User:IlL/Introduction to RTT

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Tentative title: Practical RTT

Goal: Teach a non-technical audience enough regular temperament theory (RTT) that they can use x31eq temperament finder and get started on using JI approximations in MOSes and edos.

  • Important to minimize unfamiliar terms (such as "monzo" or "val") and linear algebra jargon! Assume only knowledge of high school algebra at most. The point is not to teach a whole course on theoretical linear algebra.
  • Use plenty of examples, both from 12edo/diatonic and non-diatonic temperaments.

Lesson 0: Why temper?

  • Explain why you would want to approximate a JI subgroup in a MOS or edo.
    • Start by noting that meantone is a convenient scalar structure and approximates 4:5:6 and 10:12:15 chords.
    • Ask "What if we could do this with some other chord, such as 4:6:7 or 8:10:11:12 or 7:9:11"?
    • Note that using simple JI chords other than 4:5:6 or 10:12:15 can give novel and exotic consonances.
  • Explain how choosing a basic JI consonant chord and choosing an interval of equivalence determines a JI subgroup, by connecting these chords via common tones.
    • Visual: A 2-dimensional example showing where some 3-limit intervals live in the 3-limit lattice.
    • Visual: The 5-limit lattice mod octave (i.e. the Tonnetz), built up by connecting major triads.
  • Define JI subgroups as the set of JI ratios that are products of powers of intervals in some specified set, called a basis. Introduce the notation for JI subgroups: We write a subgroup as the basis elements separated by periods: for example, 2.5/3.7/3.11/3, and the equave comes first. Observe that the number of basis elements is the dimension of the JI subgroup.
    • Example: The 5-limit is denoted 2.3.5, because {2/1, 3/1, 5/1} is a basis. Another basis for 2.3.5 is {2/1, 3/2, 5/4}. It is the set of all JI ratios with only factors of 2/1, 3/1 and /15 in them. For example, 6/5 = 2*3/5, so it is in the 2.3.5 subgroup.
    • Example: The interval 14/9 is in the 2.9.7 subgroup.
  • Explain how to turn a JI chord X:Y:Z that you want to view as "a fundamental consonance" into a subgroup. In the case where the equave is the octave, the resulting subgroup is 2.Y/X.Z/X (with the caveat that this isn't the only JI-based chord possible in this subgroup). The fundamental consonance of a subgroup plays a role much like the generator does in MOSes. Visuals for the following examples:
    • Example: 2.3.5 means "fundamental consonance = 2:3:5 (or 4:5:6), equave = 2/1"
    • Example: 3.5.7 means "fundamental consonance = 3:5:7, equave = 3/1"
    • Example: 2.5/3.7/3 means "fundamental consonance = 3:5:7, equave = 2/1"
  • Explain that to get a fundamental consonance from the JI group, you multiply by the greatest common denominator of all the basis elements, and put together the resulting numbers into a JI chord.
    • Example: 2.5/3.7/3 -> 6.5.7 -> 5:6:7 (3:5:7 also valid, because 2/1 is in the subgroup)
  • Explain that a JI subgroup need not be a p-prime limit subgroup (e.g. 2.3.7, for 4:6:7 and equave 2), generated by primes (e.g. 2.9.5, for 4:5:9 and equave 2) or even generated by integers (e.g. 2.6/5.7/5, for 3:5:7 and equave 2).
    • An example of a non-p-limit subgroup tuning: Observe that 17edo does not have a good approximation to the 5th harmonic but is an acceptable 2.3.7 tuning.
  • Explain the benefits and limitations of this idea of temperament.
    • Pros:
      • Consonances become more common than in pure subgroup JI
      • Opens up progressions not possible in pure subgroup JI
      • Temperaments describe some ways in which various edos can be similar.
      • Viewing a MOS as a temperament (if the temperament is accurate enough) can help you find some of the consonant chords in it.
    • Cons:
      • Temperament is harder to justify for chords that are higher up in the harmonic series.
      • It's hard to "integrate" a JI interval into the temperament and connect to it via chord progressions, if the interval being approximated is too far out in the generator chain.
      • JI chords often lose some of their qualities when tempered.
      • Low complexity JI doesn't describe all of harmony in a given MOS.
      • Octave equivalence and inversional equivalence need not hold (i.e. in determining how consonant chords are).
    • Remind the reader that a temperament is just a mathematical interpretation or abstraction, and so may not always correspond to musical reality.

Lesson 1: EDO step mappings

  • Introduce interval vectors (aka "monzos") as the "address" of intervals in the JI subgroup, using however numbers you need (for 5-limit intervals you need three numbers: the 2 coordinate, the 3 coordinate and the 5 coordinate, you get those by factorizing the interval into primes). Use examples, writing some simple 2.3.5 or 2.3.7 intervals (5/4, 6/5, 7/4, 7/6) as interval vectors. An interval vector describes how many of each basis element a given interval has.
  • Introduce step mappings (aka "vals") for edos in arbitrary JI subgroups. (Be sure to label every entry with what basis interval is represented by it.)
  • Explain how to use step mappings together with interval vectors to find how many edo steps approximates a given JI ratio. (Walk through finding the inner product of two vectors of the same length, without using the term.)
    • Example: Using the 2.3.5 step mapping 12 19 28] for 12edo, given by the best approximations of harmonics 2/1, 3/1 and 5/1.
      • Let's find out how to get how many step sizes the interval 5/4 is mapped to. Convert the interval into an interval vector: 5/4 = 2^-2 * 3^0 * 5^1 = [-2 0 1.
      • Point out that in order to get down to 1/4, you need to go down two octaves. So -2*12 gives you -24.
      • Then to get to 5 you need to go up 28.
      • You can get to 5/4 by going up to 5 and then down two octaves from there, or by going down to 1/4 and then up by a 5 (ie by 28 steps) from there.
      • Then give 3/2 as a next example, then something like 5/3 that has both 5 and 3.
      • Point out that the 12, 19, and 28 you multiply components of the interval vector with are the same in each case. So that list of three numbers defines a way of mapping JI fractions to 12edo.
    • Repeat the above, but for 19edo (step mapping 19 30 44]) and 22edo (step mapping 22 35 51]).
    • Now do an example for 2.3.7, using 19edo and 17edo.
      • Find the edo approximation of 7/6 ([-1 -1 1 in basis {2, 3, 7})...
        • with 19edo's 2.3.7 step mapping (19 30 53]). Explain that we don't need 19edo's mapping for 5 because 7/6 has no 5 in it anyway.
        • and with 17edo's 2.3.7 step mapping (17 27 48]).
    • An example for a non-prime subgroup 2.9.5 using the 13edo mapping 13 41 30] (components for 2/1, 9/1, 5/1) to find an approximation for 10/9 = [1 -1 1. Here, 13*3 + 41*-1 + 30*1 = 13 - 41 + 30 = 2, so 10/9 is mapped to 2 steps.
  • Explain typical conventions for writing step mappings (they're written in limit subgroups like the p limit)
  • Explain that the patent val is the step mapping giving the best approximation of the subgroup's basis in a given edo. In some sense it's the most obvious, or patent, step mapping given by the edo.
  • Briefly explain things like "34d".

Lesson 2: Rank-2 temperaments and MOSes

(This lesson assumes familiarity with basic temperament-agnostic MOS theory: periods, generators, and the scale tree.)

  • Motivate rank-2 temperaments as JI-based interpretations of MOS scales and/or generator chains (Motivating examples: 2.3.5 meantone and 2.3.7 archy for diatonic, and porcupine as a non-diatonic example.).
  • Explain that a rank-2 temperament specifies period-and-generator mappings (aka 'mapping matrices') of JI intervals. Like a step mapping, a period-and-generator mapping only needs to be specified on the basis of the JI subgroup.
    • (Explain that rank means dimension. Edos (properly, step mappings) are rank-1, or 1-dimensional, temperaments, the 1 dimension being along the edo step. Rank-2 scales such as MOSes are 2-dimensional scales, where the period and generator are the two dimensions.)
    • Progressively label the period-generator lattice with JI ratios for each example, for sake of illustration.
  • Explain that a given rank-2 temperament will always assign a given JI ratio in the subgroup to the same number of periods and generators.
  • Explain that a rank-2 temperament comes with some optimum size for the generator that minimizes the tuning's error from JI.
  • Walk through finding the period-and-generator combination that represents a given JI ratio, just as for the rank-1 case.
    • Do this with 2.3.5 meantone, 2.3.7 archy and porcupine.
  • Introduce the & operation: using two edo step mappings A and B to get a rank-2 temperament A&B. (It is the temperament represented by both m\A and n\B, assuming period 1\k, where m\A and n\B are generators for the rank-2 temperament in question.)
    • 12edo's (1\1, 7\12) and 19edo's (1\1, 11\19) do that for 2.3.5 meantone. This makes meantone a 12&19 temperament.
    • When you say 12&19, you're saying that any period-generator pair with the same period and a generator similar to 7\12 or 11\19 is a valid tuning for meantone temperament. Specifically, any generator between 7\12 = 700c and 11\19 = 694.737c can be used.
    • To get other edo generators for meantone, any Farey-sum of the form (a*7 + b*11)\(a*12 + b*19) can be used, for positive integers a and b, and any such generator will have size between 11\19 and 7\12. (I'm assuming that the audience knows Farey sums and scale trees from MOS theory)
      • For example, (7+11)\(12+19) = 18\31, (2*7+11)\(2*12+19) = 25\43 and (7+2*11)\(12+2*19) = 29\50 are all meantone fifths.
      • (The audience should already know that you can tweak generator sizes towards an EDO generator by repeated Farey addition with the EDO generator in question.)
    • Repeat the above description with 2.3.7 archy = 17&22 and porcupine = 15&22.
    • This is a useful description for a non-mathy audience since it's very concrete; it suggests a range of possible generator sizes for a rank-2 temperament.
  • Explain that Farey addition can also be used for getting edo tunings (generator sizes) for rank-2 temperaments. Thus rank-2 temperaments are threads relating many different edos; for example, 12, 19, 31, 43 and 50 are all meantone edos.

Lesson 3: Commas

  • Explain that declaring a JI interval to be a comma (aka "unison vector") specifies what combinations of simple JI intervals we want to equate.
  • Explain what it means for a given EDO (properly, a step mapping) or rank-2 temperament to temper out a comma.
    • Illustrate with examples (12edo and 19edo temper out 81/80 but 22edo does not, 17edo tempers out 64/63, 19edo tempers out 49/48, 22edo tempers out 250/243, 13edo tempers out 81/80).
  • Introduce comma pumps. A comma pump is a chord progression that uses the fact that the comma causes paths of intervals on the JI lattice to close to a loop.
  • Demonstrate some 2.3.5 (meantone, augmented, porcupine, negri, magic) and 2.3.7 (1029/1024) comma pumps.
  • Explain that if two edos A and B both temper out a given comma in a subgroup, then so does the rank-2 temperament A&B. This is in fact one of several equivalent ways to define A&B temperament: if you know that a step mapping for an edo tempers out all the commas of the rank-2 temperament A&B, then you know that the edo supports the rank-2 scale in that temperament.
  • Explain that a rank-2 temperament on a JI subgroup of dimension n is defined by tempering out n-2 commas. An edo mapping on a JI subgroup of dimension n is determined by n-1 commas.
  • For example, 12edo is the unique 5-limit edo that tempers out both 81/80 (syntonic comma) and 128/125 (augmented comma).
  • Explain that from a math POV, the reason that Farey sums work for EDO generators is that
    • Farey addition corresponds to adding step mappings, and
    • the property of tempering out a comma is preserved under addition of edo mappings.

Lesson 4: Using the x31eq Temperament Finder Tool

Explain how to use the x31eq temperament finder, with examples.

  • Describe the data shown on each temperament's data page. The most important data are:
    • "Reduced mapping": the period-and-generator mapping, in terms of the generator given
    • "TE/POTE Generator tunings": Period, optimum size of generator
    • "TE/POTE Tuning Map": Tuning of each basis element
    • "TE/POTE Mistunings": Mistuning of each basis element relative to JI
    • "Show TE tunings/POTE tunings" switches the tuning between TE and POTE.
      • POTE optimizes the generator keeping octaves pure.
      • TE additionally lets the octave stretch or compress in order to optimize the basis intervals slightly more.
    • "Unison Vector(s)": The commas tempered out by this temperament
  • How to use "Temperament class from ETs"
    • "limit": The subgroup information (not even necessarily JI): prime limit, subgroup, or a harmonic series chord corresponding to the subgroup.
    • "list of steps to the octave": edos you want to do the & operation on, i.e. edos that you know already has a certain generator size or JI subgroup representation.
    • Explain that if two edos are both good tunings on a given subgroup, you get a rank-2 temperament on that subgroup.
  • How to use "Unison vector search"
    • "limit": already gone over.
    • "Put your commas in this box:" Put the commas in the subgroup that you know are tempered out.
    • If the commas you enter don't pin down a rank-2 temperament, the temperament finder will give a list of multiple rank-2 temperaments, and you'll need to either find some more commas or know what edos support the temperament.