Man in Space wrote: ↑28 Sep 2021 04:03
I would like to look into musical scales based on P5 equivalence. My brief googling and scanning Wikipedia on my phone is leaving me confused. Anybody know a layperson-friendly discussion of these?
Perhaps if you explained what you meant by "P5 equivalence", it might help? I can't find anything for this, and very little for "perfect fifth equivalence" (which I assume is what you mean by 'P5').
Do you mean a tradition in which the 3:2 ratio ('the perfect fifth') is considered perfectly harmonic, making the fifth degree the harmonic equivalent of (and interchangeable with) the first? If so, I'm not sure any such tradition exists, and it wouldn't be a distinct musical scale if it did. It IS common in vocal cultures to treat recitation on the fifth as equivalent to recitation on the first - that is, singers sing in parallel fifths - as a harmonic approximation to the difference in tessitura between men and women/boys, but that's best seen as non-harmonic monophony (one line mirrors the other, but does not harmonically interact with it), and once you start breaking the mirroring (developing an organum-style practice) the 'equivalence' likewise breaks down. These traditions can lead to notational systems that treat the fifth as equivalent - that is, that repeat their notation after the fourth - as iirc was done in early mediaeval music - but this is a notational/conceptual issue rather than a musical one per se. To the extent that they're musical it's probably better to see them as a reflection of organum?
Or do you mean a tradition in which the 3:1 ratio (an octave PLUS a fifth, also known as a twelfth or a tritave) replaces the 2:1 ratio? This exists, marginally, in 20th century art music, and although it's not itself a scale, there have been scales created to facilitate it. Specifically, music that tries this is generally written in a "Bohlen-Pierce" scale - actually a gamut (from which multiple scales may be derived) based on 13tet tritaves. This has the benefit of having inherently more consonant intervals than 12tet octaves (it uses 27:25, 25:21, 9:7, 7:5, 75:49, 5:3, 9:5, 49:25, 15:7, 7:3, 63:25, 25:9 and 3:1, so it's both tet and 7-limit - in other words, you can construct it entirely from octaves, fifths, just major thirds and septimal minor thirds). However, not only will everything sound horribly out of tune to normal ears, but because you're taking away the most powerful harmonic relationships you're left with something with very little sense of tonality - ideal for modernism, but not so much for actual music.
That said, if you use synthetic instruments, you can artificially warp the harmonic partials to match the tritave, which should make the music sound less ugly, if still weird. With natural music, only the the chalumeau register of clarinets (and chalumeaux themselves, obviously) approximates this - the first strong partial of a chalumeau is the tritave, not the octave - although even this doesn't match up with the BP subdivision of the tritave. However, because of the approximation, you can actually buy (for only money!) physical Bohlen-Pierce clarinets (clarinets tuned to the Bohlen-Pierce scale). If you want to, you know, perform bohlen-Pierce music to conclaves of Bohlen-Pierce enthusiasts. If you want that to be your life. Hey, no judgment...
(anyway, if you want to hear BP music, I'm sure youtube will accomodate you)
[there are real musical traditions with wolf octaves, but I don't think these have just twelfths. There are also culture with double octaves - the upper octave is tuned differently from the one below - so in theory (since two tritaves is three double octaves) a sort of double-BP might be possible, but I don't think they ever actually work like that.]
Or do you mean a tradition in which perfect fifths are equivalent to one another? The problem with this is that if all fifths are of equal size, one of three ugly things needs to happen: either the gamut loses octave symmetry (the scale is different in different octaves) and hence octave equivalence (notes an octave apart are no longer consonant); or there are an infinite number of notes in each octave (not practical for many instruments, or for human brains); or the fifths must be out of tune. 12tet is what you get when you assume fifths are equivalent, accept imperfect fifths, and then remove the need for infinitissimal division of the octave by assuming that twelve fifths are equivalent to seven octaves (the fundamental pythagorean approximation, aka the tempering of the 531441:524288 pythagorean comma to zero). There are of course other solutions - most sophisticated mathematical cultures have at least toyed with the idea of instead tempering the 19383245667680019896796723:19342813113834066795298816 Mercator's comma to zero (i.e. equating 53 fifths with 31 octaves), which in many ways solves many of the traditional musical problems - it gives you 53tet with near-perfect fifths AND juster thirds than Pythagorian tuning. However, having 53 notes to the octave is, frankly, a right bugger organologically, and it's only been substantially adopted in Ottoman music theory.
(again, I'm sure you can find samples on youtube)
I don't know which meaning of 'P5 equivalence' you mean. But also, I don't know what you mean by 'layman' - I don't know if what I've said will be too confusing, or too patronisingly simplistic. Did you have a more specific question you wanted an answer to?