Why sound a Dur scale better than a random collection of notes???

[JumpingJackFlash]

The 7 tone diatonic scale (virtually the same as "C Major" on your keyboard)
24/24 = 1/1 (C)
27/24 = 9/8 (D)
30/24 = 5/4 (E)
32/24 = 4/3 (F)
36/24 = 3/2 (G)
40/24 = 5/3 (A)
45/24 = 15/8 (B)
48/24 = 2/1 (C)

Sounds great in theory... until you learn that this isn't actually what you get on your keyboard at all. More likely, you will be in equal temperament, which isn't anywhere near as neat and simple. This is one reason why simple "ratios" as the solution to the OP is actually very misleading.

[djtrancendance]

The 7 tone diatonic scale (virtually the same as "C Major" on your keyboard)
24/24 = 1/1 (C)
27/24 = 9/8 (D)
30/24 = 5/4 (E)
32/24 = 4/3 (F)
36/24 = 3/2 (G)
40/24 = 5/3 (A)
45/24 = 15/8 (B)
48/24 = 2/1 (C)

Sounds great in theory... until you learn that this isn't actually what you get on your keyboard at all. More likely, you will be in equal temperament, which isn't anywhere near as neat and simple. This is one reason why simple "ratios" as the solution to the OP is actually very misleading.

Warning...technical discussion ahead (all the theory you really need to make your own scales using SCALA and VSTi's that support it, IMO, are in my previous posts...don't let this post confuse you!)


Firstly, I'll agree the "ratios" solution (also known as "Just Intonation" doesn't tell the whole story, as 12EDO simply estimates such ratios and occasionally the estimates are far enough off that errors are formed (the main one being that 10/9 and 9/8 are treated as equivalent).
But that alone really isn't a big deal...it causes a slight harmonic purity gain in some places, like eliminating a sour fifth from JI diatonic...for loss in others, like a particularly sour 5/3 (major 6th) and 5/4 (major third)...but nothing enough to change the mood of the 7-tone diatonic scale or how you compose in it.

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Of course, 12EDO (on virtually all keyboards by default) is not the same as Just Intonation or the Just Intonation notated scale I used above. I avoided additional terms such as 12EDO, commas, mappings, the chromatic scale...needed for perfect accuracy to avoid scaring people off. My point is...try converting virtually any track to the above diatonic scale from that on your 12EDO (equal temperament) keyboard, and the chords and moods will still sound virtually identical a huge majority (think 95%+) of the time and you can, again most of the time, simply look at the fractions and simplify to find where your chords are.
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However, looking at my JI estimation of the for virtually all practical purposes, the chords are the same, if with a bit of added error.

For example 1/1 5/4 3/2 = 4:5:6 = a major chord and E G B = 5/4 3/2 15/8 = 10:12:15 = a minor chord.

However, there are some exceptions as a result of the temperament, such as 9/8 4/3 5/3 acting as a minor chord despite not reducing to 10:12:15. However, you assume an 81/80 (very small) error is "equal", you can set 10/9 and 9/8 as "equal", and thus make the 9/8 act as 10/9, showing you that you still have a 10:12:15 minor chord, though a more sour one (it's still easily normal enough to sound/feel like an obvious minor chord).
Yes, 81/80 is the comma and so on and we can delve more into theory...but the point is, most of the time, the above JI "approximation" gives you a good idea of how the 7-tone diatonic scale in an EDO works most of the time without scaring people off.

Which, I swear, is the main problem with Xenharmony and why it's not catching on...how the experts often phrase it in terms that take huge efforts to understand.
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Of course, you can also say things like the 9/8 and the 5/3 don't form a perfect fifth or near-perfect fifth like 12EDO, but instead a sour "wolf" fifth of 40/27 (the only one in the scale) and that's a large part behind why we use 12EDO instead of JI, minus the ability to modulate between keys. Or that 12EDO distributes that error between 40/27 and a perfect 3/2 fifth among all 12 tones.

In general, we use 12EDO (the chromatic scale, all 12 notes on the keyboard) as a compromise, losing purity in many places, but keeping the fifth and the octave very pure.

If you're going with (or stuck to, due to keyboard hardware limitations and/or no SCALA micro-tuning file support) the EDO (equal tempered route), you'll find 19EDO, 24EDO (go figure, it's a multiple of 12EDO), and 31EDO are some easy choices.

Those temperaments (19,24,and 31 tones per octave) do a better job at keeping ratios used to form the diatonic scale pure without destroying/"wolfing" the fifth than 12...but they are larger and trickier to manage (imagine using 31 keys per octave on you keyboard for 31EDO...it may be virtually identical to the generally purer/more-harmonious old-school quarter comma mean-tone, but boy that's tricky to play)! And then you get systems like Adaptive JI, which often round 12EDO notes to the nearest 31EDO notes and then move notes slightly so they form perfect JI chords, with no approximations used e.g. a major chord becomes exactly 4:5:6, not the slight estimate of 4:5.04:5.98 or whatever the exact value of a major chord in 12EDO is.
31EDO (equal temperament) I actually would recommend as just about perfectly pure (maximum error for ratios there is about 5 cents, vs. 14+ cents in 12EDO) for both the Western/diatonic and Middle-Eastern "Rast" scale systems...it's just flat-out good at just about everything...if you have the patience to deal with 31 tones per octave.


I'll leave it at this...it my not be perfect but, a large majority of the time, it's fairly safe to summarize what's going on in 12EDO as its estimate in (5-limit) JI.[/b][/u]

[jancivil]

there are some semantic, and some historical problems with *rationalizing* ET if we're going to be proper pedants.

ET is not an estimate or a roughing of a rational temperament, it is a compromise so that on an instrument with *12* fixed tones you can go from eg., 'key of C' to a 'distant' key and it's less different than. the idea of a 'wolf fifth' comes out of these problems, ie., the term doesn't exist before these problems. Strictly speaking it belongs to quarter-comma meantone tuning. The so-called equal distance of the twelve tones in a row inside an octave means an irrational result per se because we are dealing with the twelfth root of 2. The tempering of the compounding of 3:2 to get intervals that fit in a 2:1 'octave' use corrections known as commas; ultimately we arrived at the compromise in the service of full modulation without all the deviation on the fixed keyboard or fretted instrument.

I would like to reinforce that 'the major scale' IS cultural, as it is not as prevalent as one would think (given some of this math) in cultures that use rational temperaments.
That would make an interesting paper, to ask why.
For instance I have seen too many times the use of 'Bilaval thaat' [which corresponds with 'Ionian mode'] in Hindustani music attributed as 'western key' etc; it is an assimilation and relatively novel.

[jancivil]

You can put the halfsteps anywhere anyway, that's what modes are. The major scale seems to be the most "natural" or neutral and the further you deviate from it, the more "coloured" the resulting music seems to be, with the scale consisting of the white notes starting from B (I never got around to memorizing mode names, I know, I'm terrible ) being the most coloured, to the point that it sounds off-colour or ill.

it doesn't seem the most natural universally though. the modes pre-existed the major scale; the 'ecclesiastic', ie., 'church modes' derive from greek theory. Arabic musicians went somewhere quite different with that kind of theory. Western Europe sought to reduce and restrict things, the Arabic theory becomes pretty involved.

the major scale was found desirable by the Holy Roman Catholic Church over time, for proper religious expression. the practice evolved over quite some time; for instance a practice of modal counterpoint employed the use of 'musica ficta' to correct modes; eg., raising the seventh to obtain a striving, 'uplifting' or what-have-you feel that was desirable in the expression of your worship. (The augmented second in this Church was very much repressed. NB its prevalence in Arabic music. ) One was subject to 'diabolus in musica', you could go very wrong in music then!

this 'most natural' is purely your conditioning inside a culture that has received this kind of thing and reinforced it over the ages. The so-called Mixolydian mode is 'more natural' to me .

[Lazos]

Western Europe sought to reduce and restrict things, the Arabic theory becomes pretty involved.

the major scale was found desirable by the Holy Roman Catholic Church over time, for proper religious expression. the practice evolved over quite some time; for instance a practice of modal counterpoint employed the use of 'musica ficta' to correct modes; eg., raising the seventh to obtain a striving, 'uplifting' or what-have-you feel that was desirable in the expression of your worship. (The augmented second in this Church was very much repressed. NB its prevalence in Arabic music. ) One was subject to 'diabolus in musica', you could go very wrong in music then!

The "diabolus in musica" refers to the tritone or three adjacent whole tones; this was culturally viewed as a very dissonant and undesirable interval and thus was shunned by church composers.

Altogether different is the "augmented second" in Arabic, Turkish, Persian, and other Near East music which is a property of the hicaz(hijaz) tetrachord. It contains the characteristic second interval of an augmented second. This is a result of it having the structure of tonic, minor second, major third, perfect fourth. In the Turkish and Arabic systems, these intervals tend to be tuned in just tuning(notably Pythagorean for the Turkish); additionally this interval of an augmented second is lessened by shaving off a comma (ditonic) on both ends. Not sure how the early church thought about hijaz makam, but the basic tetrachord survived in tempered Western art music in the harmonic minor scale.

[jancivil]

The "diabolus in musica" refers to the tritone or three adjacent whole tones; this was culturally viewed as a very dissonant and undesirable interval and thus was shunned by church composers.

Altogether different is the "augmented second" in Arabic, Turkish, Persian, and other Near East music.

I was only after a general point in a kind of rant: these are two frowned-upon intervals in music that had a religious purpose; ultimately the practice of which resulted in molding the 'church modes' as we receive them into a prevalence of a major/minor system. The tritone was frowned on by musicians in certain practice I think before the church propagandized such things.



interval derivation in magam is described in various systems and you get various ratios to pinpoint things...

https://docs.google.com/viewer?a=v&q=ca ... 41nGnHZXpQ

[Lazos]

Hmm. I have never heard of the augmented second being frowned upon. Can you point to any research to back this up? As I mentioned, its structure is embedded in the upper tetrachord of the harmonic minor scale, well used by composers in the West. The tritone is also called an augmented 4th; perhaps this is what you are referring to?

Yep, interval derivation is definitely different in all these systems, but with modern players from different traditions coming to general consensus about things through collaboration and inherent commonalities. But theory (both modern and historical) is one side of interval derivation. Playing with and talking to actual proponents of the living traditions is where actual musical practice meets theory, and they aren't always in accord. NB comparing the way Arabic musicians may tune the second degree of Beyati to the way that Turkish players might. Or how in Turkish notation this same written note called "segah" may get played by actual musicians (in the same composition) as one comma flat, two commas, or even 2.5 commas flat.

Lots of food for thought for the OP.

My two cents: Random collections of notes are just that. We organise notes into scales to facilitate creativity. We creatively go beyond their boundaries to further add elements of tension and change, and hopefully our music is the better for it. Scales are a way to give perceived purpose and meaning to our musical ideas and allow us to share them more readily with others. We can do without them whenever we like because there are no rules in art. The caveat with any of this (using scales or not) is that it doesn't guarantee quality

[JumpingJackFlash]

Hmm. I have never heard of the augmented second being frowned upon. Can you point to any research to back this up? As I mentioned, its structure is embedded in the upper tetrachord of the harmonic minor scale, well used by composers in the West.

Classically, the melodic interval of augmented second is hardly ever used (save for a specific effect, as with other augmented intervals). In minor keys, the sixth is sharpened when it rises to the 7th (and the 7th is not sharpened when it falls), hence the melodic minor.

Which brings up another point; it is generally a mistake to view real music in minor keys in terms of scales (thanks to the variable nature of the 6th and 7th degrees). The music came first, the "scales" were derived afterwards as a theoretical tool for teaching and so on. Unfortunately, this is often lost on musicians today who are conditioned to think only in terms of "harmonic minor" or "melodic minor" as though they are rigid concepts; they aren't.

[fmr]

Hmm. I have never heard of the augmented second being frowned upon. Can you point to any research to back this up? As I mentioned, its structure is embedded in the upper tetrachord of the harmonic minor scale, well used by composers in the West.

Classically, the melodic interval of augmented second is hardly ever used (save for a specific effect, as with other augmented intervals). In minor keys, the sixth is sharpened when it rises to the 7th (and the 7th is not sharpened when it falls), hence the melodic minor.

Which brings up another point; it is generally a mistake to view real music in minor keys in terms of scales (thanks to the variable nature of the 6th and 7th degrees). The music came first, the "scales" were derived afterwards as a theoretical tool for teaching and so on. Unfortunately, this is often lost on musicians today who are conditioned to think only in terms of "harmonic minor" or "melodic minor" as though they are rigid concepts; they aren't.

Completely agree with JJF. And to add something to what was written by the OP, there is plenty of music that's written, even today, outside of the major/minor universe, and doesn't sound wrong (at least to me, and to some others). Some even do not compose with notes AT ALL, but just sounds.
Also, nothing prevents music written within the major/minor modes to sound wrong (actually, it's dead easy).
Unfortunately, for the vast majority of people, the music universe is seen (or listened) through a lock hole. It's like the cave alegory by Plato.

[Lazos]

So, having done my own research (since no one cited anything), yes the augmented second appears to have been avoided in the common practice era in the West. Fortunately, we are well beyond the common practice era (except for some folks). It's been enough years since my undergraduate in music, but I'm pretty sure I've analysed a fair amount of music with augmented seconds, but what the hey...

As far as it being avoided, that is purely a Western cultural thing. Near Eastern singers and musicians from other traditions didn't have any trouble singing this interval, thus hicaz, zirguleli hicaz, hicaz kar, nihavend and other such makamlar are quite common in the Turkish repertoire, for instance.

Viewing real music in minor keys in terms of scales is a mistake? Not really, but if so, you must say the same about major scales as well. I would say it's not a mistake to think in terms of scales but it's rather advantageous to think in terms of the mutability of scales as by mutating they provide the flavours that we are hoping to hear in a given moment. This is why: Just to be clear, JJF, your reasoning that because of the nature of melodic minor having a raised sixth and seventh on the way up and those notes flattened on the way down points to the mutability of the scale. The same thing occurs in Turkish music in many many makamlar (not just the ones we in the West would call of a minor tonality). It occurs in makamlar that do not exhibit either tonality or both within the same compositions They call this ascending/descending attraction and is more related to properties of melodic modulation (modulating between makam) than any need to give "proper" voice leading to a singer.

[A.M. Gold]

So it seems to be somewhat independent of the musical environment in which one grows up. It seems to be telling us something fundamental about the human auditory system."

Unless you grew up in China, where everyone loves a five note scale, or in some parts of Eastern Europe, where they use microtonic scales. Or if you are Arnold Schoenberg. Do they have different brains?

BTW, what in the world did Pythagoras know about brain physiology?

[fmr]

Just to be clear, JJF, your reasoning that because of the nature of melodic minor having a raised sixth and seventh on the way up and those notes flattened on the way down points to the mutability of the scale. The same thing occurs in Turkish music in many many makamlar (not just the ones we in the West would call of a minor tonality). It occurs in makamlar that do not exhibit either tonality or both within the same compositions They call this ascending/descending attraction and is more related to properties of melodic modulation (modulating between makam) than any need to give "proper" voice leading to a singer.

This is what happened in what you call (somehow depreciatively?) western culture, before the advent of tonality. When composers used the modes, they changed form one mode to the other (that's where the term "modulation" comes from). The turkish "modes" (I think I can call them that), being probably inherited from the byzantine tradition, have similar behaviours.
Unfortunately, tonality leaded to a simplification of the modes, but tonality evolved, and in the XIX century (with Liszt, Wagner, Bruckner, Mahler, Max Reger, until Schoenberg and the second Vienese school) the tonality turned in "progressive tonality", reaching complete dissolution with Schoenberg.
Later (but almost at the same time), composers like Debussy, Ravel, Stravinsky, followed by Messiaen and others, returned to the modes, and (specially with Messiaen) expanded very much that universe. It's amazing how people keep talking about tonality as if the "westerners" (means US) didn't got way beyond that - we did, many, many years ago. Already.

[JumpingJackFlash]

Viewing real music in minor keys in terms of scales is a mistake? Not really

It is important not to confuse scale with key.

if so, you must say the same about major scales as well.

Sort of. But the major scale is much closer to the major key than with minor equivalents, so it tends not to matter so much. The point is that no Classical composer ever sat down and said "I think I'll use the melodic minor here" - that's just not how they thought.

And I also cautioned about students thinking they are "in the harmonic minor" (similar to saying they are "in G major" or whatever). Again, it doesn't work that way.

It's amazing how people keep talking about tonality as if the "westerners" (means US) didn't got way beyond that - we did, many, many years ago. Already.

While what you say it true, it's important not to use loaded terms like moving "beyond" tonality, as though everything that came later was better. - You may think it is of course, but that's entirely subjective (the same is true of modality as well).

[fmr]

It's amazing how people keep talking about tonality as if the "westerners" (means US) didn't got way beyond that - we did, many, many years ago. Already.

While what you say it true, it's important not to use loaded terms like moving "beyond" tonality, as though everything that came later was better. - You may think it is of course, but that's entirely subjective (the same is true of modality as well).

Not implying that, of course. But I'm amazed how lack of knowledge people are, in what regards western music. Lots of people tend to think that jazz is the "most advanced" achievement of western musicians, ignoring completely the works of Stravinsky, Messiaen and Ligeti (to just mention three composers) all of them are firmly in what is the "western civilization" tradition.
But of course that tonality has it's place, as has modality (in modern terms), and atonality, and everything in between, and nothing is better just because it is written in some specific style. Point is modern composers are not writing in terms of "keys" os "major/minor" anymore (not JUST, at least). And you're right - as I wrote in another thread, the term "scales" is competely out of place here. A scale, in itself, is a meaningless term, only used to define a succession of notes, which may even not be in any key (like the chromatic scale, for example, which may start and end in any note, is a scale, but doesn't belong to any tonality)

[Lazos]

Viewing real music in minor keys in terms of scales is a mistake? Not really

It is important not to confuse scale with key.

if so, you must say the same about major scales as well.

Sort of. But the major scale is much closer to the major key than with minor equivalents, so it tends not to matter so much. The point is that no Classical composer ever sat down and said "I think I'll use the melodic minor here" - that's just not how they thought.

And I also cautioned about students thinking they are "in the harmonic minor" (similar to saying they are "in G major" or whatever). Again, it doesn't work that way.

I'm not confusing scale with key at all, though. Remember that key is a Western concept not necessarily present in other cultures. So, saying that the major scale is closer to the major key is actually doing a bit of what you're counselling against. I'll use Turkish music as an example, as it's the non-Western system I'm most familiar with: modern Turkish Art music has co-opted and added to the Western notation system to suit its needs; so even though there are what look like key signatures at the beginning of a piece, there are definitely not keys in the Western sense. Makamlar are sets of scales or flavours that are played in particular sequences based on individual interpretation that set a mood. Thus the makam is realised because a particular mood is present. Again, with a makam like Rast, the behaviour of the progression exhibits the aforementioned ascending/descending attraction found in the melodic minor scale, though the third note of Rast is of a major tonality (though a fair bit flat of a Western major third).