A brief introduction to Regular Temperament Theory: Difference between revisions

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Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>and where "approximating" means something like "having errors less than 30 cents"</ref> a simple<ref>and "simple" means something like involving a pair of integers whose product is less than 105</ref> ratio.

#REDIRECT [[Dave Keenan & Douglas Blumeyer's guide to RTT/Introductions#Dave's brief introduction to RTT]]

We like to have different flavours of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough flavours, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.

[[~~regular temperament theory|Regular Temperament Theory]] is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced~~ to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the [[generators]]. In one extreme, the generators are the small prime numbers themselves, giving [[just intonation]] (JI) as a lattice, having [https://en.xen.wiki/w/Tenney-Euclidean_temperament_measures zero errors but high complexity]. In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an [[equal temperament]] (ET), having low complexity but high errors.

~~These two extremes were well explored prior to RTT. What~~ RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).

When the generation of tunings is formulated in this way, the tools of linear algebra can be applied (or the slightly more powerful but less well-known and less available tools of a related algebra called exterior algebra, Grassman algebra or geometric algebra).

The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.

Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators. For example, the first generator alone should generate the lowest prime using a positive count. Then the first and second generators alone should generate the next prime using a positive count of the second generator, and so on. This is equivalent to requiring the mapping matrix to be in a certain form. I don't know if this form has a name. I note that an RT mapping matrix is never a square matrix (otherwise it would be JI).</~~ref> This is called the temperament~~'s ~~[[Temperament_Mapping_Matrices_(M-maps)|mapping]], and can be represented as a matrix.~~

~~We can then institute computer searches~~ to ~~find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments|many resulting temperaments named and catalogued~~]].

~~<references/>~~

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-Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>and where "approximating" means something like "having errors less than 30 cents"</ref> a simple<ref>and "simple" means something like involving a pair of integers whose product is less than 105</ref> ratio.
+#REDIRECT [[Dave Keenan & Douglas Blumeyer's guide to RTT/Introductions#Dave's brief introduction to RTT]]
-We like to have different flavours of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough flavours, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.
-[[regular temperament theory|Regular Temperament Theory]] is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the [[generators]]. In one extreme, the generators are the small prime numbers themselves, giving [[just intonation]] (JI) as a lattice, having [https://en.xen.wiki/w/Tenney-Euclidean_temperament_measures zero errors but high complexity]. In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an [[equal temperament]] (ET), having low complexity but high errors.
-These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).
-When the generation of tunings is formulated in this way, the tools of linear algebra can be applied (or the slightly more powerful but less well-known and less available tools of a related algebra called exterior algebra, Grassman algebra or geometric algebra).
-The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.
-Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators. For example, the first generator alone should generate the lowest prime using a positive count. Then the first and second generators alone should generate the next prime using a positive count of the second generator, and so on. This is equivalent to requiring the mapping matrix to be in a certain form. I don't know if this form has a name. I note that an RT mapping matrix is never a square matrix (otherwise it would be JI).</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)|mapping]], and can be represented as a matrix.
-We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments|many resulting temperaments named and catalogued]].
-<references/>