Yog-Sothoth

Thanks for sharing your thoughts! I experimented with 17et in Scala, and it was quite interesting: - The 5th is reasonably accurate with 4 cent above. - The 3rds are similar to the ones in the pythagorean tuning, but even smaller (minor third) and wider (major third), respectively. - There's a neutral 3rd with about 353 cent, which corresponds to the augmented 2nd or the diminished 4th - both are the same in 17et! - The minor 2nds are very small (70.588 cent), and are extremely close to the interval between major 3rd and minor 3rd in JI (70.672 cent). Still, they're not as small as the augmented primes in 19et. - Augmented primes are very wide (141 cent), which is almost 1 1/2 halftones in 12et! Like in 19et, 'chromatic' passages like C Db D (in 19et: C C# D) sound more like a scaled-down C Db Eb in 12et because of the (logarithmic) 1:2 ratio of minor 2nd and augmented prime. - The tritones are similar to the ones in 19et, and close to some kind of #4 with ratio 25:18 = (5:4) * (10:9) and b5 with ratio 36:25 = (6:5)^2, respectively. (usually, in JI the #4 is 45:32 = (5:4) * (9:8), and the b5 is 64:45 = (6:5) * (32:27)) If you experiment with this tuning, try the short sequence C E Db C, which includes a very wide major 3rd C - E, a #2 between E and Db (aka the neutral 3rd), and a very small b2 between C and Db. ;) ...all in all, I'd characterize 17et as similar to the pythagorean tuning, but with sharper leading-tones, caused by the slightly wider 5th. Edit: Wow!!! Did anyone try the major scale / melodic minor with a neutral 3rd in 17et? :)

With "spiral of the temperament", do you mean something like the "spiral of fifths" (which was by the way what I intended to say in my last post, instead of "cycle of fifths" - sorry! :whistling:)? The spiral of fifth works as a representation of all notes in pythagorean tuning, but not in 5-limit just tuning, or p-limit just tuning with p prime and greater than 5. The reason is that by accumulating fifths, you could only generate intervals in the form 2^x * 3^y (at least if octave equivalency applies), which excludes ratios in the form 2^x * 3^y * 5^z (like 5:4 = 2^(-2) * 3^0 * 5^1). It's possible to represent all tones (octaves ignored) of 5-limit just intonation with a 2-dimensional lattice, an example would be: http://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Quint-Terz-Schema2.jpg/800px-Quint-Terz-Schema2.jpg (German note names are used so "h" means "b", "b" means "bb", "fis" means "f#" and "ges" means "gb") Here, notes are connected horizontally if the distance equals a fifth (like notes in a spiral of fifths), vertically if it's a major third, and diagonally for minor thirds (which can be derived since a fifth minus a major third equals a minor third). An equivalent representation is this one (a screenshot of the program "Scala"), using a hexagonal lattice: http://www.xs4all.nl/~huygensf/scala/snapshot2.png As you see, the note names "Bb" and "A#" aren't sufficient in just intonations, they must be extended using comma notations (e.g. like "\" for "syntonic comma downwards"). This syntonic comma (81:80) represents the difference between a pythagorean major third (81:64) and a pure major third (5:4 = 80:64), and is different from the pythagorean comma (-> the difference between 12 fifths and 7 octaves, between A# and Bb, or the interval "augmented seventh" in pythagorean tuning = (3/2)^12 : 2^7 = 3^12 : 2^19 = 531441:524288).

There are multiple reasons why I decided to interpret the Dm in the cadence C Dm G7 C as "D F A": - The notes D and F are part of the subsequent G7, and I think it sounds very irritating if two consecutive chords possess notes with a differerence of about 22 cents. - One voice moves from E\ to F, holds F, and returns to E\. In just intonation, the step between F and E\ is already greater than in pythagorean tuning or 12et, changing the F to a F/ would make this step even greater, and the fourth between C and F/ wouldn't be a pure one. However, it reminds slightly of meantone temperament, so it may appeal to you to play it as D F/ A and G B\ D F/. - I think the pure perfect fifth between D and A, or between D\ and A\ is important. I don't like the latter, because of the small 10:9 whole tone between C and D\, and because D (and not D\) is already part of the G7. - I don't mind using pythagorean thirds or sixths (C - A instead of C A\, or D - F instead of D - F/) if this ensures a smoother chord progression. This may be due to listening habit, as the thirds and sixths from E12 are closer to the pythagorean ones than to the pure ones, but I'm not sure. ...of course it's also an interesting option to modulate from C to C\, but this is something I'll have to try before I can judge about it. By the way, if you modulate from C major to C minor, you can either simply use minor thirds for the three main triads (I IV V -> i iv v), or you can modulate from C major -> A\ minor -> F major -> D\ minor -> Bb major -> G\ minor -> Eb major -> C\ minor, so in fact you end in slightly different keys here. ;) Playing melodies over a chord progression is an interesting point, but unless I have a real instrument for 53et it's always exhausting to try things like these. I think it will be funny to play notes like C Db Eb Fb over a C major chord (C E\ G), which should work since in E53, Fb and E\ is the same note. The three accumulated major / minor chords describe the exact structure of the major / minor scale in 5-limit just intonation, just like accumulated perfect fifths describe the structure of the pythagorean tuning. That's why the cycle of fifths is useful for modulations in the latter case, and a 2-dimensional grid (perfect fifths and major thirds axis) is useful in the first case. Both kinds of representation can be used for many equal temperaments, but 53et is here different from 12et and 19et, since in 53et the major third is different wether it's derived by accumulating 4 fifths, or wether the interval closest to the pure major third is used. Edit: Sorry, I forgot to mention that the three accumulated triads can be described in a 2-dimsional way, like A\ E\ B\ F C G D or D\ A\ E\ B\ ... F C G Hey, I'm glad if you ask questions. :thumbsup:

I have a fretless guitar, and I'm planning to mark the "frets" according to 53et with lines. For 13 of them (including #4 and b5) I'll use colors, the remaining "frets" will be in grey or black. I fear I will encounter several problems, let's hope I err. :whistling: I haven't composed music in 53et yet, just experimented a bit with existing pieces. I guess the 5-limit approximation in 53et will be very challenging, so I won't do much with 7-limit intervals the next time. I mainly use the lattice player for experimenting with harmonies in 53et, and use the "example" command to convert .txt files into .mid files. Indeed - maybe we should have two threads: one as an introduction for microtonality, and one for extended discussions.

Sorry, I just realized the major second which I didn't like was the one with a ratio of 10:9, which is a syntonic comma smaller than the "normal" major second (9:8). I still think the 9:8 major second sounds slightly better than the one from 19et, but don't think that the latter sounds dissonant. I like the semitones in 19et, too. It's only natural that the interval between minor third and major third is smaller than other semitones are. :) I have to admit I mainly studied intervals with prime numbers not greater than 5 in the ratios. I know the harmonic seventh, as well as the tritone with the ratio 7:5, and both inversions, but now I hear the first time about these septimal major (9:7) and minor (7:6) thirds. Now I'm really becoming curious about 31et. :cool: The best approximation for the harmonic seventh (7:4 => 968.83 cent) in 53et is the 43rd step (973.58 cent). The septimal major third (9:7 => 435.08 cent) relates to the 19th step (430.19 cent), and the (7:6 => 266.87 cent) relates to the 12th step (271.70 cent), which is always an inaccuracy of almost 5 cent. It may not be as precise as the fifths and thirds, but it's acceptable, I think. ...and you're right, modulation in just intonation is more complex than it is in 12et, 19et or pythagorean tuning. C-Dur consists of 3 major triads: F A\ C, C E\ G, G B\ D, where "\" means "1 syntonic comma lower". A\ minor, on the other hand, consists of the triads D\ F A\, A\ C E\, E\ G B\, which means the D is lowered to a D\ when modulating from C major to A\ minor. When modulating from A\ to F major, the B\ is replaced by a Bb. When modulating from C major to G major, you first have to modulate to E\ minor (F -> F#\), then to G major (A\ -> A). If you're interested I could write more about modulation in 5-limit just intonation. But the main problem is that even cadences like C Dm G7 C don't work smoothly - I think the best solution here is to play the Dm with the tones D F A instead of D\ F A\, D F A\, D F/ A or what else is possible. 53et works about the same when interpreting it as an approximation to 5-limit just intonation. When using 53et to approximate the pythagorean tuning (the tuning created by accumulating perfect fifths), the diminished fourth is a pythagorean comma smaller than the pythagorean major third (81:64). Since 1 step in 53et relates to both the pythagorean and the syntonic comma, the diminished fourth in approximated pythagorean tuning equals the major third in approximated just intonation. So if we have the sequence C# F E C# D (for example D harmonic minor) in approximated pythagorean tuning, there are both sharp leading-tones between C#-D and E-F, and the diminished fourth between C# and F sounds like a pure major third (5:4), which is nice since normally, sharp leading-tones and pure thirds exclude each other. ;)

Hi all, I found this forum when I searched for the term "microtonality", and this is my first post here (except for the one in the "I'm new here" thread). I think 19et is interesting, though I admit I don't like the major seconds much. Intervals become more dissonant when closer to the root, and I think it happens fast when going below 200 cent. Of course it might be due to listening habit, who knows? ;) Previously, I didn't care about 31et, but it seems the major third is approximated much better than in 19et, and it's slightly larger, rather than smaller, which fortunately also increases the major second slightly. The musical temperament I'm currently most interested in is 53et, an almost perfect approximation of both the pythagorean tuning and the 5-limit just intonation, and it's up to the musician wether he prefers leading-tones or pure thirds and sixths - or if he's clever, he uses both in scales like the gipsy minor scale, since pythagorean augmented seconds and pure minor thirds are almost identical - in 53et even the same! I think 53et is very close to how classical musicians play on string instruments, and therefore not only a temperament, but also helpful when imagining the approximate size of pure intervals (at least those based on 3:2 fifths and 5:4 major thirds). In 53et, the perfect fifth equals 31 steps, and the perfect fourth equals 53 - 31 = 22 steps. The major second equals 2*31 - 53 = 9 steps, which can be devided into a pythagorean minor second (4 steps) and a pythagorean augmented unision (5 steps). As we see, the pythagorean comma (difference between 12 perfect fifth and 7 octaves: (3:2)^12/(2:1)^7 => 23.46 cent) relates to 1 step (2^(1/53) => 22.64 cent), which is also close to the syntonic comma (difference between pythagorean major third and pure major third: (81:64)/(5:4) = 81:80 => 21.51 cent), so 1 step relates to both commas in 53et. ...so the pythagorean major third equals two major seconds = 2*9 = 18 steps, while the pure major third lies a syntonic comma below, which means it's 17 steps. Further intervals can be derived similarly. Alright, that's it for today, I hope at least some of you are interested in the topic. :) P.S.: I'm currently using Scala. In addition, I tried Tonescape, but it doesn't run on my PC. Do you know any other software which is useful when working with microtonal music? P.S.S: I'm from Germany, and my English is not perfect. If you find major typos, please give me a hint so I can learn from my faults, and you don't have to guess all the time what I'm trying to write. :whistling: