why our scales have seven notes, part 2

[TallKite]

jancivil: "prejudiced"? "ignorant"? "narrow worldview"?

Let's keep it civil, shall we? I won't make negative comments about your character or motives. Can you please do the same?

It seems the crux of our disagreement is you guys feel any mathematical analysis is suspect from the get-go.
JJF: "Can you use mathematics to explain these things? - You might well try, but that won't actually answer the question "why"."
IncarnateX: "used as a source of explanation (of causes to the 7 note scale) rather than a mean of description (of the already established 7 note scale) [the math] becomes misleading and also very ethnocentric"

This seems a matter of opinion, not fact. I don't think there's any way to prove or disprove that the mathematical validity of 5 notes or 7 notes has to do with their historical prominence. So perhaps we can agree to respect each other's opinions?

Finally, I'm not making any value judgements about 8-note scales or scales with large gaps or anything. I can see where you might have gotten that impression from my comment about the tonic - m2 - M2 - P4 - tritone - P5 - m7 - M7 - 8ve scale. I'll withdraw that comment, since it's not my main point anyway.





Musicologo: Interesting theory. It's really hard to pick out the 11th and 13th harmonic in a string or wind instrument by ear. I've done it, but it's hard. Do you have any evidence that the ancient Greeks or whoever used these intervals?




murnau: Nice link. This guy is saying much of the same things I'm saying, that 12 fifths add up to about an octave, as do 19 fifths, and therefore they are useful frameworks.




D.H. Miltz: Point taken. Sorry about that!

[IncarnateX]

IncarnateX: "used as a source of explanation (of causes to the 7 note scale) rather than a mean of description (of the already established 7 note scale) [the math] becomes misleading and also very ethnocentric"

This seems a matter of opinion, not fact. I don't think there's any way to prove or disprove that the mathematical validity of 5 notes or 7 notes has to do with their historical prominence. So perhaps we can agree to respect each other's opinions? !

Tallkite, if the etnocentrism is not by given by statement then it is by consequence the same moment you not just describe the mathematical properties of the 7 note system but attach value to it, which is exactly what you do when stating that there are "sound" mathematical reason to "prefer" 5 or 7 note scales. Consequence: Any preferences of other scales like in other cultures are not as "sound" as choosing 5 or 7 note scales.

Now with regard to evidence about historical prominence I think the burden is on you to disprove it and not us to prove it. There are evidence that basically most of human inventions, even math itself, are based on development of goal oriented practise (e.g. the wheel in relation to transport), experimentation, trial-and-error, unforeseen consequences and even coincidence and not ready-to-go perfect abstract plans given by the universe itself and just waiting to be implemented. Why should this be different for musical scales? I will leave to jancivil and others to provide you with the historical facts in this respect.

And as a scientist myself I do respect idealism but I still think it is a strange and upside down way of thinking once you take any notion of development of human activity, culture and evolution into consideration. However that was also late in history (more specific the nineteenth centure) where this turn took place within the natural sciences, so I fully understand if it does not stick everywhere yet. Especially within linear mathematics as this has been one of the most prominent weapons of idealists.

To understand my view, you would have to understand the wide scientic, philosophical and historical gab between a statement like for instance

1) The aesthetic goals of an architect is determined by maths

and

2) An architect is achieving his aesthetic goals by use of maths (among other things)

And it doesn't seems to me that your approach would allow for such a distinction as far as e.g. musical expression concerns.

Cheers

[Gamma-UT]

Two observations:

1) If simple arithmetic was important to music, equal temperament would have been strangled at birth. Kepler had a good old go at making music conform to mathematical rules and he wasn't the first. It didn't go so well.
2) Almost all practical music, other than the simplest nursery rhymes, is not made purely from scales. Accidentals and mode shifts are part and parcel of most popular and classical music. Just the melodic minor scale has nine notes.

However, I'd recommend Bill Sethares' work on why certain intervals dominate most music as some light bedtime reading. It's all in the harmonics, which I think one or two people mentioned (but not so perfect that the difference between natural temperaments and equal can't be papered over).

[jancivil]

jancivil: "prejudiced"? "ignorant"? "narrow worldview"?

Let's keep it civil, shall we? I won't make negative comments about your character or motives, these words are objective and I stand by them. Can you please do the same?

Yes, yes, and yes. That has nothing to do with talking about you or your character. That is my interpretation of your remarks. You can't have a broad musical worldview and be this in the dark about it. You have to have ignored quite a lot and the word 'ignorant' is appropriate. Your thesis is not really competent. You can characterize that as 'you're mean' but it doesn't change matters.

You come in here brand new to the forum, didn't read the other thread or you would have expected some resistance and this very tack, but felt you must start a new thread because you wanted more exposure to your ideas. Did you expect a lot of people would be impressed, or say "OH NOW I GET IT"? That's not terrifically respectful, is it. This objection looks like a tactic to shut down criticism of your ideas and a dodge of addressing such. And it appears to me you're the one imputing motive to me.

murnau: Nice link. This guy is saying much of the same things I'm saying, that 12 fifths add up to about an octave, as do 19 fifths, and therefore they are useful frameworks.

Yes, THOSE are. And again, had you read the other thread you will see that I have fairly thoroughly presented that material as a grounding for the discussion.

You on the other hand have reached ad culum and pulled out a thesis that is really a description posited as a reason. The cart does not pull the horse, really. It's circular logic.

This is something you could stand to learn.

[jancivil]

IncarnateX: "used as a source of explanation (of causes to the 7 note scale) rather than a mean of description (of the already established 7 note scale) [the math] becomes misleading and also very ethnocentric"

This seems a matter of opinion, not fact. I don't think there's any way to prove or disprove that the mathematical validity of 5 notes or 7 notes has to do with their historical prominence. So perhaps we can agree to respect each other's opinions? !

Tallkite, if the etnocentrism is not by given by statement then it is by consequence the same moment you not just describe the mathematical properties of the 7 note system but attach value to it, which is exactly what you do when stating that there are "sound" mathematical reason to "prefer" 5 or 7 note scales. Consequence: Any preferences of other scales like in other cultures are not as "sound" as choosing 5 or 7 note scales.

This is why I thought at once of Java. They did things differently according to their view of the world. I do believe when they made the instruments of the gamelan they had solid principles to draw from that served that idea. Do they have a different and lesser math? The implication is that your understanding is the right one if not the only one available, it is limited to your information and seems incurious to me. It is a judgement preceding your investigation, and there is a single word I chose for it that is accurate. I didn't call you a racist or any names, but you do present us with that problem of worldview plainly enough.

[jancivil]

It seems the crux of our disagreement is you guys feel any mathematical analysis is suspect from the get-go.

Well, here is another sign of ignoring thread 1. I'm very interested in the maths of rational intonation particularly. I find straight maths as a rationale for the decisions and human intent that created the systems tends to fall apart upon investigation and that your particular argument is fraught with circular logic and as such not useful or persuasive.

As hit upon more than once by IncarnateX, this resembles Platonic Ideal and it is just really bogus. That cart doesn't pull any horse.

[crazyfiltertweaker]

I also could use 10 notes and it still sounds nice as melody. As example I transpose a melody and voila 10 notes were used!

maybe a modulation is occured? And if yes how can I recognize it to harmonize it correctly?

and why sounds it nice using 10 notes?

[PurpleCatfishBettie]

one thing is: the tritone is a big deal. i can't imagine the perfect 5th being 'more important' than the tritone. these two intervals - tritone and perfect 5th - really work off of one another. imo ymmv

i've got an aquaintance who worked at finding more 7-note ('diatonic') scales than the major-natural-minor/harmonic minor/melodic minor. he managed to find 3 more scales. one of his rules was no 'clusters' (3 or more consecutive half-steps), and no neighboring intervals larger than 3 frets.

now, the 3 scales that this guy worked out were quite interesting, but it was when i started experimenting with 'clusters' and 'bigger neighboring intervals' that things 'opened up even further'. ...oh, that and also arriving at 6-note (sexatonic), 8-note, 9-note, 10-note scales.

i think it all depends on what you're after. ymmv

[MadBrain]

If you start with the following rules:

- No clusters (ie no consecutive half steps)
- Scale isn't a subset of some other scale that doesn't have clusters (so pentatonic scales are excluded because they're subsets of the major/minor scale)

...you end up with 7 scales, with a total of 33 modes:

4 asymmetric scales with 7 notes (7 modes each)
- Major scale
- Minor melodic scale (upwards variation)
- Minor harmonic scale
- Major harmonic scale (major with b6)

3 symmetric scales:
- Whole tone scale (6 notes, 1 mode)
- Augmented second/half-tone scale (6 notes, 2 modes)
- Half-tone/Tone scale, aka diminished scale (8 notes, 2 modes)

If you require scales to have a 5th, then your total number of modes falls to 20. 18 of these are 7 note, with the remaining 2 being 6 and 8 notes.

Of course, you could have scales with quarter-tones or other microtonal intervals (see: Arabic music) or consecutive half-tones (see: Arabic music) but these tend to blow up any attempts at harmony (just try it), although they are usable for melody (which is why they work in Arabic music, which isn't traditionally polyphonic). Ironically, Arabic, Turkish and Persian is essentially always 7-tone too.

[jancivil]

If you start with the following rules:

- No clusters (ie no consecutive half steps)
- Scale isn't a subset of some other scale that doesn't have clusters (so pentatonic scales are excluded because they're subsets of the major/minor scale)

But they aren't necessarily subsets of major/minor. In too many cases they have no particular relationship even, having predated that paradigm by centuries if not millenia. Do you not see that dismissing pentatonics during this 'essentialist' argument for heptatonics is begging the question?

You're trying to describe a cart pulling a horse. The entire exercise you're about here is carts asked to pull the horse. WHY would we start with the cart of 'no clusters'? It follows another cart, which was pulled by events and culture. What use does this have? Just as the person originating this thread did, you're acting like a description of a selection of things that went on is essential and rule-making itself. This is an error.

4 asymmetric scales with 7 notes (7 modes each)
- Major scale
- Minor melodic scale (upwards variation)
- Minor harmonic scale
- Major harmonic scale (major with b6)

If you require scales to have a 5th, then your total number of modes falls to 20. 18 of these are 7 note, with the remaining 2 being 6 and 8 notes.

What??

Of course, you could have scales with quarter-tones or other microtonal intervals (see: Arabic music) or consecutive half-tones (see: Arabic music) but these tend to blow up any attempts at harmony (just try it), although they are usable for melody (which is why they work in Arabic music, which isn't traditionally polyphonic). Ironically, Arabic, Turkish and Persian is essentially always 7-tone too.

First of all, while there are a couple of systems referred to, mainly a 25-to-the-octave set by al Farabi which has been dumbed down to a 24-to-the-octave 'quarter tone' set, the scales do not really employ quarter tones. In a vocal performance, there will be bends and slides so in a sense there is 'microtonality' but the tunes are modal and the maqam is made of seven tones in an octave. Actually when a magam modulates, there are more.

just try it

Yeah, do that.
As I have said, I think twice now, make a symmetrical octatonic. Does this 'blow up' all attempts at harmony? & WHY will a scale with consecutive semitones do that? No, the practice in jazz is about decorating a harmony specifically. Often called 'the diminished scale'. C Db Eb E F# G A Bb on a C7 [confer: b9 #9 #11 13]. There are two diminished seventh chords in it, in fact. Then you prove the opposite of the point you're after: the Arabic system isn't about harmony, so they can have the 'medium thirds' etc. So, we now see that the western scales you have pulling your argument are tied to harmony. "just try it". While I have certainly done quite some harmony with linear material you will find exotic (including what you would call microtonal), there is a truism here: the scales are not essential, they are a product of a Particular Thought in a Cultural Context. They weren't always around, you know.

Ok, we have Arabic heptatonic and Western Europe heptatonic.

'Ironically' (or coincidentaly) they arrive at seven through a different thought (albeit not in mutual exclusion to one another). What of cultures that just don't do it? In the primary thread I went into this. I guess they need you to get in the Wayback Machine and set them right with your rules? So we're back to an obvious thrust towards hegemony.

What are you doing with this? You're reverse-engineering what happened, over a long time for a whole slew of reasons, in order to fabricate rules to justify a belief that seven had to happen. This is ass-backwards. I think one is going to learn more about music really looking at it rather than reducing it.

[aciddose]

I like jancivil's answer where there are no answers and we all ought to just shut up and stop trying to rationalize anything because it just isn't rational, just read the music bible.

Jancivil; actually some people have rational minds unlike yourself and you might want to take a step back and appreciate that everyone who ever accomplished anything meaningful did so rationally.

If you want to view music as an irrational art-form you're free to delude yourself, just go do it in some corner somewhere away from the rest of us please.

[MadBrain]

If you start with the following rules:

- No clusters (ie no consecutive half steps)
- Scale isn't a subset of some other scale that doesn't have clusters (so pentatonic scales are excluded because they're subsets of the major/minor scale)

But they aren't necessarily subsets of major/minor. In too many cases they have no particular relationship even, having predated that paradigm by centuries if not millenia. Do you not see that dismissing pentatonics during this 'essentialist' argument for heptatonics is begging the question?

You're trying to describe a cart pulling a horse. The entire exercise you're about here is carts asked to pull the horse. WHY would we start with the cart of 'no clusters'? It follows another cart, which was pulled by events and culture. What use does this have? Just as the person originating this thread did, you're acting like a description of a selection of things that went on is essential and rule-making itself. This is an error.

Clusters of semitones are rather very dissonant (you can't normally use them in jazz harmonization for instance, even though you can use practically any other chord that doesn't have a minor 9th), and are rather strange melodically (though blues scales make really good use of them). IMHO because of those limitations, they end up being rather rare around the world (same reason why there are very few languages with clicks - they are kinda hard to pronounce).

Obviously you absolutely don't _have_ to completely "fill in all the notes" of the scale, which is why pentatonic scales are so common. But if you do fill in all the notes, then 7 notes is the point where it starts making a lot less sense to an extra note more, which I think is why 7 note systems are so common around the world.

4 asymmetric scales with 7 notes (7 modes each)
- Major scale
- Minor melodic scale (upwards variation)
- Minor harmonic scale
- Major harmonic scale (major with b6)

If you require scales to have a 5th, then your total number of modes falls to 20. 18 of these are 7 note, with the remaining 2 being 6 and 8 notes.

What??

What I'm saying is that there are only 4 possible 7 note scales that don't have clusters:

- Major: C D E F G A B
- Minor melodic: C D Eb F G Ab Bb
- Minor harmonic: C D Eb F G Ab B
- "Major harmonic": C D E F G Ab B

Every other possibility is either a mode of one of these 4 scales, or it has a cluster somewhere. For instance, if you take a variant of the Hijaz Arabic Maqam that starts on G and that goes G Ab B C D Eb F, this is a mode of the minor harmonic scale above, starting on G. Or, if you want a variant of the Saba Arabic Maqam in E that goes E F G G# B C D, this is a mode of the major harmonic scale above, starting on E.

Of course, you could have scales with quarter-tones or other microtonal intervals (see: Arabic music) or consecutive half-tones (see: Arabic music) but these tend to blow up any attempts at harmony (just try it), although they are usable for melody (which is why they work in Arabic music, which isn't traditionally polyphonic). Ironically, Arabic, Turkish and Persian is essentially always 7-tone too.

First of all, while there are a couple of systems referred to, mainly a 25-to-the-octave set by al Farabi which has been dumbed down to a 24-to-the-octave 'quarter tone' set, the scales do not really employ quarter tones. In a vocal performance, there will be bends and slides so in a sense there is 'microtonality' but the tunes are modal and the maqam is made of seven tones in an octave. Actually when a magam modulates, there are more.

True, there's a zillion variations on Arabic and Turkish tuning systems, including virtuoso Qanun players that will move a string up or down a Comma during a song! But there are real instruments that are based on the quarter tone approximation (for instance the Turkish Saz with equal temperament frets, plus 5 frets that are lowered by 40 cents).

But afaik melodic motion is very stepwise and diatonic, even moreso than occidental music (which has gotten more jumps thanks to pentatonic scales and jazz scales and whatnot). When a Maqam modulates, the notes get shifted up or down, but you've still got only 7 notes at the same time...

just try it

Yeah, do that.
As I have said, I think twice now, make a symmetrical octatonic. Does this 'blow up' all attempts at harmony? & WHY will a scale with consecutive semitones do that? No, the practice in jazz is about decorating a harmony specifically. Often called 'the diminished scale'. C Db Eb E F# G A Bb on a C7 [confer: b9 #9 #11 13]. There are two diminished seventh chords in it, in fact. Then you prove the opposite of the point you're after: the Arabic system isn't about harmony, so they can have the 'medium thirds' etc. So, we now see that the western scales you have pulling your argument are tied to harmony. "just try it". While I have certainly done quite some harmony with linear material you will find exotic (including what you would call microtonal), there is a truism here: the scales are not essential, they are a product of a Particular Thought in a Cultural Context. They weren't always around, you know.

Ok, we have Arabic heptatonic and Western Europe heptatonic.

'Ironically' (or coincidentaly) they arrive at seven through a different thought (albeit not in mutual exclusion to one another). What of cultures that just don't do it? In the primary thread I went into this. I guess they need you to get in the Wayback Machine and set them right with your rules? So we're back to an obvious thrust towards hegemony.

No idea but there's just a huge continuous area stretching from Europe, through all the former Ottoman empire (including Greece, Christian Arabic church music, Turkey, Arabic), Persia (Iran) and India, where all the music cultures are heptatonic (except some Indian Ragas are 5 or 6 notes), so for that zone I think it's just millenia of common influence.

What are you doing with this? You're reverse-engineering what happened, over a long time for a whole slew of reasons, in order to fabricate rules to justify a belief that seven had to happen. This is ass-backwards. I think one is going to learn more about music really looking at it rather than reducing it.

I think 5 and 7 tone systems are so common around the world simply due to the fact that a 5th is just a little bit wider than half an octave, so you can easily get it by splitting the octave at 3/5ths or 4/7ths. It's in the same category of effects as why there are so many languages with 5 vowels, rather than 4 or 6 (the reason: it forms a well balanced symmetric system with equal gaps).

[IncarnateX]

The wheel was invented because it is round

The telescope was invented because it enhances sight

The airplane was invented because it can fly

The A-bomb was invented because of Hiroshima

KVR was invented because it is a perfect dumping ground for great pretenders and their pseudo-scientific nonsense

[IncarnateX]

If you want to view music as an irrational art-form you're free to delude yourself, just go do it in some corner somewhere away from the rest of us please.

More pseudo-scientific nonsense. First of all, indeterminism does not equal irrationality. Secondly, most forms of historicism are only partly indeterministic to the extent that they believe the course of the future cannot be predicted (and guess what: No predictions of the future from determinists have yet disproved this premise). However causes are believed to exist, namely those of human activity in which experimentation, trial-and-error, coincidence, fantasy and imagination as well as purely pragmatic reasons grounded in the needs of daily living play a major part.
And then your link to the Cartesian mind-body problem: What does this has to do with anything? Actually the materialistic historicism of people like Marx and Darwin did indeed became a revolt against any kind of idealism. Including that of Descartes and mathematically minded Platonists, which paradoxically would apply to yourself to the extent you think development of human culture is determined by mathematical ideas.

Do you even know that disciplines such as nonlinear mathematics or quantum mechanics deal with indeterministic systems from a strictly scientific point of view? (the article you linked actually mentions it but you only read the headlines, don't you?)

Unlike Jancivil I have no personal issues with you but you have indeed more than once proven that you are no educated scientist at my time at KVR despite your efforts to pretend you know all by linking random wiki articles in random threads. Stick to coding and compressors and stop making a fool of yourself.

This is actually a kind advice and no personal attack.

[Aroused by JarJar]

If you play (in 12-tET):

C, Db, Ebb, F, Gb, Abb, Bb, c
(that's "c, c#, d, f, f#, g, Bb, c" for those who learned from, say, trackers)

You'll be playing a chromatic scale of ancient Greece (in the mode with two conjunct tetrachords followed by a whole tone, the original Mixolydian mode).

Over centuries the chromatic tetrachords fell out of use and the diatonic took over. Here's the diatonic Mixolydian:

C, Db, Eb, F, Gb, Ab, Bb, C

(it's what's called the "locrian mode" these days).

Consecutive semitones, aside from trills, in modulations, etc.,are rare in contemporary middle eastern music. I've heard that they don't happen at all as parts of scales, but I've also heard that there is Persian classical music that still has this ancient feature. Don't know if this is true and sadly the way the world has been messed up for so long there's probably a lot of stuff from that part of the world that's either hidden or even lost forever.

Anyway you *can* use scales with consecutive semitones in harmony, if you work in more ways than tertian-chords-on-every-other-note-of-a-scale.