Відео

Geometric mean happens to sound pretty decent even though it's a seemingly ugly ratio of square root of 5 (2.236:1). It's very close to the just tempered major ninth (9/4 or 2.25:1) and even closer to the equal tempered major ninth (2^(14/12) = 2.245:1).

my understand is it would be 2 / (1/5 + 1)

That's arithmetic mean

You can also check harmonic and arithmetic means of 1 and X by dividing X into an isoharmonic chord of three notes, that is a chord consisting of harmonics M, M + N, M + 2N where M, N are integers. Here we have such a chord as 1:3:5. Now, harmonic and arithmetic means here are just two intervals with the middle note: 5/3 and 3/1. I don’t remember which is which but that’s honestly not an issue because you can always check later; and if you’re certain that one is the arithmetic mean then the other should be the harmonic mean.

thats sort of the basic idea! its a little more complicated though - if you'd the LONG COMPLICATED explanation though, this is The Helmholtz Ellis Notation System This is my microtonal notation system of choice, designed for JI, by Marc Sabat, Wolfgang von Schweinitz, and others. I’m gonna have to make a whole video about this for anyone new to actually follow along but here is the fast version of how it works - WHY it works will have to come later: We start with a pythagorean diatonic scale tuned to A as the root - that is to say, a diatonic scale derived from the circle of fifths. When we talk about pythagorean tuning, this is what we mean. Pythagoras did not invent or discover this system, but as with many other things he gets the credit. I’d prefer to call it something like 3-limit tuning, but the name pythagorean tuning is so ingrained that we really have no choice. Anyway. [Graphics Explain the tuning) Every sharp adds 114¢ to this diatonic base tuning, every flat deducts 114¢ (graphics) <<<FIX THIS This might seem like it leads to having to remember a lot of numbers, but for now, just keep in mind that as we go up the circle of fifths into the territory of the sharps, our notes get gradually sharper than their equal tempered equivalents, and as we go down in the realm of the flats, the notes get flatter. That’s not so hard to remember, is it? [Graphics] What this means though is that by the time we reach A-Sharp, all the way up there, we have a note significantly sharper than B-Flat, all the way down there. But in a lovely coincidence, this A-Sharp happens to be tuned nearly the same as a B-Flat raised by a 5-Comma. And that leads us to our next point. In order to modify notes by a 5-comma, we add a little stem to the accidental or natural sign or flat. As we see with our B-Flat, an upward stem raises the pythagorean base pitch 21.5 cents, whereas a downward stem, like in this diagram of the harmonic series, lowers our base pitch by 21.5 cents. It’s worth noting that, since the pythagorean major third is about 8 cents sharper than an equal tempered major third, our 5-Comma Down Major Third, tuned to the harmonic series, is only 14 cents flat, rather than the full 21.5. But don’t dwell too much on that if you’re a beginner. Speaking of our diagram of the harmonic series, what’s this next funny little shape? It, and all the other funny shapes you see here, are further prime modifiers. Every time we reach a new prime in the harmonic series, we basically need a new symbol to accurately modify our pythagorean base pitch. Some people consider this a significant limitation of the HEJI system, and it is, especially if you want to use a lot of different primes. All the same, *I* use a lot of different primes, and I love the HEJI system, so really if you want to do it, go for it. Just, if you are master this system and use it yourself for your own compositions, and give them to a performer, make sure you have sufficient explanation and maybe a lot of annotation. Marc Sabat provides a PDF that explains the system more thoroughly, which you can find in the description. It would be prudent to include this with ANY piece you write using HEJI meant for performance. Why do I use HEJI? I love it’s systematic nature and aesthetic beauty. I love that I can use it in Dorico, although it takes a lot of work. More on that later. Anyway, some JI notation systems are not very good at all - we don’t have time to talk about how Ben Johnston, despite being widely and righteously regarded as the greatest JI composer, just wasn’t any good at coming up with a useful notation system. Most useful types of JI notation are similar to HEJI in that they start with a pythagorean base, but other substitute different kinds of abstract symbols, or numbers, as their modifiers. If you want to know more about HEJI, maybe talk to Thomas Nicholson! He’s done a lot of work on it and knows it front to back.

in a 5-limit context its definitely the normal way to go - you can hear it in this piece ua-cam.com/video/a58vXuZYDj0/v-deo.html or this piece ua-cam.com/video/CSL_Axohw94/v-deo.html - but there's a lot of stuff out there. these two videos show some of them: ua-cam.com/video/N8Aa_-dZEvA/v-deo.html and ua-cam.com/video/ZnZ82uGtYUo/v-deo.html

@@mannfishh What do you think about using 19/12 as an augmented fifth? I've gotten into the habit of primarily making my augmented chords from the idea of 12:15:19.

@@PastenMusic it sounds really interesting! i've never used a 12:15:19 myself but I'll have to give it a try

Search HEJI ( Helmholtz Ellis Just intonation) and you’ll find your answer.

Each down arrow on an accidental brings it down by one syntonic comma (81/80) or about 21.5 cents from the Pythagorean interval

@@MaxIsBackInTown jeez what a rabbit hole 😅

haha nice one

Fr

ah I'm glad you asked, though it means revealing a little secret - i.imgur.com/z2jaQ2d.jpeg

it means that a C and an Ev together could be a 5/4, 5/2 or 5/1

thats what i mean by like harmonic family, its just including all the octave permutations, since thats what the video is about

normally i label the axes like 5th Harmonic (Major 3rd) but that isnt really accurate since of course a major 3rd is a 5/4 so in the interest of clarity i figured I'd expand that to include all possibilities of what the axis could contain. but in trying to make it clearer it seems to have made it more confusing! oh well.

ah, i see what you did. feels a little unnecessary since you still represent the octaves with colour. think of a coordinate system in 3d space: if you've taken care of the octaves dimension on the 'colour axis', then you don't need to consider it moving in 5/4-3/2 plane

9/8 was excluded because it falls within the critical band

​@@mannfishh what critical band?

@@rainbowbloom575 en.wikipedia.org/wiki/Critical_band

@mannfishh i still don't understand, but thatnks

This topic interests me or lot, or maybe I could say it annoys me, lol. That it is difficult to communicate many dimension in a few... it gets worse when you try to show changes over time like in sheet music as well as want it to be interactive like a keyboard but *still* use multiple spatial dimensions to arrange pitches... because isomorphic keyboards and microtonal lattices are just so nice.

5:58 Well don’t you say that! It’s indeed really simple when constrained to how you used it here, so why bother with other primes while they’re not in use. I think understanding “up/down by 1/2/3 syntonic commas” doesn’t need to be complicated for a brain if you explain that it’s precisely this difference between a Pythagorean M3 and 5/4, and IIRC you explained it in another video. Then having more arrows is just natural because of course we might want to modulate and because there are no closed loops in JI so we may need to add more and more accidentals... ...well okay I see that it looks not as simple when I try to expand all the tidbits one needs to know it gets quite long. Still I think the concepts _are_ still quite simple, and you of all people are communicating them very much alright; there are worse ways. I forgot what I was trying to say but it was something in vein of being more optimistic about how complicated it might be to explain JI, particularly 5-limit. And I mean, there are notations that are less consistent or easy to parse that HEJI. This is quite a comfortable and consistent one. I dunno maybe Ben Johnston’s was harder (I forgot what it was, exactly). So it’s almost as transparent as it gets (well, FJS can be an alternative but I don’t think it changes anything in regard of reading what pitches are). Sorry, this tangent is quite tangential.

8:37 Ooo do share! I think I have an inkling but no more. Like if the music uses another interval as a prescribed equivalence, like 3/1, then if the composer is careful enough, small quantities of gentle 2/1 usage should be able to end up _not_ messing up this sense of equivalence (by being simpler and culturally the sole equivalence interval for many). But this isn’t even a method, it’s an “try and see if it works”: what’s gentle enough, how to make it work?

my viewers tend to be pretty busy, they're like, professional athletes, astronauts, world famous MCs, medics in conflict zones, etc

@@mannfishhOh yeah, definitely

@@mannfishh also like 99% of them have ADHD as well, solid.

Yess btw i have a lumatone im experimenting with Infinitone 2. If u ever wanna chat about it we can hehehe

hmm I DO have a single grandstaff with all the keyboard parts but it was written with two keyboards in mind. I''ll show you the score at some point!! (also 19edo song is nearly there.... for real... this time... lol!)