Microtonality

That said, the distinction between the two major seconds is intriguing; I don't really fear pitch drift as much as I do keeping track of so many notes. I hope one day to own a keyboard on which I can play microtonal music with all the virtuosity I expect from the Renaissance and early Baroque; split sharps sound manageable but 31 or 53-ET is too far.

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:

So if you are, like me, fairly interested in producing something like standard harmony but in a new guise, I'd like to hear how you deal with these special qualities of 53-ET. Or even if not, tell us more about how you write microtonal music. Could we hear some of your microtonal compositions or experiments? And have you experimented with 7-limit intervals?

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 use Scala's GUI keyboard and lattice to experiment with chord progressions so I have a good feel for the (very consonant!) sound of 53-ET. I edit MIDI files manually by making 19 tracks for each note of 19-ET and tuning every one. But I have not gotten around to writing any actual music, just experiments and speculation. As for Tonescape, I don't think you're missing out on much yet; maybe it will become more useful in later versions.

I mainly use the lattice player for experimenting with harmonies in 53et, and use the "example" command to convert .txt files into .mid files.

EDIT: For those not into microtonality, the vocabulary can sound intimidating. It is :). Almost sounds like another language. I might explain more soon.

Indeed - maybe we should have two threads: one as an introduction for microtonality, and one for extended discussions.

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.

Well this does make sense, to have say

C E\ G (I hope I'm understanding the notation right!)

D F/ A

G B\ A

Isn't the F/ fairly important? Or is the ditone F A less dissonant when it is above the (slightly large) minor third?

And if I do use F/ is there some sort of ugly cross relation between that and the C of the previous chord?

And to go further, if I add passing tones and embellishments, should I use D\ in the first harmony and the D in the next? Wouldn't that be too noticeable?

And if we have something like I vi ii V I, don't we get:

C E\ G

A\ C E\

D\ F G\

G\ B\\ C\

C\ E\\ G\

I guess that could be altered at some point to

C E\ G

A C/ E

D F/ G

G B\ C

C E\ G

which means some very odd shifts from the first to the second chord since we're jumping further away on the lattice (which BTW is how I visualize all this :D). Are those shifts C - C/ and E\ - E noticeable in actual music? Truthfully I prefer the first version with its drift... in my mind that is. I'm not sure how it would actually sound.

But like I mentioned, if we add ornaments or passing tones etc... to a progression like I ii V I, won't they too have to shift through new patterns of large and small tones, even without modulation?

...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.

So I guess in modulation it's convenient to think mostly of the principal triads I V and IV or i v and iv? That makes sense since there is no risk of pitch drift within the key as long as you stick to those triads.

Sorry, it sounds like I'm quizzing you :). I'm just curious and a bit confused. Maybe instead of asking so many questions I should actually go try some actual music with these.

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.

So I guess in modulation it's convenient to think mostly of the principal triads I V and IV or i v and iv? That makes sense since there is no risk of pitch drift within the key as long as you stick to those triads.

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

Sorry, it sounds like I'm quizzing you :). I'm just curious and a bit confused. Maybe instead of asking so many questions I should actually go try some actual music with these.

Hey, I'm glad if you ask questions. :thumbsup:

I've been reading a wonderful book entitled "Music and Mathematics" edited by Fauvel, Flood, and Wilson. There first chapter deals with tuning and temperament. These split sharp keyboards were apparently created to facilitate the playing in distant keys, because when using just tuning, your Bb must be different than your A# to keep the ratios between the intervals constant. This continuing process of having different frequencies for enharmonics is called the spiral of the temperament and the distance between the enharmonics is called the comma. The different pitches for Bb and A# in such a system are not really microtones, because the music of the period never would have wrote for the music to switch between the two close frequencies abruptly. The 12-tone scale still was king with the minor second with the ratio of 16:15 being the smallest used.

By the way, the book has a cool diagram of Mersenne's 31 note octave which has two keys each for F, F-sharp, G-flat, and G. Handel actually played an organ built with such a design in the Netherlands; just imagine how many additional pipes were needed!

Actually the 31 note octave would have had keys for F Gbb F# Gb F## G... which is even more amazing! Or maybe you're thinking of a keyboard with 19 notes, which would be F F# Gb G like you say. The latter was fairly common on both organs and harpsichords in the 16th and 17th centuries, though many only had a couple of pairs like D#/Eb and G#/Ab.

In fact, not only Handel but a whole group of composers from the generation before the previous were taught on such instruments... composers like Frescobaldi, Luzzaschi and Valentini and presumably many of their pupils as well.

And you're right that they were only used to make the traditional harmonies of their times more in tune. But we do know that even further back in time, in the early 16th century, composers had an intense interest in reviving what they thought was ancient Greek theory.

You see, the Greeks had a tetrachord, a half-scale of four notes, of two quarter tones and a major third. Various Greek authors and the early Christian writer Boethius carefully described it. And in the early Renaissance, classical thinking was the model for many of the arts. Naturally, however, no musician could get the enharmonic tetrachord to sound melodious; it sounded too foreign and could not integrate into the mostly consonant polyphonic style of the times.

But the theorist Nicola Vincentino and various of his followers thought they could revive the enharmonic tetrachord and create truly microtonal music, which of course predates what you're referring to Musicthor. These early composers were actually looking for new sonorities and bizarre intervals. In fact the movement even predated Vincentio. The style was called musica reservata and composers as eminent as Lassus and I think Cipriano de Rore wrote occasional pieces as chromatic as Gesualdo's famous madrigals. I think Lassus's collection of musica reservata was called Prophetiae Sibyllarum.

But anyway Vincentino was the first to advocate true microtonality. In his famous book (and he was among the two most famous theorists of that century along with Zarlino) L'antica musica ridotta alla moderna prattica, he includes a chapter on his inventions, the arcicembalo and the arciorgano (oddly enough it is modern authors that spell that arcHicembalo) both which include 36 keys to the octave. Today we speculate that a subset of those 36 notes included something like 31 note equal temperament.

As you say, Musicthor, the main use for the instrument was to play triads in perfect tune, but we know that Vincentino advocated going even further into true microtonality, becuase in his book, he encourages the reader to experiment with lowering and raising all the pure intervals to reproduce the subtle inflections of the human voice. I think, for example, he specifically mentions the neutral third which is right between the minor and major thirds. And finally he published his collection of madrigals intended to showcase his techniques. And indeed they do employ microtonal progressions. I know that L'antica musica ridotta alla moderna prattica has been translated into English, though I doubt much of his music has been published in moder editions.

Vincentino's radical ideas led to a famous debate with his rival Lusitano. In fact, the last few chapters of Zarlino's Le institutioni harmoniche are dedicated to discrediting Vincentio's ideas. Vincentio's rivals objected because the complex microtonal system of L'antica musica ridotta alla moderna prattica had little to to with the enharmonic genus of the "antica musica" Vincentio claimed to be reviving. In that sense their criticism was certainly valid, and Lusitano won the debates in Rome, but to microtonalists like me he remains a source of inspiration.

EDIT: You can see a facsimile of Vincentino's book here Gallica - Vicentino, Nicola. L' antica musica ridotta alla moderna prattica, con la dichiaratione : et con gli essempi de I tre generi, con le loro spetie, et con l'inventione di uno nuovo stromento, nelquale si contiene tutta la perfetta musica, con. Of course I can't read Italian, and much less dense Renaissance Italian :(. And at http://www.cipoo.net/downloads/scores/ProphetiaeChromatico.pdf you can see the first piece of Lassus's Prophetiae Sibyllarum, Carmina Chromatico, which begins with the progression C Maj, G Maj, B Maj, c# min!!! And Wikipedia of course has interesting articles on both Vincentino and the archicembalo. Finally, at Four enharmonic madrigals can be bought Vincentino's madrigals, edited by the eminent musicologist Alexander Silbiger; I don't own this but I think it must be a quality edition if Silbiger's name is attached to it.

I've been reading a wonderful book entitled "Music and Mathematics" edited by Fauvel, Flood, and Wilson. There first chapter deals with tuning and temperament. These split sharp keyboards were apparently created to facilitate the playing in distant keys, because when using just tuning, your Bb must be different than your A# to keep the ratios between the intervals constant. This continuing process of having different frequencies for enharmonics is called the spiral of the temperament and the distance between the enharmonics is called the comma. The different pitches for Bb and A# in such a system are not really microtones, because the music of the period never would have wrote for the music to switch between the two close frequencies abruptly. The 12-tone scale still was king with the minor second with the ratio of 16:15 being the smallest used.

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).

Hello,

I was bit by the microtonal bug five years ago. When one is a musician and also in the habit of questioning the assumptions provided to you by your inherited culture, a rigorous questioning of "How and Why do we tune the way we do?" along with "What if it were different?" yields a rich hidden valley of music & idea. Some thoughts I have:

1. There are myriad tunings, each one different. And I mean perceptibly different, if you took the time to compare them side by side. And why wouldn't you, you careful craftsperson and artist, take the proper time to choose your tools? Each one is capable of doing different things, much like different languages are. I recommend reading the writings of undersung microtonal pioneer Ivor Darreg. But:

2. Tunings have no inherent sound of their own because what we hear is always what we think to play in them! Many a broad dismissal have been made by people who never heard tunings, or heard them used in particularly unflattering ways. I hear talk of 19-equal and wonder if anyone took the time to try 17-equal? (Having been part of piano concerts in both tunings, I decide I like both tunings, though on pianos they are perhaps not at their best.)

3. Seek an alternative to a hierarchical view of tunings. Please. Don't make 12 tone equal temperament (the Western status quo for the past 100 years only) THE standard against which to measure every other scale, and take care that you don't overthrow it and simply place the Harmonic Series in its throne. Ivor Darreg said "There are no bad scales."

4. Synthesizers are easy to retune. Scala is an excellent piece of free software, and there are plenty of other programs and softsynths to satisfy.

5. Acoustic instruments are not so easy! Some you have to modify extensively or redesign completely (seen the double-barrel quarter-tone clarinet?) and the ones you don't require new techniques which require patience and dedication to learn how to hear differently. Toward that end, I'm holding a Thirty-one-tone singing camp in NYC in two weeks. Figuring out what's possible is a collaborative project I'm interested in and have barely started.

6. Hundreds of people have experimented with microtonality, some of them with results they aren't too embarrassed about to share. This list can give you a taste, but as always, so much more is possible! (Microtonality is not a style.)

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? :)