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Microtonality


echurchill

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A while ago there was a member called Dunael who wrote microtonal music. He seems to have stopped visiting Young Composers, but I am wondering whether anyone else on our forum has experience or knowledge with microtones. Lately I have been reading and experimenting with microtones a lot - it's really the only avant-garde idea I have investigated carefully and plan to one day write music with.

The reasons I like microtonality are hard to describe. Microtonality is to me somewhat like alchemy - lots of abstract math and magical charts (look at articles like http://www.xs4all.nl/~huygensf/doc/terp31.html) and philosophizing about the nature of the universe, with comparatively little music written supporting the grandiose theories.

That said, it seems that many young composers are unaware that microtonal music is composed and performed by many modern musicians. It may be somewhat rare, but it can be performed without too much difficulty, and, as I mentioned, much literature and theory surrounds the subject. Some of the music sounds totally crazy, but certain types of microtonality are almost imperceptible to modern ears and even tonal.

Something else many composers forget is that microtonality is a very old idea in the West. From the Greek philosophers and medieval authors to the theorists of the Renaissance and Baroque, we find tons of treatises concerning the theory of tuning. Some went as far as proposing systems of more than 12 notes to the octave. In Italy a fair amount of Renaissance harpsichords were built with "split sharps" providing separate keys for, say, C# and Db. Look at DENZIL WRAIGHT - Roman harpsichords.

Oftentimes quartertones are the first sort of microtonality to come to mind, since they fit into our modern systems so neatly, but in the Renaissance more authors were interested in 19 or 31 notes to the octave. These may seem like fairly random numbers; I could try explaining them a bit if anyone's interested. Lately I have been experimenting with 19 tone music in particular.

And I need not remind anyone that cultures outside of Europe deal with microtonality extensively. This is yet another field entirely and also very interesting.

So basically, I want to know if you use microtones, and learn from your experiences and ideas. Or if you have never given any thought to them, I could answer questions or explain some of my own ideas. Lots of the theory can get very mathematical, but at its worst you only deal with fractions :P (and some logarithms and roots, but I don't want to scare anyone away).

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microtonality to music is like a spice to food. it sure may make it tastier :)

if i'm not wrong, if you take a black piano key and then use it as white one, you already done some tuning. also, i used to have a guitar, so the only thing i really liked was to play a string while tuning it. plus it's nice effects to export some chord/sound textures in a normal key, and then use it in a different one, making some really odd/strange sounding all around.

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Ahh, a response!! So have you heard much microtonal music? Any specific composers?

Of modern music, I have been listening to Harry Partch. You can hear and read about some of his music at Art of the States: Harry Partch (You do need RealPlayer unfortunately). It's strange, almost ugly, yet fascinating, sort of addictive. He works with just intonation and tonality diamonds (I could explain these a bit more too if anyone cares).

if i'm not wrong, if you take a black piano key and then use it as white one, you already done some tuning. also, i used to have a guitar, so the only thing i really liked was to play a string while tuning it. plus it's nice effects to export some chord/sound textures in a normal key, and then use it in a different one, making some really odd/strange sounding all around.

Yes, tuning is always a fun way to discover new sounds. I have about 100 strings to tune around three times a week, so I would know :D.

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well, there is one rytis mazulis,a lithuanian composer, who made some really nice microtonal compositions. the there's carlos seraphini, italian one, his works you may find here electro-music.com :: View Forum - Seraph.

Yes, tuning is always a fun way to discover new sounds. I have about 100 strings to tune around three times a week, so I would know :D.

and i have a piano, and everytime the guy comes to tune it i ask him to let me play a bit with that :toothygrin:

but mostly i do these kind of things just by exporting chords/sounds and using them differently. they give more color to the work, yet never done truly pure microtonal composition.

is glissandi microtonal procedure? cause it just goes so smotthly through one key to another. it should be, or am i of track? if that's the case then i've something of that as well :)

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but mostly i do these kind of things just by exporting chords/sounds and using them differently. they give more color to the work, yet never done truly pure microtonal composition.

Not to worry, I have not tried my hand at creating any actual 19 tone pieces, but it is something that will always fascinate me. Like so many theorists before me, one day I will succumb to the lure of never heard before intervals and harmonies....

Anyway, this Seraphini's music sounds very interesting, not exactly my style but I'm certainly open to everything! I am listening right now.

EDIT: About the glissandi, well, yes, there are new tones in them, are there not? :) But you do have a point in that you are most likely just connecting normal 12 tone equal tempered notes with those glissandi, so in a way such microtones are just ornaments to normal harmony. That is a sort of non-essential microtonality. I am personally interested in a full-blown break from 12 tone music, but what you are describing certainly has its place and is worth studying! And it can approach full microtonality, say when three or four string instruments gliss. really slowly to another chord: some really new sounds will appear long the way! Besides, even outside of 12-note-per-octave some systems are more like ours and others totally alien. For example, 24 ET (I will be using this abbreviation. Simply put you squeeze 24 equally spaced notes into the octave.) contains our standard 12 ET, so naturally it has many similar structures. 31 or 19 ET have better approximations of pure intervals like thirds and fifths and such, the ingredients of much common practice harmony.

This may surprise many people, but 13 ET is something else entirely! There is hardly a consonant combination to be found, and thus nothing like our traditional harmonies.

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Not to worry, I have not tried my hand at creating any actual 19 tone pieces, but it is something that will always fascinate me. Like so many theorists before me, one day I will succumb to the lure of never heard before intervals and harmonies....

Anyway, this Seraphini's music sounds very interesting, not exactly my style but I'm certainly open to everything! I am listening right now.

EDIT: About the glissandi, well, yes, there are new tones in them, are there not? :) But you do have a point in that you are most likely just connecting normal 12 tone equal tempered notes with those glissandi, so in a way such microtones are just ornaments to normal harmony. That is a sort of non-essential microtonality. I am personally interested in a full-blown break from 12 tone music, but what you are describing certainly has its place and is worth studying! And it can approach full microtonality, say when three or four string instruments gliss. really slowly to another chord: some really new sounds will appear long the way! Besides, even outside of 12-note-per-octave some systems are more like ours and others totally alien. For example, 24 ET (I will be using this abbreviation. Simply put you squeeze 24 equally spaced notes into the octave.) contains our standard 12 ET, so naturally it has many similar structures. 31 or 19 ET have better approximations of pure intervals like thirds and fifths and such, the ingredients of much common practice harmony.

This my surprise many people, but 13 ET is something else entirely! There is hardly a consonant combination to be found, and thus nothing like our traditional harmonies.

yes, i do get it :)

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Well, the realm of microtonality brings up several issues. For example, Harry Partch didn't *quite* write in microtones, not in the way Haba and Carrillo, who explored Microtonality for the sake of it (and although it might have been avant-garde back in the early 20th century, I don't think it is so avant-garde now). Partch re-wrote a musical system, in fact, he invented a whole musical culture, with new instruments, harmonies, scales, and he basically wrote music closer to just intonation than the equal temperament (I think he based many instruments and scales off of what we know about ancient greek music, or something like that). Ligeti, in his Ramifications, for example, has two string quartets, on of which is tuned a quarter of a tone sharper than the other, so this creates a very rich atmosphere. Xenakis and Penderecki also use lots of quarter tones.

So yeah, when you say "microtonality" you must define it more properly. You can just add quarters of the tone, or you can write whole scales based on other divisions of the tone, like 6ths, 8ths, 21sts, 26ths etc.

The other issue is notation. You can go with graphic notation (like SSC's piece and a piece I've written, but I haven't uploaded mine) or if you want to include microtones in normal notation, you can use one of many symbols.

A very good book on notation of microtones (and some basic stuff about its history and development) is Gardner Read's "20th-Century Microtonal Notation" (or something like that). As with the other Read books I've read, it's easy to digest, concise, and provides you with everything you want to know if you want to start writing using microtones.

Partch has also written a book, called "The Genesis of Music" (or something like that - again, I am not too good with book titles :P ), in which he explains his music, including his 43-tone scale (another thing that plays part in microtones - you may wonder on how many divisions there are between a tone, or between an octave - you'll get different sounds if you divide the octave in 43 tones than dividing the tones in 4 :P )

Hope this helps :)

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I confess that microtonality is way over my head for the most part. I'm accepting of the idea now, but it still seems a bit impractical to me.

My understanding and use of microtonalism is really limited to historic temperaments I use on my keyboard instruments, yet that's not really microtonalism as it's meant in the context.

I am newly fascinated by Arabic music, though I really don't understand its systems yet.

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I confess that microtonality is way over my head for the most part. I'm accepting of the idea now, but it still seems a bit impractical to me.

My understanding and use of microtonalism is really limited to historic temperaments I use on my keyboard instruments, yet that's not really microtonalism as it's meant in the context.

I am newly fascinated by Arabic music, though I really don't understand its systems yet.

I do realize that it is an intimidating subject... especially to composers like you and me who express themselves best in historical styles. Like I've said, its difficult for me to explain why such a conservative composer as myself is attracted to such a novel idea (well, far from modern but certainly foreign to the Baroque period). Of course the composing and performing does present new challenges. But I hope that if you ever feel inclined to experiment, don't feel that it is too far out of your reach! Indians manage to perform their "microtonal" music very nicely.

And to go slightly further with a more complete catalog of the Renaissance attempts at microtonality.... I think if you deal with tuning your own keyboards, you may have heard of meantone tuning.

I will briefly explain just in case you are unfamiliar with it: It was the Renaissance standard. If you have ever tuned equal temperament, you know that a series of twelve consecutive fifths along the circle of fifths (forming our chromatic scale) will actually exceed the octave by a tiny (microtonal!) interval, the pythagorean comma. Thus we temper the fifth very slightly, each fifth narrowed by exactly a twelfth of that comma, and we get a cycle of 12 almost perfect fifths. For our purposes let's call this fifth the perfect fifth; after all, the pythagorean comma is already small and a twelfth of it is almost nothing; the fifth hardly beats at all. But the third... the third is another matter entirely. The perfect third is the fifth note of the harmonic series (counting the fundamental as the 1st): 5/4 (the /4 lowers that harmonic back down to the same octave as the fundamental... when subtracting intervals you divide numbers). But when we tune in perfect or near perfect fifths, we are actually trying to get four fifths to add up (that is multiply!) to a third. Moving up four fifths is 3/2 x 3/2 x 3/2 x 3/2 and we move back down two octaves /2 /2, to get 81/64. Our third made of perfect fifths, 81/64, equals about 1.27... our pure beatless third from the harmonic series, however, is 1.2. Very different! The equal tempered third is very wide, and of course you must know this if you use historical temperaments. (I used to use Kirnberger, which has one perfect major third and the rest wide. Now I use Werkmeister III... what temperaments do you use?)

The Renaissance solution was formulated at a time when the (major) third and (minor) sixth was replacing the fourth and fifth as the dominant interval. Their idea was to temper the fifth much more... much, much more, a fifth so narrow that our giant third is reduced to a perfect major third. That is, we narrow each fifth just enough so that when we stack four of these fifths we don't get our modern wide third.

The downside is that the narrow fifths make a circle of fifths that doesn't close up at 12 notes. So they chose to sacrifice the fifth from Eb to G#... this is the famous wolf fifth and it is too dissonant to use in the Renaissance style. They had beautiful thirds yet narrow fifths, and enharmonic equivalence was also sacrificed. If they tuned the note in between D and E to Eb (a perfect third below G), that note could no longer serve as a D# (a perfect third above B) in, say, a B Major chord.

Here is where the Renaissance theorists began experimenting. As I described, 12 perfect fifths almost reach the perfect octave... by narrowing each by a minimal amount, we get a chromatic cycle of 12 notes. It turns out that an analogous procedure exists for the narrowed fifths of meantone. We saw that 12 meantone fifths falls significantly short of the octave, leaving the wolf fifth. But if you go a bit further... at 19 narrowed fifths the cycle almost closes. By narrowing a touch more, you have an equal tempered sixteen note scale. This scale forsakes enharmonic equivalence... now every whole step is divided in three, for example the step A - B is filled in A - A# - Bb - B. The accidentals must always be used literally... the major third must be written E - G#, because in this system E - Ab is another interval entirely... as it's written, a diminished fourth.

Entirely new sounds appear on the keyboard. The minor seventh of the dominant chord has an entirely different flavor from the augmented sixth of the German sixth chord. The various augmented triads and diminished seventh chords no longer invert into transpositions of themselves. And some sounds are entirely new. The aforementioned augmented 6th comes very close to approximating the seventh note of the overtone series. If our series begins on C, the seventh harmonic is nearest to our modern Bb, but not close enough to sound consonant. Oddly enough, the augmented sixth or "septimal seventh" of the 19-note scale sounds almost consonant when compared to the dissonant minor seventh. Another example is the "septimal third." In 19-note equal temperament the interval, written as an augmented second like C-D#, actually produces a fairly consonant sort of third, narrower and darker than our minor third.

A handful of Italian harpsichords survive with fully enharmonic keyboards. This system was described by theorists as eminent as Zarlino. The extra notes were not really used to produce microtonal music per se but instead allow a wide range of triads with perfect major thirds, like both C minor and B Major. On a 12-note keyboard tuned to meantone, only the one or the other may be used.

Clearly such keyboards aided the chromatic explorations of composers towards the end of the Renaissance like Gesualdo, Frescobaldi, Luzzaschi, etc... Frescobaldi was known to have learned to play the enharmonic keyboard with Luzzasco Luzzaschi, and Kerll, for example, was also taught to play such harpsichords by his teacher Valentini (apparently the court at Graz was very much into enharmonic music). By the middle of the 17th century various temperaments and modified meantones allowed chromatic harmony to fit on the 12-note keayboard, but today many historians forget the vigorous microtonal tradition of the late 16th century.

EDIT: I think it might be interesting for me to post small audio examples of these exotic intervals in 19-note equal temperament. I will make some when I have time.

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I eschew microtonality simply for the fact that most audiences are so used to musical baby food that they can't even deal with major 7th chords.

Well, this is a perfectly good reason. And a perfectly bad reason too, if you know what I mean.

You are right, a very specialized audience is often required for microtonality, at least today. Going back to my Renaissance examples, Gesualdo, who was a rich noble, wrote for his own personally funded singers. Luzzasco's virtuosic, chromatic music was written for a private ensemble of female singers payed by the prince of Ferrara; they performed for only the wealthy merchants and visiting nobility.

And in more modern times Partch's music, for example, could only be performed towards the end of his life with university sponsorship. Before that he lived as a hobo.

I certainly do not want you to live as a hobo :(. But I still personally feel the new sounds deserve a chance. Also remember that some forms of microtonality are much less extreme than others, to the point that an audience untrained in music might not even recognize it immediately.

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I wonder how this relates to 'perfect pitch'.

I must say I believe in equal temperament. The Pythagorean ratios are approximations much in the sense that you cannot 'square the circle' (pi is transcendental). Trying to 'square' the circle of fifths is likewise futile.

On a slight tangent, it feels unsettling hearing Ton Koopman's recording of Bach Trio Sonatas on the baroque organ. Wonderful playing notwithstanding, everything sounds a semitone lower. No. 1 in E flat sounds like it had been transposed to D.

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Daniel: You're quite right. A = 415 (a half-step lower than modern 440) is now considered more or less standard for baroque music by early music specialists, which is why the Koopman recording sounds half a step low to cygnusdei.

echurchill: I've read a lot (and understood little, mainly because I hadn't the inclination) about temperaments, but your description of meantone and the rationale for 19-note is the most concise and sensible I've ever read! I'm going to print it out and keep it for ready reference.

I have also seen and heard examples of Italian 19-note harpsichords with their multiple split sharps (!), and I must admit they sound wonderful. As a choral singer specialising in early music, I and my colleagues are intuitively familiar with the difference between E as the tonic of an E minor chord, say, and the "same" note as the third of a C major chord. As a string player, I tend to play wide pythagorian major seconds and narrower minor seconds as a matter of course, unless I have to play with a piano.

This doesn't really feel like microtonality to me, though, as most people mean it; it's just proper intonation of intervals. I have trouble listening to modern pianos now, sometimes; as a matter of fact, though I own a harpsichord, a fortepiano and a clavichord, I don't own a modern piano yet and am in no hurry, largely due to that fact that I really don't care for equal temperament anymore.

The systems of temperament I use are Kirnberger III (on the harpsichord) and Vallotti (on the fortepiano). My clavichord is a late 1920s antique in the midst of renovation, and I have trouble getting it to hold tune at all, but I'll eventually probably tune that to Kirnberger also. I don't really play much Renaissance music on the keyboard, so I don't have much use for Meantone. I chose these systems because to my ear they sounded like the compromises that worked best in the most situations, given that I play mostly Baroque music on the harpsichord and clavichord, and mostly Classical on the fortepiano. I really fancy Valotti for the fortepiano, and everything up through early Beethoven sounds great on it. I must admit that I don't tune "the old fashioned way;" I have an electronic tuner that helps me with the bearing octave, and I tune by ear from there.

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Ton Koopman, yuck.

In either case, since I don't have perfect pitch nor any of that I don't notice stuff is tuned lower or higher, nor does it really interest me much. The interval relations are almost the same, though on baroque tuning something like C sharp major sounds awkward because of the small differences.

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Boy, the number of things we need to fistfight over increases more and more! I like Koopman :(

He adds, what, an ornament every quarter note? Plus, at 1000 mph. Bleh, his renditions of Bach's organ works are basically the worst I have ever heard.

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  • 1 month later...

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:

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It's good to find someone else interested in the microtones. I was planning to come back to this thread and write about Vincentino's experiments with true microtonal progressions. As I left it, it seems like 19-ET and 31-ET were only tools to improve the consonance of the traditional renaissance harmonies, but some, like Vincentino, took it further.

So later I will come back and mention the instruments of Vincentino and (I think?) Fabio Colonna.

So you don't like the major seconds of 19-ET? It has never been an issue for me because as a harpsichordist, I already know meantone with its relatively dissonant fifths but good thirds. And 19 and 31-ET are logical extensions of meantone. Of course to get to just intonation there are no good solutions before your 53-ET.

And in 19-ET the major second is 3/2x the diatonic semitone and 3x the chromatic semitone. For me those are some very logical (if unusual) proportions, to have the diatonic semitone be twice the chromatic. I have been looking into 19-ET because

1. There were such keyboards in the Renaissance and they are reproduced today by some builders.

2. Renaissance theory survives the transition intact, although with a wider range of progression. I like the Renaissance style... one day I want to make a style something blending late Renaissance with 19 or 31-ET.

3. Obviously I am not interested in writing music in 19 new, different keys. Rather I am looking for more distant and colorful ("enharmonic" as Vincentino would have called them) chord progressions as well as new melodic resources like the chromatic semitone as in for example C - C# - Db - D. I'm not sure about using other new melodic intervals however (except the septimal minor third- I love it).

Also I am intrigued by the 7-limit intervals. I cannot get the septimal major third (I think it's called) to sound consonant in my ears, but I love the dark sound of the narrower septimal minor third. Have you heard these before? And the septimal (or overtone) seventh (or the augmented sixth) sounds practically consonant to my ears. I am open to exploring all these intervals further (in 19-ET I guess, although it's 31-ET that has the near perfect match).

But from what I understand 53-ET does not approximate the 7-limit intervals well. Furthermore from what I understand you have separate major and minor tones because the syntonic comma is not tempered out. I am scared of pitch drift in almost any progression and trouble with modulation. From what I understand a simple modulation from C Major to G Major would require more than just one new note to reproduce the pattern of major and minor tones (and diatonic semitones of course). With this I must add that in a meantone temperament there is no schisma and a major third is four fifths as usual; in 53 ET the thirds and fifths belong to different systems. How would it affect progressions if technically the consonant third is a diminished fourth?

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.

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 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.

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

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So you don't like the major seconds of 19-ET? It has never been an issue for me because as a harpsichordist, I already know meantone with its relatively dissonant fifths but good thirds. And 19 and 31-ET are logical extensions of meantone. Of course to get to just intonation there are no good solutions before your 53-ET.

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.

And in 19-ET the major second is 3/2x the diatonic semitone and 3x the chromatic semitone. For me those are some very logical (if unusual) proportions, to have the diatonic semitone be twice the chromatic.

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. :)

Also I am intrigued by the 7-limit intervals. I cannot get the septimal major third (I think it's called) to sound consonant in my ears, but I love the dark sound of the narrower septimal minor third. Have you heard these before? And the septimal (or overtone) seventh (or the augmented sixth) sounds practically consonant to my ears. I am open to exploring all these intervals further (in 19-ET I guess, although it's 31-ET that has the near perfect match).

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:

But from what I understand 53-ET does not approximate the 7-limit intervals well. Furthermore from what I understand you have separate major and minor tones because the syntonic comma is not tempered out. I am scared of pitch drift in almost any progression and trouble with modulation. From what I understand a simple modulation from C Major to G Major would require more than just one new note to reproduce the pattern of major and minor tones (and diatonic semitones of course). With this I must add that in a meantone temperament there is no schisma and a major third is four fifths as usual; in 53 ET the thirds and fifths belong to different systems.

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.

How would it affect progressions if technically the consonant third is a diminished fourth?

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. ;)

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