Regular Mappings and Marvel Temperaments Graham Breed This is the annotated version of the slides for the presentation I gave at Beyond the Semitone in Aberdeen, October 2013. The annotations cover things I said, or should have said, or might have said if I’d had more time. 0.1 Motivation Goals of Regular Temperament Regular temperaments are systems that approximate just intonation using fewer notes. This assumes there’s something special about just intonation that you want to preserve, and if you don’t feel this, temperament won’t be useful to you. The ”regular” part means each just ratio is tuned in a consistent way, and more than one just ratio will typically approximate to the same tempered interval. The list below is roughly prioritized, and the tuning is the least important thing. There’s a trap in microtonal literature that you focus on the tuning, and people think it’s all about tuning, although the tuning is only a tool in the service of exploring new harmonies, or making just intonation scales more manageable. • New harmonic systems • Frugal scales • Notation • Generalized keyboards • Tuning Beyond Temperament These ideas come from the study of temperaments, but can still be useful if you don’t use temperament. This is why I talk about a ”regular mapping paradigm” rather than ”regular temperament theory”. 1 I added some citations for the academic audience, and also to avoid giving the impression that everything I was talking about was original to me. Christian Kaernbach, who gave the talk after mine, discussed the idea that musicians might not play the (simplistic) theoretically correct pitches, and that doesn’t mean they’re wrong. That fortuitously agrees with my first point here. • Support flexible pitch instruments. • Leave the tuning as just intonation. (Fokker 1969) • Write just intonation as tempered notation. (Secor 1975) The mapping from just to tempered pitches remains. What Took So Long? This is the defeatist slide that excuses why everybody isn’t using these temper- aments and related techniques already. It furthers my agenda regarding the regular mapping paradigm in that part of Kuhn’s description of paradigm shifts is about the resistance that a new paradigm encounters. It’s also worth pointing out that these ideas won’t be equally valuable to everybody. • New pitches are disruptive. • Meantone is good. • Not all music needs pure harmony. • Good mappings are easy to recognize but hard to find. • Computers make searches and tuning easier. 0.2 Theory The Marvel Mapping Tempering out 225:224 (7.7 cents) relates the 7th harmonic partial to two fifths and two thirds. These lattice diagrams show how the prime number 7 ties into the 5-limit lattice. With proper spelling, the 7 should be written as an augmented sixth rather than a minor seventh. (You may choose to add a diesis¨ accidental so that the 7 can be a kind of seventh.) 2 Because the 7.7 cents get shared out among four 5-limit intervals, each 7-limit interval can be tuned to within 2 cents, which will sound as good as just intonation in most contexts. I copied the diagrams from my Tripod Notation document. The thirds are shown shorter than the fifths so that subminor thirds giving magic approxima- tions don’t end up as huge triangles. Without showing magic approximations, the triangles should be roughly equilateral, but it doesn’t really matter. 14 5 4 6 9 A] E (F]) C G D Interval Arithmetic I like equations, so I had to bring some equations into my music theory talk. Each shows a musical relationship, and in each case the equations fail to be exact by the 225:224 that’s being tempered out. The first is an augmented triad showing two 5:4 major thirds and a 9:7 su- permajor third adding up to an octave. (Intervals add where fractions multiply.) The Wikipedia page on augmented triads mentions the 225:224, so it must be standard theory. The second equation ties the new septimal interval to familiar extended 5-limit intervals including the larger whole tone of 9:8. The third equation is the same but with the 9:8 split into two 3:2 fifths, and covers another octave. This is the equation that describes the lattices on the previous slide. Finally, we can see that two diatonic semitones approximate to being the same. This means that you can take any marvel temperament, pile up two semitones, and call the result an 8:7. In fact, you can do more than see. You can hear audio examples of these equations in ”fractions.pdf” in the accompanying bundle. For each equation, notes of an ascending scale overlap to give each dyad on the left hand side and then the first note comes back to give the right hand dyad. 3 5 5 9 2 × × ≈ 4 4 7 1 5 5 9 7 × × ≈ 4 4 8 4 5 5 3 3 7 × × × ≈ 4 4 2 2 2 16 15 ≈ 15 14 Question This was an attempt to catch the audience out, but they were too canny and asked what kind of semitone I was asking about. How may semitones are there to an octave? The point about focusing on semitones was partly to subvert the ”beyond” in the conference title, but also to show that we’re expecting generalzed diatonics of around 10 notes. I could have put this at the introduction to miracle temperament, but I decided to have it to introduce the step size list instead. Answer I showed the logarithms so that people can see how easy the calculations are. The last one is the geometric mean of the two just intervals. It’s a rough approximation of the optimal marvel semitone. log(2) 15 = 10:04 ::: log 14 log(2) 16 = 10:74 ::: log 15 log(2) 2 8 = 10:38 ::: log 7 Scale Sizes These are scale sizes rather than equal temperaments so that I don’t have to worry about ambiguous mappings. (Where there’s ambiguity, this is the 4 marvel option.) It also helps draw attention to the small numbers that represent diatonics rather than tunings. 10 and 9 both come out better than 7, so we might still expect 7 note diatonics, but also 9 and 10 note scales. For a bit more complexity, 12 is very good. There’s a notable absence of pentatonics. This is because the things that make pentatonics good in a 5-limit context put 10 and 15 into this list, but 5 itself doesn’t work so well in the 7-limit. The ranking follows cangwu badness with a dimensionless parameter of 0.001. This gives simpler scales than the default my Python module uses, and means pajara gets near the top of the rank 2 list later on. It also gives 19 as the number one division to tie in with Graham Hair’s work that was featured at the conference. I filtered out temperaments with contorsion (where each JI approximation is an even number, or something like that) because they aren’t very interesting. A lot of the information in the list ties in with that rank 2 list. If you can compare the two, you may like to do so. Note that the top three are all meantones. 19 can than pair with 41 or 22 to give magic. You can find multiple representatives of the other top rank 2 classes in the top row. The bottom row is less interesting. Because I don’t target excessively small errors, near-optimal tunings for planar marvel don’t show up. If you want one, you could try 269. The planar audio examples use 1273 ”marvel cents” to the octave. This week’s Marvel Top 40: 19 31 12 41 22 53 10 72 50 9 34 29 7 60 43 3 84 63 65 21 2 94 91 75 32 51 28 103 48 15 40 69 81 46 113 17 79 26 16 74 0.3 Harmony Substituted Dominant Cadences These audio examples (sdc.mp3) are to illustrate the use of tempered augmented triads. They’re written in musical notation, but they aren’t musical examples. They only show some chord progressions you could use to build music out of. What I know about augmented triads is that you can use one to connect any chord to any other chord. So, given that, I don’t know what a typical progression 5 would be to use as an example. I thought about finding one in Gesualdo, to tie in with some other things at the conference, but that would have meant work so instead I went to Wikipedia. That mentioned this ii-V-I progression with the V substituted. The accidentals are mixed 72-tone Sagittal. The sharp hooks are commas and the more rounded hooks are larger deviations. The first example has the augmented triad written as two consecutive 5-limit thirds. That’s the way Wikipedia writes it, but in marvel I find it the worst tuning. The whole chord covers an interval of 14:9 which isn’t very digestible. In the second example, the higher third is a 9:7 approximation. The chord as a whole covers 8:5. This is the best version if you like 5-limit intervals from the root. The final example has the 9:7 as the lower interval. That’s the best version if you like chords that imitate the large/small pattern of a major triad. Whether or not that’s important, it sounds fine because the span is still 8:5.