The equally tempered scale is dirty

[IncarnateX]

Let it go. I find that this thread took a turn out of something I feel was troll-like. You're not making me like it better this way.

My general agenda... my actual statements are pretty nuanced, though. I opined that music that isn't rooted for very long tends to be suited by equal temperament, for instance, there may not be that much to gain by worrying about this.
I'm the opposite of an ideologue (and I'm not even any 'music theorist'). I think that road leads to unnecessary problems.

This isn't me lecturing, there is a context running through a thread. I saw the OP reply to 'aciddose' "it isn't pointless" and this was a new plot point. Maybe I should shut up, but I wanted to correct course.
I think it's good stuff to know about to have more choices. It's material.

I do not think you should shut up. Again: I was explaining why I think a statement like "this mode will sound pleasing to you" is a theoretical (hypothetical) statement.

My error seems to be that I used too many words doing that thus soiling the ground for further misunderstandings

You can have whatever stance you prefer, I was just exlaining my own point of view on basis of your point (how hypothetical or rethorical that may be).

If it really is of relevance (which I personally do not think it is) my own stance toward music theory is that it is a wonderful tool of discription but not of prescription to what anyone should do or like. If you are a professional music composer and someone orders a work in this or that style it can be a perfect tool, but it would be rather oppressive as a mean to control people's freedom of creation or musical preferences. My impression so far is that you agree to the latter, so I am not accusing you of anything here.

[MadBrain]

Oh god, another futile thread ahead.

My opinion on this is that for harmonic music (ie western music), splitting the comma (which leads to the 22-tone system of indian classical music or the 25 "actually used tones" of turkish classical music) isn't worth it - having twice as many keys makes playing harder, you have to remember to play just major 3rds by inverting the accidentals (so D major becomes D Gb A) and a lot of times the pythagorean 3rd sounds better than the just 3rd anyways, which defeats the whole point. If you want a full circle of 5ths, you end up with 53-tone equal temperament.

The truth is, while equal temperament isn't perfect, it's really, really damn good.
- The 5th is 700 cents instead of 702. This is damn close and the slow beating actually makes it sound better.
- The 4th is an inverted 5th so it also benefits from this.
- The major 2nd is two stacked 5ths minus an octave so it also benefits from this.
- The minor 7th is an inverted major 2nd so it also benefits from this.
- There are at least three "just" minor 3rds ("just" proper 6/5, pythagorean 32/27 and 19/16 from the harmonic series). Equal temperament's 300 cents minor 3rd is a great compromise.
- Same for major 6ths.
- Major 3rds are arguably the "worst" interval. But IRL musicians on freely tunable instruments have major 3rds that go all the way from just 385 cents, up to pythagorean 408 cents, and the average size in one experiment was 395 cents (about half way between). So the whole argument that the major 3rd "should" be 5/4 is potentially totally wrong in the first place anyways. Using the 395 cents compromise as the "ideal" major 3rd, equal temperament is actually really close.
- Minor 6ths have the same issues as other 3rds and 6ths (can be pythagorean or just 5-limit or whatnot and equal temperament is a good compromise between these).
- Minor 2nds and major 7ths are a 5th apart from 3rds and 6ths so the same arguments apply to them.
- The tritone is just some dissonant leftover to plug the gap between the 4th and the 5th.

You could also argue that well-temperament shares a lot of these characteristics and is probably a good system, and has the benefit of having 12 different keys instead of 12 shifted versions of the same key, which is true... but it also happens to sound really close to equal temperament so it's barely worth the effort at all anyways.

I personally think the only microtonal intervals worth the trouble are the melodic ones from arabic (quarter tones) and turkish pop music (equal temperament + extra frets that are 40 cents flat on some Baglamas).

[D.Josef]

Lol I agree, equal temperament is awesome, though I'm kind of curious about well temperament, for the different emotional characters of each key.

To be honest, pure thirds tend to sound really sterile to me. You know, like a room that's so clean and bright that it's disconcerting and upsetting. Like this:

[jancivil]

Oh god, another futile thread ahead.

Futile, meaning what? No one wins by total persuasion? Of course not, but I think this subject is the most interesting one in music theory, because it brings so much in.

- Major 3rds are arguably the "worst" interval. But IRL musicians on freely tunable instruments have major 3rds that go all the way from just 385 cents, up to pythagorean 408 cents, and the average size in one experiment was 395 cents (about half way between). So the whole argument that the major 3rd "should" be 5/4 is potentially totally wrong in the first place anyways. Using the 395 cents compromise as the "ideal" major 3rd, equal temperament is actually really close.

This I don't like, because of this 'wrong'. You may not be cognizant of it, but you're trying to justify ET and 400 cents by it, so for me this is a circular type argument.
I don't care about the experiment, whatever it is, I'm just skeptical of anything of the sort, it's probably bad science. There is nothing 'ideal' about 'this compromise', that's a terrible use of language.

If you like it, you do, if not, do the other thing. I think the 13.69¢ is glaring and it has made me shy of pianos. Before I knew anything about it I would get weary of working with a piano. I think 5¢ is a big deal, too. I'm a guitarist. I cannot stand to use that tuner thing. I tune the third string flatter than ET because G# is wildly too sharp in an open E chord to begin with. I did this before I knew any of this stuff. I have an idea of what it should be to be in tune and it's a compromise that drives me nuts.

I think of all of this as material. I'm not pushing any of it as 'this is the thing to do per se'. 5:4 is where major third comes from. This temperament of it was a compromise, through partial commas and ultimately 12th root of 2 making a supposedly equal 12. But the equal 12 only happens truly in virtual reality or a synthesizer. NB: Pianos are_not_actually tuned like this. It doesn't work. If you don't have this, apprentice as a piano tuner some time. Try being automatic with this 'theory' some time and tune a piano by it absolutely. You won't find it a usable instrument. It isn't going to sound right. Instruments are not faithful to math like this, anyway. There are physical things which occur. The nut on a guitar, there is an art to setting up a guitar. Wind instruments, sample them some time, there are all kinds of things which crop up. I work with samples extensively, I need ways to deal with problems. This is all of it a matter of practicality for me.

The point of, example given Hermode Tuning, is that in concert, musicians dealing with harmony will seek out this thing which is not as sharp as 400 cents. It has been said more than once here that 700 is only 2¢ off of 3:2, but I found a problem I couldn't stand that was an order of magnitude smaller than that in a piano track vis a vis another part (it involved a bad 'string' I was always going to notice, but there was a best compromise). OTOH for a time I kind of liked the out-of-tuneness of a thing, until I didn't.
So I'm actually using different rational intonations all the time. I enjoy it. I change my mind a lot. I prefer ratios out of theory, out of the fundament of fifths, to cents. That is not something automatic for me, I studied various things for a long time. There is an acoustical basis rather than this cents which justifies a product of a certain impetus, modulation of tonal harmony, which I'm not even dealing in.

[jancivil]

I personally think the only microtonal intervals worth the trouble are the melodic ones from arabic (quarter tones) and turkish pop music (equal temperament + extra frets that are 40 cents flat on some Baglamas).

Thanks for sharing. Which ones from Arabic music? "Quarter tones" isn't more than a conventional term as per notation; that doesn't indicate half a cent, the actual intervals could be anything out of a myriad of choices and are contextual per a maqam. I have to question "equal temperament + extra frets that are 40 cents flat", too. That isn't how they proceed.

Baglama intervallic basis

https://en.wikipedia.org/wiki/Arabic_maqam

[jancivil]

[IMO] for harmonic music (ie western music), splitting the comma (which leads to the 22-tone system of indian classical music or the 25 "actually used tones" of turkish classical music) isn't worth it - having twice as many keys makes playing harder, you have to remember to play just major 3rds by inverting the accidentals (so D major becomes D Gb A) and a lot of times the pythagorean 3rd sounds better than the just 3rd anyways, which defeats the whole point. If you want a full circle of 5ths, you end up with 53-tone equal temperament.

Well, the cart of the piano is supposed to pull the horse of music here. There were keyboards made to accomodate the conceptions of sharps vs flats then, and there still are. I wouldn't want to cope with it myself, but there are things that aren't a piano that I like better anyway. I don't know what point is defeated by 'a lot of times the pythagorean...' and I have no idea where you got 'inverting accidentals'. I have noted quite some disagreement through the ages as to what they mean, some believe the opposite of some others.

"splitting the comma" was a temperament; 'equal' temperament, 'octave' divided into 12 'equal', was the ultimate compromise for modulating harmonic music. There was a lot of debate on how good the new ideas were during the process, as per harmonic music. There is an interesting book out recently 'How Equal Temperament Ruined Harmony (and why you should care)'

and
Discovering that equal temperament was possible did not make it acceptable, however. Indeed, this tuning must have sounded very odd [to many of Willaert's contemporaries]. It is wondrously free of "wolves," inconsistencies, or unpleasant surprises. To our modern ears, attuned to today's pianos, it is a perfectly beautiful tuning. Yet, its sound was a radical departure at the time. Creating twelve equal steps in an octave changed the proportions that had been used for the various musical intervals, some more drastically than others. Equal-tempered fifths, for example, are closer to pure (3:2) than the fifths used in mean-tone tuning. But major thirds in equal temperament are tempered seven times as much as the fifths. Minor thirds are tempered eight times as much. In the end, these alterations are not intolerable: None of these intervals sounds bad. But in comparison many seem robbed of their original character. This was readily noticed in the sound of equal-tempered major thirds: Lustrous and calm in their pure form, they were now slightly rough and somewhat bland. [Stuart Isacoff, Temperament, Alfred A. Knopf, New York, 2001, p118.]

When music, conceived on the piano, is played by an orchestra a new difficulty appears: that of notation. The chords of the piano have introduced into music new intervals that accommodate themselves in some way or other to temperament. But if those chords are to be played on untempered instruments just as they are written, the result may be completely discordant. This is the case with Wagner, for example, whose music, conceived on the piano, was much more dissonant in his time, when the musicians of the orchestra made some distinction between sharps and flats, than it is nowadays when practically all musicians play in the tempered scale. [Alain Danielou, Music and the Power of Sound, Inner Traditions, Rochester, Vermont, 1995, p134.]

Anyway.
Splitting the comma does not lead to 22-tone in ICM. That's backwards.
The syntonic comma was split into eg., quarter-comma, sixth-comma meantone.

Danielou's system (which is not necessarily a foundation for anything that happens) proceeds by creating 10 alternatives [by syntonic comma] to a JI of 12, leaving the [1:1 and] 3:2 (as inviolate) alone.
The reasoning is to obtain 3:2 where it is compromised in that JI 12 [16:15; 9:8; 6:5; 5:4; 4:3; 45:32; 3:2; 8:5; 5:3; 9:5; 15:8; 2:1].
16:15 x 3:2 = 8:5 ('Db -> Ab')
8:5 x 3:2 = 6:5 (back within 2:1) ('Eb')
6:5 x 3:2 = 9:5 " " ('Bb')
9:5 x 3:2 = 27:20 " " ('F')
IE: NOT 4:3. So 80:81 x 27:20 = 4:3.

9:8 x 3:2 = 27:16, NOT 5:3, difference of syntonic comma.

5:4 x 3:2 = 15:8 ('E -> B')
15:8 x 3:2 = 45:32 ('F#')
45:32 x 3:2 = 135:128 ('C#')
NOT 16:15.
PROBLEM:
16:15 x 80:81 = 256:243, not 135:128.
So we would need to exceed this system to make 'cycle of 3:2' perfectly complete.

As to 53 equal, what happened was 31/53rds of an 'octave' amounts to something very close to 3:2. So there was a justification of just intervals to correspond with the notion of 'equal', although I'm hard-pressed to see the actual musical point, you can do this through the thing itself to begin with.

Ching Fang (78–37 BC), a Chinese music theorist, observed that a series of 53 just fifths ((3/2)^{53}) is very nearly equal to 31 octaves ((2/1)^{31}). He calculated this difference to be 177147/176776

Similarly, here is Danielou's 53 compared with 72 ET:
http://www.72note.com/danielou/danielou.html

Danielou - Sémantique_musicale

Danielou's work was not necessarily based in prior Indian 'theory' yet was part of a sort of thrust towards a consistent philosophy of sound and part of his religion, essentially. However he was very interested in the practical nature of it all. There is no single 'Indian Classical Music' 22 srutis. The one thing I know from '22' is that work, which is a reduction of his theories that fits 22 and makes a certain sense.

A difference of one comma in a fifth or an octave is not only perceptible but extremely disagreeable even to an untrained ear. The same difference in a third or in a major second (it is then the difference between the major and the minor tone) completely changes the color of the note and its expression. One can even say, as a rule, that such differences are the very basis of vocal and melodic expression, ... [Daniélou, Alain, Music and The Power of Sound, Inner Traditions, Rochester, VT, 1995, p55]

<Danielou is an ethnomusicologist from a less empirical era. Most criticism
of his writings that I have seen focus on the fact that the justifications
for his derivations of scales and tunings are based on his own mystical
precepts and not actual practice. (Danielou denied this, claiming that
modern practice supported his theories, but Mark Levy pretty clearly showed
that this was not the case in his empirical study of North Indian
intonation.)> - Bill Alves /mills3.txt

I should point out, too, that actual tuning ratios are not found in Indian
music theory until the late 17th century, and that it is rare to find two
treatises that agree. - ibid.

[Sendy]

The thing is, a lot of modern music doesn't even modulate much (if at all), let alone change key. That's not a bad thing in-and-of it's-self, but it renders ET a bit pointless. A lot of the best music has some kind of tuning thing going on - even Eminem has experimented (or rather, his producer has?) with tuning. You're missing a trick by staying entirely in ET, in my opinion.

It's not a bad thing even to mix up tuning systems in one work. It can make certain parts of the music beat and stand out against others, I read somewhere a metaphor comparing a well chosen tuning adjustment doing for a melody line what a dash of perfume can do for a person.

It's quite a fun exercise to take a synth that lets you detune notes in the octave (Like FM7/8) and tune each interval yourself to be beatless. You can do it by tuning all notes to be beatless against the root, or you can go around the circle of fifths or fourths, or whatever. This can show you in a very visceral way that what you're doing is pushing the discrepancies around the octave, because some chords will sound fantastic, and others will howl like a wolf.

[MadBrain]

Oh god, another futile thread ahead.

Futile, meaning what? No one wins by total persuasion? Of course not, but I think this subject is the most interesting one in music theory, because it brings so much in.

No, not futile in the sense that no one "wins" (what's the point of that anyways?). Futile in the sense of having large amount of words, but barely any actually useful musical advice... too much theory, not enough practice.

I personally think the only microtonal intervals worth the trouble are the melodic ones from arabic (quarter tones) and turkish pop music (equal temperament + extra frets that are 40 cents flat on some Baglamas).

Thanks for sharing. Which ones from Arabic music? "Quarter tones" isn't more than a conventional term as per notation; that doesn't indicate half a cent, the actual intervals could be anything out of a myriad of choices and are contextual per a maqam. I have to question "equal temperament + extra frets that are 40 cents flat", too. That isn't how they proceed.

Baglama intervallic basis

https://en.wikipedia.org/wiki/Arabic_maqam

I'm not really qualified to go deep into the subject of Arabic intonation, of course, but this guy (unfortunately the video is in French) says the quarter tone should be exactly 50 cents for Arabic music specifically (and not for Turkish or Byzantine church music), not only in theory but also in practice. Then again I'm sure there are zillion other ways to do intonation in Arabic music, so yeah the intonation can be anything, but it also CAN be 50 cent quarter tones IRL.

The "equal temperament + 40 cents flat" figure for saz/baglama I can't find the source of anymore, sorry (was something about how one particular maker would setup the frets this way by default I think), and is probably one example in a million. This guy has actual measurements, and they're clearly not equal-temperament but they also seem to be a bit unreliable (the octave fret is at 1189 cents for instance). As far as I can tell, the system can be described as the usual 12 notes + 5 most common quarter-tone-ish notes (B-, E-, A-, F#-, C#-). Sometimes, you can also see a separate B- (apparently the B- in Rast in the key of G is higher than the B- in Bayati in the key of A).

Afaik, the theory for Byzantine chant is that the octave is separated in 72 commas, and this site lists them in 4 different types:
- "Enharmonic genus" 12-12-6-12-12-6-12 (same as western scales)
- "Diatonic genus" 10-8-12-12-10-8-12 (similar to Rast or Bayati)
- "Soft chromatic genus" 8-14-8-12-8-14-8 (similar to Hijaz Kar, "narrow")
- "Hard chromatic genus" 6-20-4-12-6-20-4 (similar to Hijaz Kar, "wide")
That would make the Byzantine equivalent of the Arabic quarter-tone 33 cents (effectively a sixth-tone).

Turkish classical theory is based on 53-tone equal temperament, but not all notes are used, and actual notes used in practice can be quite different. See here for a glimpse of the system in its full glory.

Sometimes you see videos of Qanun players with a system of levers to raise/lower each note, and they have LOTS of levers - practically one lever for each comma!

[...] and I have no idea where you got 'inverting accidentals'. [...]

By "inverting the accidental", I mean that in 53-TET, you can get a pretty good approximation of just 3rds (5-limit) by lowering/raising them by a comma. One way of notating this raised/lowered comma is to go through the circle of 5ths, which has the effect of turning notated sharps into flats: in an E major "E G# B" chord, lowering the G# by a comma nominally gives you "E Ab B". Minor 3rds show the reverse effect: the Eb in "C Eb G" can be brought to 6/5 by raising it one comma, which you can notate as "C D# G". That's what I mean by "inverting accidentals" - it's an artifact of how you can use the circle of 5ths to move notes up and down by a comma.

This idea pops into many theoretical systems: one earlier theoretical system for Arabic music (before 12-TET + quarter-tones) simply stringed together 24 pythagorean 5ths to get 25 notes which gives more or less exactly this system, and the Turkish system could also be described this way (though in practice they simply use different smaller or larger sharps and flats).

As to 53 equal, what happened was 31/53rds of an 'octave' amounts to something very close to 3:2. So there was a justification of just intervals to correspond with the notion of 'equal', although I'm hard-pressed to see the actual musical point, you can do this through the thing itself to begin with.

The nice thing with 53-TET is not only that it has a good 5th, but it has plenty of intervals that roughly correspond to perfect ratios. You could schematize the notes in 53-TET as:

Pythagorean (3 Limit): D Eb E F F# G G# A Bb B C C#
Just (5 limit): D- Eb+ E- F+ F#- G+ G#- A- Bb+ B- C+ C#-
"Wide" (7 limit): D+ Eb- E+ F- F#+ G- G#+ A+ Bb- B+ C- C#+
"Narrow" (11/13 limit): D-- Eb++ E-- F++ F#-- G++ G#-- A-- Bb++ B-- C++ C#--
"Ultra-Wide": D++ E++ G-- A++ B++

(In practice, some of these are hard to use - in particular detuned unison/4th/5th, wide major intervals, and especially ultra-wide intervals)

The "narrow" series of intervals roughly correspond to the generalized arabic/turkish/byzantine "quarter-tones" (ok I'm mixing up a lot of concepts here so I'm probably going to get crucified here). I'm calling the intervals "narrow" because they're more or less what you get if you make the 5th narrow.

When I talk about "splitting the comma" (ok, it's a bad term I admit), I mean that in 12-TET, the variants of a given interval are all conflated - for instance, there's no difference between a "Pythagorean", "just", "wide" or "narrow" minor 2nd. There's just one minor 2nd in 12-TET (roughly 66% Pythagorean 33% just), and when you go to 53-TET, you split this one interval into 4 different variants. You're essentially introducing the comma into the tone system. In fact, you could say that 53-TET essentially tempers the comma, instead of tempering the semitone like 12-TET. It combines different commas (81/80 just-intonation comma, 3^12/2^19 circle-of-5ths comma) into a single compromise comma that also lets you loop around the circle of 5ths (2^(1/53) comma).

The reason we don't use 53-TET is obviously that our ears aren't THAT sensitive to small differences in pitch (a comma is only about 1% difference in frequency) and that we don't have 22 fingers per hand. And culturally, that western music leans towards the Pythagorean versions of intervals, to the exclusion of just intervals and "narrow" intervals, and the heavy emphasis on harmony compounds this.

[jancivil]

Oh god, another futile thread ahead.

Futile, meaning what? No one wins by total persuasion? Of course not, but I think this subject is the most interesting one in music theory, because it brings so much in.

No, not futile in the sense that no one "wins" (what's the point of that anyways?). Futile in the sense of having large amount of words, but barely any actually useful musical advice... too much theory, not enough practice.]

I'm involved in the practice. I don't know a terrific lot, but I'm into this for the effects.
You have some statements which seem so conclusive but it seems like you're premature and being very selective and I don't know why. To wit:

I personally think the only microtonal intervals worth the trouble are the melodic ones from arabic (quarter tones) and turkish pop music (equal temperament + extra frets that are 40 cents flat on some Baglamas).

Which ones from Arabic music? "Quarter tones" isn't more than a conventional term as per notation; that doesn't indicate half a cent, the actual intervals could be anything out of a myriad of choices and are contextual per a maqam. I have to question "equal temperament + extra frets that are 40 cents flat", too. That isn't how they proceed.

I'm not really qualified to go deep into the subject of Arabic intonation, of course, but this guy (unfortunately the video is in French) says the quarter tone should be exactly 50 cents for Arabic music specifically (and not for Turkish or Byzantine church music), not only in theory but also in practice. Then again I'm sure there are zillion other ways to do intonation in Arabic music, so yeah the intonation can be anything, but it also CAN be 50 cent quarter tones IRL.

I'm not qualified either, but I know better than this. That guy is wrong, that is a corruption and only a stupid person acts to vacate centuries of thought with that shite. Talk about futile. The whole idea of these intervals is expression and it's founded in the voice, the accent, the ethnicity. The theory is part of philosophy also. I don't know, maybe there is some coincidence in something that winds up being more or less 50¢ to the next note, but cart doesn't pull the horse.

- actual measurements, and they're clearly not equal-temperament but they also seem to be a bit unreliable (the octave fret is at 1189 cents for instance). As far as I can tell, the system can be described as the usual 12 notes + 5 most common quarter-tone-ish notes...
Afaik, the theory for Byzantine chant is that the octave is separated in 72 commas
That would make the Byzantine equivalent of the Arabic quarter-tone 33 cents (effectively a sixth-tone).

Turkish classical theory is based on 53-tone equal temperament, but not all notes are used, and actual notes used in practice can be quite different.

You mentioned a lot of words being used and not towards practice. Clearly you've done some reading. 53-ET is theory more than practice, I think you have shown this quite well. The point, as you indicate is 'commas' and this is just an approximation of it to suit this idea 'equal'. But a list of commas isn't it. A comma is a correction essentially. A list of all the possible outcomes of commas, and smaller things out of commas, is just theory. I do doubt that it is a basis for practice.

By "inverting the accidental", I mean that in 53-TET, you can get a pretty good approximation of just 3rds (5-limit) by lowering/raising them by a comma. One way of notating this raised/lowered comma is to go through the circle of 5ths, which has the effect of turning notated sharps into flats.

This is cart pulling horse again. The thing I replied to was you going for this to make 5/4 pointless. I'm confused (that doesn't mean 'please explain it some more again').

The reason we don't use 53-TET is obviously that our ears aren't THAT sensitive to small differences in pitch (a comma is only about 1% difference in frequency) and that we don't have 22 fingers per hand. And culturally, that western music leans towards the Pythagorean versions of intervals, to the exclusion of just intervals and "narrow" intervals, and the heavy emphasis on harmony compounds this.

It isn't made as a template for use. It's pure 'music theory', it makes a whole complex thing fit a chart that a certain sort of person prefers.

one earlier theoretical system for Arabic music (before 12-TET + quarter-tones) simply stringed together 24 pythagorean 5ths to get 25 notes which gives more or less exactly this system, and the Turkish system could also be described this way (though in practice they simply use different smaller or larger sharps and flats).

Exactly.
It's not unlike Danielou's 53 (which he did and revised), or 22, people don't do things based in it, it's after the fact analysis and thinking about things. You and I find it fascinating, and you find the notation interesting enough to go into, but what's done is something else. Such as the Arabic things, they build instruments which requires them to do measurements. (On the sarod, you're taught specific distances on the neck for specific ragas.)

I studied this (towards an instrument design) for a while and at the end of the day found it beyond my real capacities and patience. So when I object to 'it's just ET with some 40 cents something' it's founded in the practical, not me wanting to argue a point necessarily.

I totally disagree about the sensitivity, you don't speak for me. I anticipated this which is why I quoted Danielou. I definitely notice this and a lot smaller. I really did have a thing I had to address way smaller than one cent.

The emphasis on harmony is why western music has no interest in these differences, following the 12tET. It used to, per harmony but it's mostly gone. I don't know, maybe string players are still taught sharps and flats as different and context-bound but I think Danielou's point is right enough to be a truism.

[jancivil]

duplicate post

[MadBrain]

I'm not qualified either, but I know better than this. That guy is wrong, that is a corruption and only a stupid person acts to vacate centuries of thought with that shite. Talk about futile. The whole idea of these intervals is expression and it's founded in the voice, the accent, the ethnicity. The theory is part of philosophy also. I don't know, maybe there is some coincidence in something that winds up being more or less 50¢ to the next note, but cart doesn't pull the horse.

I'm taking this guy as an indication because he's still at least a professional musician, and he's also an Arab musician. He could've made the extra piston 40 or 30 cents, he could've added a second piston, or he could have gone for a slide trumpet, but he didn't. He went to great lengths to be able to play Rast and Bayati on trumpet, so if this guy thinks an exact 50 cents piston is a good solution to that problem, I think that's a good clue, even though the fact that it's a trumpet and it's also used to play western music (like Jazz) forces it to be equal temperament.

[...]The thing I replied to was you going for this to make 5/4 pointless. I'm confused (that doesn't mean 'please explain it some more again').

Just saying that if you're using all "pythagorean" intervals, such as 81/64 major thirds, that's quite probably not different enough from equal temperament to justify the extra effort in building instruments.

[KoolFartWind]

I just want to chime in to say, that I really appreciate the extensive responses by many users, great discussion.

Perhaps I should have called the OP-hypothesis : The equally tempered scale is a useful concept, but nevertheless a dirty one from the purist's point of view.

I don't know, whether the approach of a jazz musician would be possible without the ET12-scale. But I am sure, there must be other strategies for modulating through different keys. What should not be forgotten is the fact, that the circle of fifths is basically a scam, if you take into consideration that a fifth has the ratio of 3:2 (obviously that statement is not true for ET12).

Also in my opinion a major part of the common terminology is too far away from the fundamental physical base. As somebody in this thread mentioned, music is an art and not science but a closer reference to the natural reality of acoustics would be more intuitive. Of course, YMMV.

[aciddose]

It's only "more dirty" relative to other systems if you're one of those PI = 3 sort of people.

Of course the vast majority of us feel that PI = 3 is a dirty cop-out while PI = irrational is awesome.

[KoolFartWind]

I'm sorry, maybe I should drop a few grams of acid to reach your superhuman realm of comprehension.

Just joking. I don't have a problem with imaginary numbers, but there are probably more natural alternatives to a scale, that just divides an intervall from one frequency to its double value into equal parts. I'd say it is lazy work. Nevertheless, you got a point, somehow.

[MadBrain]

I'm sorry, maybe I should drop a few grams of acid to reach your superhuman realm of comprehension.

Just joking. I don't have a problem with imaginary numbers, but there are probably more natural alternatives to a scale, that just divides an intervall from one frequency to its double value into equal parts. I'd say it is lazy work. Nevertheless, you got a point, somehow.

The equal-temperament 5th is 700 cents wide, instead of 702. That's pretty damn natural I'd say, especially for a tuning system based on consecutive 5ths (almost all western tuning systems are based on consecutive 5ths - they have to, because of harmony).

Considering that in "natural intonation", there are two possible "natural" major 3rds (5/4 from the harmonic series, and 81/64 from two major 2nds), there's no one real "natural" system: your major 3rds cannot be both 5/4 and 81/64 at the same time, they have to be one or the other, or some other compromise in between (in equal-temperament, they have a ratio of about 1.26).