios in use. Additionally, whole number ratios are easy to recognize by ear, as they produce an auditory effect called "concordance", associated with, on certain timbres, a buzzing effect. Concordance is, essentially, how much two notes feel like "the same note": think about notes an octave up or down in standard music theory.
3. Making Steps
One way to construct a tuning system is to take one of these easy-to-tune just intervals, for example, 2/1 (an octave), and stack it. To stack an interval, you multiply by it. So we get 2/1, 4/1, 8/1, 16/1 going up, and 1/2, 1/4, 1/8, 1/16 going down.
The interval we're stacking by is called a "generator", since it generates a scale or tuning system.
However, this doesn't really give you a set of intervals you can use, since everything is so far apart (and because of octave-equivalence). There are two ways to solve this.
Solution 1. Split the interval
Logarithmically splitting an interval involves taking roots. For example, if we split our 2/1 interval, we might choose to split it into 5 equally spaced parts. This gives us steps that are a bit larger than a 12edo whole tone, and are more usable as steps in a tuning system. 12edo results from splitting 2/1 into 12 equal parts. You can describe an equal tuning by specifying the number of divisions and the interval being divided (although "o" for octave and "f" for fifth are common compared to writing their ratios). This means that our 5 equal divisions of 2/1 can be called "5edo".
Solution 2. Another generator
An alternative option is to choose another generator, so that we have 2 generators. For example, we might choose to include 3/2, a perfect fifth slightly sharper than 12edo's perfect fifth. By stacking up and down by 3/2 and 2/1, we can reach a large variety of notes. In fact, one can approximate any desired pitch arbitrarily closely with rational numbers in 2 and 3. However, it's generally best to choose a limited subset of notes to work with, for example, the first 12 notes within an octave that we reach by this method. This also results in a system with 12 roughly (but not perfectly) equally spaced notes, called "Pythagorean[12]". "Pythagorean" means it involves stacking perfect fifths and octaves, and "[12]" is the number of notes we're using.
Solution 3. Do both at once
One might choose to employ both strategies at once. For example, if you have 2/1 and 3/2 as your generators, and you split 3/2 into 3 equal parts, you can take the first 5 notes within an octave to get a roughly equal pentatonic scale, or the first 6 notes to get a hexatonic scale with an interval that's ever-so-slightly flat of an octave as well as the octave itself.
4. Temperament
Another degree of flexibility involves "tempering" (slightly altering) intervals to achieve other effects.
Cents
When we're talking about slight alterations and precise tunings, it's helpful to use a unit of interval size smaller than a semitone. The most common way to do this is to use 1/100 of a 12edo semitone, called a "cent", so that 12edo's perfect fifth is 700c, and the ratio 3/2 is about 702 cents.
Example A: Improving function
For example, if we take our 6-note scale from before, our largest interval within the octave is about 30 cents flat of the octave. However, you might want an interval that's roughly a semitone flat instead, so that you can use it as a leading tone. According to some theories, the best interval to use as a leading tone is about 1130 cents, which is 70 cents flat of the octave. So, if we flatten our scale in general, so that "3/2" is about 678c (or a quarter of a semitone flat), that results in our largest interval being that desired 1130 cents.
For another example, take Pythagorean[12]. That system has perfect fifths on every note except one, which has a "wolf fifth" of, again, about 678c. While having 678c intervals on their own is fine (hence why our previous temperament works), when there are 702c intervals and just one 678c, it can sound unpleasant and "howl" like a wolf, hence the term "wolf fifth". An approach to solving this issue is temperament. If we sharpen the fifths we use to generate the scale (to about, say, 704-705 cents), the wolf fifth becomes flat enough (about 650 cents) that it doesn't really sound like a fifth anymore, which makes it less of a problem, and easier to use creatively in compositions. Additionally, in terms of just intonation, it can be treated as the simple ratio 16/11.
Example B: Tempering out commas
There is an alternate solution to what we described in either case, which is to temper out the "comma" (the very small interval) between the two notes we're focusing on. This means treating them as the same thing, and adjusting our tuning accordingly. For example, to temper out our ~30 cent interval between our leading tone and our octave, we could sharpen our fifth (to about 720 cents) instead of flattening it, which we call "blackwood" temperament. This means there are 5 equally spaced notes in the octave, but it's different from plain 5edo because it contains the additional information that the third step of our scale is a tempered 3/2, while in 5edo, it's just the third step of the scale. (Note that usually, we think of commas in terms of just intonation, so blackwood is more often described as tempering 256/243 (a ~90 cent interval), but that's the same thing, since that's just our comma cubed.)
Similarly, we can temper out the comma between our wolf fifth and our perfect fifths in Pythagorean[12], by flattening our fifth instead of sharpening it. If we tune our perfect fifth to 700c, our wolf fifth is... also 700c, meaning the comma is tempered out, and there are 12 equally spaced notes in the octave. This is called "compton" temperament. But again, it's different from 12edo by itself because it contains the additional information that our 700c interval is a tempered 3/2.
A temperament defined by what commas it tempers out is called a "regular temperament". A regular temperament should be thought of less as a specific tuning, and more as a set of rules or behaviors for a tuning to follow. This is the kind of temperament most extensively studied by xenharmonic theorists, hence why only regular temperaments have names.
Under this theory, Pythagorean tuning is a temperament: it's the temperament involving only 2/1 and 3/2 that tempers out no commas.
Equal temperament
An equal division combined with a temperament interpretation is called an "equal temperament", though often we'll see "equal division" used to refer to the tuning system and its temperaments taken together, basically synonymous with "equal temperament" (since the temperaments shown in example A aren't true "temperaments" according to the theory of regular temperaments). One often says that an equal division "supports" a regular temperament if, when treating relevant steps of the equal division as tempered just intervals (i.e. as an equal temperament), all of the same behaviors apply that you'd expect to see from the temperament (for example, if in an equal tuning, five perfect fifths close three octaves, it "supports" blackwood temperament).
For example, 12edo also supports "meantone" temperament, which means that our tempered 3/2, stacked four times, produces a tempered 5/1. This is useful for triadic harmony, because it means we can access both our 3/2 (perfect fifth) and 5/4 (simplest ratio for a major third) in a diatonic scale. And similarly, 5edo also supports "slendric" temperament, meaning that one of our individual scale steps of 240 cents is a tempered 8/7.
5. Can I Have A Moment?
A "moment of symmetry" scale, or MOS, is a scale generated by using two generators. Pythagorean[12] is an MOS, so is Pythagorean[7], which corresponds to the diatonic scale. Generators for an MOS need not be just intervals, however: they can be any interval. This is why people generally don't like using the names of temperaments to refer to MOSes: Pythagorean[12] is more commonly called "p-chromatic", and Pythagorean[7] is often simply just called "diatonic".
MOSes have the notable property that any interval category has at most two sizes within the scale (such as a perfect and augmented fourth, or a minor and major third). So, for example, melodic minor is not an MOS because it has augmented, perfect, and diminished fifths.
An MOS can be uniquely identified by its interval of equivalence (usually left implicit as the octave) and its step count. For example, diatonic is 5L 2s. To turn blackwood into an MOS, each of its 240c steps can be split into a larger and smaller step, resulting in 5L 5s (which is often also called "blackwood"). MOSes have various relations to each other:
- The "sister scale" of an MOS can be thought of as its "evil twin": for example, "antidiatonic", or 2L 5s, has swapped counts of large and small steps compared to diatonic, and as such, major and minor switch places.
- The "parent scale" of an MOS is always entirely contained within the MOS: for example, "pentic", or the classic pentatonic scale 2L 3s, can always be found within a diatonic scale. Parent scales can be thought of as simplifications of the MOS's identity.
- The two "daughter scales" of an MOS can be thought of as "chromatic scales" for the MOS. For example, diatonic's two daughter scales, 5L 7s and 7L 5s, refer to different tunings of the normal chromatic scale, called p-chromatic and m-chromatic respectively. They are also sister scales to each other.