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why is the major scale constructed like it is?

Posted by expuddle 
expuddle
why is the major scale constructed like it is?
October 12, 2010 06:03AM
Hi, I'm new on these forums. I used to play piano as a kid and after an absence of about a decade I am starting to pick it up again now as an adult. I am finding that I am much more curious now about 'why' music works like it does than I used to be and am trying to learn a bit more about music theory as well. I've got a pretty basic question that probably isn't that easy to answer but it would be great if you could help. Why does the major musical scale go Tone Tone Semitone Tone Tone Tone Semitone? Why doesn't it sound good when all notes in the scale are equally separated by a tone (ie in the key of C major: C, D, E, F#, G# A# C)? Is it simply because of we are "indoctrinated" with this musical scale or is there something inherently pleasing in the scale itself? I look forward to your thoughts!
Re: why is the major scale constructed like it is?
October 12, 2010 07:07AM
I'm no expert in the field, but I believe Pythagorus is credited with the earliest studies of frequency ratios in music. He observed that some of the intervals that we consider consonant and pleasing to the ear have simple integer ratios in frequency. You can read more in the Wikipedia citations below, and perhaps other folks will have some good citations as well -

Music & Mathematics --> [en.wikipedia.org]
Pythagorean tuning --> [en.wikipedia.org]

- Jim in Austin, TX
Re: why is the major scale constructed like it is?
October 12, 2010 07:21AM
One reason that we find major and some other scales ''easy on the ears" is that they comprise notes that have fairly simple frequency ratios with each other. For example, consider the scale of A major based on the lower A having a frequency of 440 hz (i.e., the piano string for that note vibrates at 440 times per second) The next A above that vibrates at 880. That forms a very simple frequency ratio of 880/440 which is 2:1. Our brains readily respond to this relationship. The 5th note of the scale E has a frequency of 660 and a similarly simple ratio with the lower A of 660/440, which is 3:2, The 4th note D forms a ratio of 4:3 and so on.
(Modern tuning systems (temperaments) have tweaked some of the notes a little, so E, for example, is no longer exactly 660, but still close enough to fool our ears into responding as if it were)

If you compare that to the scale spaced in tones that you mentioned (the whole tone scale) you find frequency ratios that are nowhere near as user friendly as those of the major scale - so it's something of an acquired taste.

There's also the harmonic series to consider. Any piano string vibrating at any frequency will also contain other (fainter) vibrations (overtones) which are multiples of the fundamental vibration. The closest and strongest of these correspond to a large degree to those simple ratios mentioned above.



Edited 1 time(s). Last edit at 10/12/2010 07:22AM by Fretsource. (view changes)
expuddle
Re: why is the major scale constructed like it is?
October 12, 2010 07:37AM
Thanks I read this explanation before but it still doesn't really address the question. That the 5th note has a ratio of 3:2 and the 4th a ratio of 4:3 may be so, but it doesn't explain why that ratio is pleasant to our ear. Why isn't 4:5 pleasant or 6:7, or 1:7? It seems a bit like circular reasoning. Surely the equally spaced frequencies of the whole tone scale should be as (or more) pleasant to our ears from a theoretical perspective because every note is regularly space and thus the whole scale very intuitive. Just like it feels 'right' to count from 1 to 10 in the form 1,2,3,4,5, etc rather than 1, 2, 2.5, 3.5, 4.5 etc.

I wonder if it's precisely because the scale is comprised of a series non-equal steps that makes it interesting and codifies the sound set in such as way that separates it from non-musical noises?
expuddle
Re: why is the major scale constructed like it is?
October 12, 2010 08:10AM
also just going by the numbers, lets say C is 100Khz and the octave higher C is 200. each half note is then 100/12=8.3333 Khz higher than the last.
starting on C:
the E (4th half tone above C) would be 133.3333 (3:2 relative to the higher C)
the F# (6th half tone above C) would be 150 (4:3)
so F# should sound right....or am I getting the numbers wrong?
Re: why is the major scale constructed like it is?
October 12, 2010 08:27AM
I think you misread the math in my first Wikipedia link. You assumed a chromatic scale spanning the octave from 100Hz to 200Hz would be divided linearly into 12 equally-spaced frequencies. It's not a linear series, it's logarithmic. In the standard equal-temperament system a note N half-steps above a root frequence R would be written generally as R * 2 ^ (N/12)

So in your example the next chromatic note above the 100Hz root note would be 100 * 2 ^ (1/12) = 105.946 Hz. The carat (^) means to raise to a power. i.e, 2 ^ (1/12) means "raise 2 to the 1/12 power). The next chromatic note would be 100 * 2 ^ (2/12) = 112.246 Hz. You would continue until you find the octave is 100 * 2 ^ (12/12) = 200 Hz exactly. Hope that helps.

If you want to say C=100Hz, then E would be 100 * 2 ^ (4/12) = 125.992, which is pretty close to the 5:4 Pythagorean ratio (125 Hz exactly). The frequency delta, btw, is referred to as a "comma", which you can learn more about here: [en.wikipedia.org])

- Jim in Austin, TX
Re: why is the major scale constructed like it is?
October 12, 2010 01:40PM
Ok guys, enough. It's got nothing to do with harmonic series, or ratios, etc. etc.

The reason the Major Scale is the way it is is, because it EVOLVED that way. Many people make this mistake - they think the Major Scale is somehow "the most important" or "the most perfect", etc. etc. and start looking for reasons why, when it's clear historically that the Major scale is but one of a multitude of scalar resources used for making music.

Before about 1600, the Major Scale did not exist.

It EVOLVED from the Ionian Mode, which itself evolved from Mixolydian and Lydian.

And those modes evolved from the Greek Modes, which evolved from Tetrachords.

If you REALLY want to understand music, look up the three Greek "genera" (that's the plural of Genus). There's Diatonic, Chromatic, and Enharmonic (names you probably recognize).

The Greek formed Tetrachords by tuning the outer strings of a 4-stringed instrument (called a Tetrachord) to a perfect 4th, and then the two inner strings to varying "genera".

They come out looking a little like this:

D-E-F-G
D-Eb-F#-G
D-Ebb-Fx-G - with the Ebb and Fx not quite being the D or G.

Gradually, the D-E-F-G one became favored, and they began combining two tetrachords either conjunctly or disjunctly:

D-E-F-G/G-A-Bb-C
D-E-F-G/A-B-C-D

This yielded different combinations, which where basically the Modes as we know them today (though they got mis-named later and those are the names we now know). This is what we call the "Gamut" (and there are similarities with the Dasian Scale).

So by the Middle Ages, composers were using 8 modes - the so-called Ecclesiastical Modes, Dorian, Phrygian, Lydian, and Mixolydian, and their "Hypo" or plagal counterparts.

As time went on, "plagal" modes gave way to Transposed Modes, and a system of Modal Mutation (see Guidonian Hexachord) and then by the Renaissance, 12 modes were recognized: Ionian, Dorian, Phrygian, Lydian, Mixolydian and Aeolian (and their plagal counterparts were included for theoretical consistency, thus 12 modes). Later it was noticed that they are "rotations" of a basic note set, and that "Locrian" could be included to complete the system.

Major-Minor Tonality evolved as a reaction against modality (reaction might be a harsh word, but usually music both evolves out of, and reacts to previous styles). Check into the Monteverdi versus Artusi argument (and see Prima Prattica).

The Major Scale is the way it is because of history. It's just like why we don't write "shoppe" and say "thee" and "thy" anymore (historical references excepted).

Steve
JonR
Re: why is the major scale constructed like it is?
October 12, 2010 03:23PM
expuddle Wrote:
-------------------------------------------------------
> Thanks I read this explanation before but it still
> doesn't really address the question. That the 5th
> note has a ratio of 3:2 and the 4th a ratio of 4:3
> may be so, but it doesn't explain why that ratio
> is pleasant to our ear. Why isn't 4:5 pleasant or
> 6:7, or 1:7?

They are, but a little less so.
When two pitches are in simple frequency ratio to each other, they share some of their overtones. They thus appear to be related. Our ears (or hearing system) recognise this. They blend.
Now it's a cultural habit to find this blending pleasant or desirable. Some cultures (eg Indonesian) deliberately seek out clashing pitches, tuning instruments to give harmonies we in the west would characterise as "out of tune".

As mentioned elsewhere, it was Pythagoras who (supposedly) first recognised the harmonic value of simple ratios. (So stevel is not exactly right in his post above: our system IS based on ratio, although we have long abandoned the precise ratios of ancient pure intonation.)
The Greek system was based - as steve says - on tetrachords, which were themselves defined as perfect 4ths, 4:3 ratios.
The intervening pair of notes in each tetrachord were apparently arrived at by ear, not by math, because the Greek system incorporated quarter tones as well.

The European modal system was based on Roman scholar Boethius's mistranslation of Greek texts, so our modes don't match the original Greek ones of the same name.

But the early Christian modal system was still based on the Pythagorean principle of ratios in factors of 2 and 3 only. [www.medieval.org]

An excellent survey of the mathematical roots of our system is here:
[www.midicode.com]

Later systems added the factor of 5, which proviided a better sounding major 3rd of 5:4. (The pythagorean one was 81:64, a little sharp of 5:4 and therefore "brighter" but also less smooth.)
This is known as "5 factor" intonation.
It's theoretically possible to use "7 factor" too (giving us a minor 7th of 7:4, eg) - and some people claim the 7 factor is the basis of the blues - but it generally produces notes too far from our 12 divisions of the octave to be of use with existing instruments. (To incorporate 7-factor ratios would mean basically scrapping our entire musical system and starting again...)

Equal temperament (every semitone being the 12th root of 2 above the one below) renders every semitone exactly equal, meaning the old pure ratios no longer hold, but most of them are very close - our hearing seems to have a tolerance threshold within several cents of "pure". (Even today, choirs and string quartets - who tune by ear as they play - tend to gravitate to pure ratios, so they do seem to be natural ideals.)

> It seems a bit like circular
> reasoning. Surely the equally spaced frequencies
> of the whole tone scale should be as (or more)
> pleasant to our ears from a theoretical
> perspective because every note is regularly space
> and thus the whole scale very intuitive. Just like
> it feels 'right' to count from 1 to 10 in the form
> 1,2,3,4,5, etc rather than 1, 2, 2.5, 3.5, 4.5
> etc.

But a wholetone scale is not equally spaced in any acoustic way. IOW, it simply doesn't sound right.
That is, you are looking for the wrong kind of mathematical relationship.

> I wonder if it's precisely because the scale is
> comprised of a series non-equal steps that makes
> it interesting and codifies the sound set in such
> as way that separates it from non-musical noises?

That's much nearer the truth. When a scale is irregular (like the diatonic modes), we hear a hierarchy of notes. As steve says, the fact we regard the major scale as primary is cultural, not natural, so Ionian is not better than any other mode in that way. But in an irregular scale, we can differentiate notes from each other, by their different interval relationships. They can thus acquire meaning. (In a wholetone scale, every note is equivalent to every other, so we don't get anywhere. Music made with the wholetone scale would all sound the same.)
The way scales are employed melodically means we can be persuaded of a "keynote": a "home note" or tonal focus around which a composition can revolve. This seems to be the way humans (in all cultures) like to hear music - we really like to have a constant keynote as a reference point, against which we can judge the other notes. In some cultures the keynote is constantly underlined as a drone (eg in Indian raga, or Scots bagpipe music). In others - such as European classical music, jazz, etc - the keynote is more in the background, something to begin from or return to at the end. It will be implied rather than overtly stated.

Of course, one could employ a drone keynote with the wholetone scale, but - other than the major 3rd - none of the other notes would sound too good harmonised with it, and they would not support the keynote acoustically. I.e., in a diatonic scale, the 5th supports the root by being in a 3:2 ratio with it (it reflects one of the main harmonics of the keynote). All the scales and modes in common use - and not only in the west - have a perfect 5th as a crucial scale note. (Indian raga drones use root and 5th.) The whole tone scale, of course, has no perfect 5th. It also has no harmonic variety: you can only make one kind of chord from a wholetone scale.

The wholetone scale IS used, of course, in western music - but for temporary effects (eg its air of mystery or "imminence").
Re: why is the major scale constructed like it is?
October 12, 2010 03:50PM
Steve, the point I was addressing does indeed have to do with frequency ratios and, to some extent, the harmonic series. Note that I wasn't explaining how the major scale came into being. I'm well aware of the development of the major scale through the Ionian mode, the diatonic 7 note set, and previous to that, the Greek tetrachord with their various genera. (chromatic, enharmonic and diatonic). Yes, I do indeed recognise those names. They're familiar to me not only in their modern usage but also in their original Greek usage as applied to the tetrachord.

The essential question I was answering was why the note relationships within the major scale are more readily acceptable to the ears of most people than are the equally spaced note relationships of the whole tone scale suggested by the OP, and I maintain that the points I made are valid. I did make a point of mentioning that the major scale was just one of several such scales, but saw no point in elaborating on the fact that the other diatonic modes share exactly the same properties, as the OP asked specifically about the major scale.
Nowhere in my reply did I intend to imply that the major scale is somehow the most 'perfect' of scales. Only, within the artificial context of tonality could any such suggestion be made. Otherwise its no better or worse than any of the diatonic modes.
But in terms of being readily accepted by ears of all persuasions, the simple frequency relationships found in scales such as the major scale and other diatonic modes is what led to their success, albeit at various times and in various regions.

The familiar pentatonic scale is an even better example as a case in point. It has no known evolution. It has been found throughout the world amongst even the most primitive of cultures. (of course, not as a scale but as note relationships found within their music from which visiting theorists could construct the scale). Last I heard, the consensus among such theorists was that nothing but the simplicity of the frequency relationships among the notes could account for them having been 'naturally' picked out of the air independently by cultures separated by vast oceans and no means of contact.
JonR
Re: why is the major scale constructed like it is?
October 12, 2010 05:44PM
One more observation about ratio and its connection with our musical system.

If we play 3 frequencies of 440 Hz, 550 Hz and 660 Hz together, we hear an A major triad. Clearly those numbers are in a 4:5:6 ratio.
Furthermore, they are all overtones of a single 110 Hz A note.
(Every musical note contains overtones or partial vibrations consisting of multiples of its fundamental (perceived) pitch. These overtones are known as the "harmonic series". That's how a bugle, or any horn with no valves, can play different notes, by overblowing to sound different harmonics. And it's why frequency ratios matter.)

IOW, a major triad can be said to represent the stronger harmonics of a single musical pitch (that of its root). That seems to explain why we feel a major triad is "basic" and "natural" - it has a sturdy simplicity.
In contrast, the minor triad has a more "mysterious" sound, which is reflected in its more complex ratios of 10:12:15, and the fact that the acoustic root of those 3 notes (the "1" of the ratios) is not the nominal root of the chord. Eg, for an Am triad, the acoustic root (of which the three chord tones form harmonics) is a low F.

Admittedly, the actual chords in equal temperament do not have exact ratios as above. The equivalent A major triad in ET is 440.0 - 554.0 - 659.2. But we hear that as "close enough". (A few sensitive people, however, do report feeling that the equal tempered major 3rd is somewhat sharp of where it ought to be.)
expuddle
Re: why is the major scale constructed like it is?
October 13, 2010 02:07AM
Thanks a lot for the excellent explanations everyone! Lots of food for thought here :o
JonR
Re: why is the major scale constructed like it is?
October 13, 2010 02:54AM
Just realised I need to correct by terminology above. The word is "limit" not "factor". So Pythagorean tuning is "3-limit" - using ratios based on multiples of 2 and 3 only.
Beyond there you have "5-limit" (multiples of 2, 3 and 5). And - in 20th century theory - 7-limit (or "septimal").

More on this concept here:
5-lmit - [en.wikipedia.org]
Limit in general - [en.wikipedia.org])
expuddle
Re: why is the major scale constructed like it is?
October 13, 2010 05:48AM
and i need to correct my smiley, I meant it to be a smiling smiley not a yawning smiley! :)
Re: why is the major scale constructed like it is?
October 13, 2010 05:15PM
Fretsource Wrote:
-------------------------------------------------------
> Steve, the point I was addressing does indeed have
> to do with frequency ratios and, to some extent,
> the harmonic series. Note that I wasn't explaining
> how the major scale came into being.

Well, I wasn't responding to your post specifically - I was responding to the thread title.
:)-D

Steve



Edited 1 time(s). Last edit at 10/13/2010 05:16PM by stevel. (view changes)
Re: why is the major scale constructed like it is?
October 13, 2010 05:27PM
JonR Wrote:

> That's much nearer the truth. When a scale is
> irregular (like the diatonic modes), we hear a
> hierarchy of notes. As steve says, the fact we
> regard the major scale as primary is cultural, not
> natural, so Ionian is not better than any other
> mode in that way. But in an irregular scale, we
> can differentiate notes from each other, by their
> different interval relationships. They can thus
> acquire meaning.

And an interesting feature of "irregularly spaced" scales is that they can create unique sets. In fact, some interesting things about the whole-half collection of Major/minor/modes is that there are 7 notes, each note can serve as a starting note, and in doing so, create a completely unique pattern of intervals from the keynote (thus our 7 modes). Additionally, it gives us the ability to create 12 unique keys (key signatures), none of which share the same exact note content. Another interesting thing about the Major/minor/mode set is that there is a different number of each of the basic types of intervals - one tritone, 2 minor 2nds, 3 major 3rds, 4 minor 3rds, 5 major 2nds and 6 perfect 4ths.

If anything, I feel that a large part of why the Modes, and eventually the Major/minor system were so well-utilized by composers is that they have a great deal of "sonic resources" built in, and thus can be used repeatedly but maintain a uniqueness/freshness due to their complexity/diversity.

Add the ability to incorporate chromatic notes that are "out of the scale", but "within the system", you have a veritable cornucopia of sonic resources.

Steve
Re: why is the major scale constructed like it is?
October 14, 2010 05:54AM
stevel Wrote:
-------------------------------------------------------
> Fretsource Wrote:
> --------------------------------------------------
> -----
> > Steve, the point I was addressing does indeed
> have
> > to do with frequency ratios and, to some
> extent,
> > the harmonic series. Note that I wasn't
> explaining
> > how the major scale came into being.
>
> Well, I wasn't responding to your post
> specifically - I was responding to the thread
> title.
> :)-D
>
> Steve

No worries. I felt your "Ok guys, enough..." comment was a bit dismissive of Jim's and my replies, but the offer of a beer more than makes up for it. Next round on me.
Cheers! :)-D
modellpq
Re: why is the major scale constructed like it is?
January 11, 2013 06:24PM
You musical theorists in the know: is there anything interesting to be made of the fact (which I just noticed, to my great surprise) that a major scale consists of two identical phrases, one following the other? It struck me that perhaps such a scale sounds pleasing partly because it is a very simple call and response sequence.
ttw
Re: why is the major scale constructed like it is?
January 11, 2013 10:54PM
Not every culture utilizes the major scale as we know it. Pelog and Slendro come to mind.
Re: why is the major scale constructed like it is?
January 12, 2013 02:27AM
modellpq Wrote:
-------------------------------------------------------
> You musical theorists in the know: is there
> anything interesting to be made of the fact (which
> I just noticed, to my great surprise) that a major
> scale consists of two identical phrases, one
> following the other? It struck me that perhaps
> such a scale sounds pleasing partly because it is
> a very simple call and response sequence.

Only if a response a 5th higher (or lower) is especially pleasing. ;-)
In general, call-and-response in music tends to work in the same register, not jumping up or down a 5th (or any other large interval).

The fact the major scale has that repeated pattern - the same tetrachord (2-2-1) top and bottom separated by a tone - is useful for memorising the scale structure, but IMO has no musical meaning or value.
Eg. the half-steps resolve in two opposite directions. The 4th comes down to the 3rd, while the 7th goes up to the tonic.
And the most significant scale division is the 5th, so it kind of makes sense musically to see it as the 5th (1-2-3-4-5), plus the 4th (5-6-7-8) on top.
There's one possibly interesting observation to make about the tetrachords (1-2-3-4 and 5-6-7-8). The 4th of the scale is (arguably) the acoustic root of those 7 notes. (Lydian mode is more stable than Ionian, if we stack all the notes up.) But the major key kind of forces the root of Ionian to be the tonic, with the 4th becoming the main source of tension.
IOW, the fact that the 4th makes the scale unstable is an important factor in how "key" works. We don't want a totally stable tonality, because then we can't play consonance off against dissonance, to create all the narrative movement we get from key harmony; we use the dissonance of that tritone (4-7) to resolve into the 3-8 consonance in the tonic triad. The "diabolus" of the tritone (avoided in the pre-major scale period) is thereby incorporated in order to be defeated... as it were. ;-)

Just my $0.02 :-).

Also, as ttw says, the major scale is not a universal phenomenon. It's a European invention of a few centuries ago. It's true that it's proved remarkably successful and popular - it must have something! - but I doubt that's down to the similarity of its upper and lower tetrachords.
Re: why is the major scale constructed like it is?
January 12, 2013 11:01AM
modellpq Wrote:
-------------------------------------------------------
> You musical theorists in the know: is there
> anything interesting to be made of the fact (which
> I just noticed, to my great surprise) that a major
> scale consists of two identical phrases, one
> following the other?

Yes. It points to the origin of our scales and modes as "tetrachord-based".

The Greeks used Tetrachords - a 4 note group (and an instrument with 4 strings, hence the name) that spanned a 4th. Eventually, they combined tetrachords such that a group like E-F-G-A would combine with a transposed version of itself B-C-D-E to make a complete "scale" (what we now call the Phrygian mode).


It struck me that perhaps
> such a scale sounds pleasing partly because it is
> a very simple call and response sequence.

The Major scale sounds pleasing because you've been conditioned to believe it's pleasing. It's no more pleasing than the Phrygian Mode (which as pointed out above, has those same characteristics with the half step in a different place) and the Dorian Mode (ditto, but even more importantly, is palindromic and symmetrical which by that logic *should* make it more important - which it was in the past - it was "Mode One" for a while ;-)

I'll quote John:

"The fact the major scale has that repeated pattern...has no musical meaning or value. "

A person wishing to write a piece that took advantage of the intervallic structure of any scale or mode might find some value in it as well, but like John's "memorizing the pattern", there is very little meaning behind it other than "hey look what I noticed..."

Best,
Steve
brian hemsley
Re: why is the major scale constructed like it is?
February 01, 2013 11:37AM
hI major scales c major is the perfect scale it has no sharps or flats this is because the 3-4 which is e-f and the 7-8 which is b-c occur natuarlly (ef) (bc) are natural semitones
cd(ef)ga(bc) it follows then that every scale after c major that the semitones must occur between the 3rd and forth and seventh and eighth notes of every major scale.
Order of sharps( FCGDAEB ) Order of flats ( BEADGCF)

C MAJOR C D (E F) G A (B C)
D MAJOR D E (F*G) A B (C*D) f*- g semitone c*-d semitone a sharp decreases the interval ie f-g =tone f*-g =semitone see how e-f and b-c have moved in the key of d major
EMAJOR E F*(G*A) B C*(D*E) Are you seeing the picture now the brackets are the new semitones the keep the the semitones in order.If you like what you see contact me and I' ll go through the flat keys see if you can complete the rest of the keys
Re: why is the major scale constructed like it is?
February 01, 2013 01:47PM
brian hemsley Wrote:
it follows then that every scale
> after c major that the semitones must occur
> between the 3rd and forth and seventh and eighth
> notes of every major scale.

Brian, if you understand this, again it means you understand modes. You just need to learn where the half and whole steps are in each mode.

For example, in Dorian, the half steps are between (2 and 3) and (6 and 7)

So all you have to do if you want A Dorian is to make the half steps fall in those places starting from A and you'll have A Dorian.

I don't know if English is your first language but this is an old thread and had been covered pretty extensively by the responses given. Try to refrain from adding superfluous information to old threads.

Best,
Steve
oliTUTilo
Re: why is the major scale constructed like it is?
February 04, 2013 10:52PM
JonR Wrote:
-------------------------------------------------------
> modellpq Wrote:
> --------------------------------------------------
> -----
> > You musical theorists in the know: is there
> > anything interesting to be made of the fact
> (which
> > I just noticed, to my great surprise) that a
> major
> > scale consists of two identical phrases, one
> > following the other? It struck me that perhaps
> > such a scale sounds pleasing partly because it
> is
> > a very simple call and response sequence.
>
> Only if a response a 5th higher (or lower) is
> especially pleasing. ;-)
> In general, call-and-response in music tends to
> work in the same register, not jumping up or down
> a 5th (or any other large interval).
>
> The fact the major scale has that repeated pattern
> - the same tetrachord (2-2-1) top and bottom
> separated by a tone - is useful for memorising
> the scale structure, but IMO has no musical
> meaning or value.

The C major fugue of Bach's first Well-Tempered Clavier book really takes advantage of that fact. The first motive of the subject is literally C-D-E-F, as if the key were actually F major and the subject started, as is common, with a dominant to tonic push. The rest of the voices enter with G-A-B-C (a direct transposition up a fifth), G-A-B-C, and C-D-E-F, respectively, playing with the ambiguity of key that each segment is actually in.

Since the diatonic scale is two tetrachords joined by a tone, keys a fifth apart are dissimilar by only one note. Not just useful for pretty much all fugue expositions, this enables an ease of modulation that we associate with so much of tonality. This also enables some melodic tactics to work at multiple places on the scale.

Btw, this is a great site. Thanks for making your knowledge available.
Re: why is the major scale constructed like it is?
February 05, 2013 12:33PM
This is a good question that I've asked myself often. And as other people point out, is it the spacing pattern that matters, and not the starting point (i.e. are all church modes equally "important")? I don't know. But one explanation that I sort of liked was in Schoenberg's "Theory of Harmony" (check out the beginning of chapter IV, in [books.google.com]); it has a little bit of hot air, but (and this is similar, essentially, to arguments above) it goes like this:
1) the series of overtones (tones whose freqs are multiples of the fundamental) of a note C1 are very close to, in modern equal-tempered language:
C1, C2, G2, C3, E3, G3, Bb3, etc.
2) The "most important" note is clearly C. The next "most important" is G, in that it occurs first (after C) in the overtone series.
3) If we consider the overtone series of G itself, we get: G1, G2, D2, G3, B3, etc.
4) (Now here it seems a little hand-wavy to me) Now G is related to C (a fifth above) like C is related to F. So F should be equally relevant to the tonality of C as G is. So if we then look at the overtone series of F, we get: F1, F2, C2, F3, A3, etc..
5) If we put these notes that we've found in the overtone series' of C, G, and F, we get C, D, E, F, G, A, B...

I think that this does somewhat just boil down to the reasons mentioned above about the note frequencies being simple rational fractions times the fundamental frequency. But it does sort of seem like an argument that the major scale is a little more "important" than the other modes.
Re: why is the major scale constructed like it is?
February 05, 2013 01:00PM
nickc Wrote:
-------------------------------------------------------
> This is a good question that I've asked myself
> often.
> I think that this does somewhat just boil down to
> the reasons mentioned above about the note
> frequencies being simple rational fractions times
> the fundamental frequency. But it does sort of
> seem like an argument that the major scale is a
> little more "important" than the other modes.

NIck, go back and read my first response in this thread.

The Major Scale is constructed like it is because it EVOLVED that way. It's got absolutely nothing to do with harmonics, overtones, frequency ratios, etc.

The Major scale evolved from the Ionian Mode, which in turn evolved from the 8 Ecclesiastical Modes, which in turn evolved from the Greek modes, which in turn evolved from the Tetrachords, which are about the earliest direct ancestors we can find.

Anyone who tries to explain the "superiority" of the Major scale is starting with a faulty premise to begin with. Anyone who tries to explain the "origin" of the Major scale with psuedo-scientific mumbo-jumbo is fooling themselves.

All one has to do is look at the history - it's all right there. Unfortunately, for some reason, people have this "there's got to be another reason" thinking and then they start inventing data to support a faulty hypothesis just because they can't accept evolution as the answer.

Steve
Re: why is the major scale constructed like it is?
February 05, 2013 08:18PM
stevel Wrote:
> NIck, go back and read my first response in this
> thread.

I already read it. Are you asking that I re-read it until I agree?

> The Major Scale is constructed like it is because
> it EVOLVED that way. It's got absolutely nothing
> to do with harmonics, overtones, frequency ratios,
> etc.

Hey, I was just quoting something I read recently from Schoenberg, someone I perceived to be one of the heavyweights in music theory. Sorry, I didn't realize that he's just some guy who spouts "pseudo-scientific mumbo jumbo"; I should have gone immediately to the _actual_ authority on all things music, "stevel".

Ok, internet ranting aside, I find your historical details interesting; and I can't deny that what you say is true, in the sense that it's almost tautological, in that the answer to any question of the form "why is such-and-such arena of art or design the way it is today?" is "because it evolved that way." But that kind of doesn't really answer the question, as it just begs another: why did it evolve that way?
And why did the major scale evolve the way it did? In general I think things evolve for two main reasons:
(1) whim and fashion changes over time,
and (2) to make improvements.

now I can't speak to (1), but it does seem to me like the major scale has some nice properties; for one, the notion of consonance and dissonance are pretty key to a lot of music; I'm not saying consonant music is inherently "better" than dissonant music, but certainly a lot of people react well to hearing consonance. Even psychoacoustic experiments show that people inherently hear notes which are an octave apart in a similar way, and this even extends to other mammals! So anyway, check out the wikipedia page on consonance and dissonance. Notes with frequencies that are separated by simple rational fractions tend to sound more consonant; to some degree this is a subjective label that is a result of cultural evolution, but to some degree I think it might not be. And the major scale, as schoenberg argued, does have some nice properties in terms of consonance.

Are there other possibilities? To be sure. We could have just stopped at a 3-tone scale, with C, F, and G. Not enough notes to be interesting. Or one could cram a whole lot more notes and include more and more overtones. The major scale is a compromise there, and where the compromise lies is probably a result of taste. But I believe that the consonance of the major scale is a relevant factor in why it flourished. But I also agree that because that was the scale people were playing with at the time of the Renaissance , when knowledge and information could spread in ways that it couldn't before, is also a reason for its success too.
oliTUTilo
Re: why is the major scale constructed like it is?
February 05, 2013 08:48PM
Hi all,

Steve wrote:
"The Major Scale is constructed like it is because it EVOLVED that way. It's got absolutely nothing to do with harmonics, overtones, frequency ratios, etc. "

The evolution of the major scale certainly describes its condition and success. But a conception of evolution is rather empty without an idea of the processes that determine it. Just as the eye needs function to have evolved from the simpler photo-receptors before it, the major scale possesses some rather clear and useful traits. In fact, the story of how its traits and its progenitor's traits were discovered and utilized largely is its evolution.

Steve summarized his first post nicely:
"The Major scale evolved from the Ionian Mode, which in turn evolved from the 8 Ecclesiastical Modes, which in turn evolved from the Greek modes, which in turn evolved from the Tetrachords, which are about the earliest direct ancestors we can find."

There are many objective milestones in the history of the major-minor system like the ones you've described. But while it's less clear how they were reached, it's absurd to say that socio-personal motivations didn't exist. Considerations of the harmonic series and frequency ratios, justified or not, absolutely shaped musical thought and practice. Even from this rather objective, historical point of view, you'd be hard pressed to argue that there is nothing relevant in these acoustic phenomena.

But music isn't done evolving. We should continue critical discussion of the major scale's qualities as we see them, including the frequency ratios of its notes and any ties to the harmonic series, to better continue its evolution.
Re: why is the major scale constructed like it is?
February 06, 2013 04:16AM
I find myself in a middle position here.

I agree that the harmonic series MUST have some impact on how and why certain scale structures are chosen and preferred over others - and therefore (in some way at least) how scales evolve. It governs how we perceive pitch relationships, outside of context - even though it's difficult if not impossible to remove all context when it comes to something like music. (If we hear two steady tones, even as unmusical as two sine waves, we instinctively search for some kind of musical meaning.)

The so-called "Greater Perfect System" was built on Pythagorean concepts of sonic "purity" associated with simple ratios (2:1, 3:2). The "harmony of the spheres" was an important religious principle in Ancient Greece, conflating mathematics, music and astronomy (which they would not have seen as separate doctrines).
BUT - it seems they did not make their scales entirely according to mathematical ratio, adjusting intervals by ear. They used 4ths (in tetrachords), which suggests they recognised the 4:3 ratio, but other intervals were not fixed mathematically (AFAIK).

Also:

1. Simple ratios need to be "tempered" to produce usable musical scales (avoiding the "Pythagorean comma" that arises if you build a scale by calculating solely in 5ths), so it's lucky that our perceptions of consonance have a tolerance either side of "pure". There is nothing mathematically simple about equal temperament. (The 12th root of 2 is what's known as an irrational figure, I believe.)

2. The major scale contains a built-in dissonance: the perfect 4th from the root. In isolation, a 4th is consonant, but only because the top note is perceived as the root (because it's an inverted 5th, with the 5th representing an overtone of the root). In the context of a major scale (or key), the 4th is dissonant.
This doesn't - however - mean the major scale is "imperfect" (nor that - as some argue - lydian mode is more "ideal"). It is, in fact, ideally suited to the music that is (and was) made with it.
Naturally that's a chicken-and-egg argument, but it doesn't really matter whether the music ("tonality") or the scale came first, or both evolved together. The point is that it enables certain kinds of musical expression. Tonality (major and minor keys) - more so than the previous modal system - is about particular kinds of musical narratives, told through harmony rather than (or as well as) melody. This fitted with the society in which the tonal system evolved: a Christian one, then in the process of embarking on dangerous imperialist adventures. They needed a music which expressed both the increasing sophistication (as they saw it) of their own societies, and the dangers and challenges that nature and other cultures increasingly presented.
Narratives require conflict, to show how good can triumph over evil, and the built-in dissonance of the major scale (precisely the 4th, esp in combination with the 7th) provides that conflict. The essential defining element is the resolution of the half-steps: 7>1 and 4>3. The old "diabolus in musica" is confronted and soundly defeated.
Against the tonic triad, both the leading tone and the 4th are dissonant, but the 4th is worse, more intolerable. (Jazz has done much to make the maj7 tolerable in popular music, even sweetly consonant. But that's not possible with the 4th. I'm discounting quartal harmony, of course, which is by defintiion outside the tonal system built on 3rds.)

3. The psychological perception of music is highly complex, and the harmonic series seems to have a vanishingly small role to play. Research has shown that even the octave is not as simple as you'd think. Subjects seemed to prefer the sound of an octave tuned slightly wide, to one tuned to a perfect 2:1. If that's true, it doesn't suggest math will help us much in explaining far more complex musical phenomena.
Other musical cultures enjoy scales we'd consider to be badly "out of tune", such as Indonesion pelogs, pentatonics with no relationship to our divisions of the octave.

IOW, culture is all important. Perceptions of dissonance vary over time (eg the jazz maj7, or the medieval tritone), as well as between cultures. We become acclimatized to new sounds, so that what seems harsh to one generation can seem normal and pleasant (even boringly bland) to future generations. (In 1963 - I remember it well - people over 25 or so thought the Beatles made a terrible noise. In the previous decade, some people thought rock'n'roll was the devil's music, and would destroy the world as they knew it. Luckily they were right on the 2nd count at least :-).)


Caveat: if stevel disagrees with any of this, then he is right. Just to save me posting again when he does. :-)



Edited 1 time(s). Last edit at 02/06/2013 04:17AM by JonR. (view changes)
Re: why is the major scale constructed like it is?
February 06, 2013 07:22AM
nickc Wrote:
-------------------------------------------------------
> I believe that the consonance of the major scale is a relevant factor in why it flourished.



Scales do not have consonance and dissonance, intervals do.
There is just as many consonant intervals in the Ionian mode as there is in any other mode, and just as many dissonant ones too.

It should also be pointed out that the overtone series wasn't discovered until centuries after the major scale became common.

Check out this textbook on Four Part Harmony.
Re: why is the major scale constructed like it is?
February 06, 2013 01:41PM
JonR Wrote:
-------------------------------------------------------
> I find myself in a middle position here.
>
> I agree that the harmonic series MUST have some
> impact on how and why certain scale structures are
> chosen and preferred over others - and therefore
> (in some way at least) how scales evolve.

Why?


It
> governs how we perceive pitch relationships,
> outside of context - even though it's difficult if
> not impossible to remove all context when it comes
> to something like music. (If we hear two steady
> tones, even as unmusical as two sine waves, we
> instinctively search for some kind of musical
> meaning.)

Remember Jon that scales developed before any concepts of harmony (at least as we know it).


>
> The so-called "Greater Perfect System" was built
> on Pythagorean concepts of sonic "purity"
> associated with simple ratios (2:1, 3:2). The
> "harmony of the spheres" was an important
> religious principle in Ancient Greece, conflating
> mathematics, music and astronomy (which they would
> not have seen as separate doctrines).
> BUT - it seems they did not make their
> scales entirely according to mathematical ratio,
> adjusting intervals by ear. They used 4ths (in
> tetrachords), which suggests they recognised the
> 4:3 ratio, but other intervals were not fixed
> mathematically (AFAIK).

Exactly. Which tells us they weren't listening to overtones to determine which notes to pick, or how to tune them (or, if they were, they certainly weren't the simpler ratios).

>
> Also:
>
> 1. Simple ratios need to be "tempered" to produce
> usable musical scales (avoiding the "Pythagorean
> comma" that arises if you build a scale by
> calculating solely in 5ths),

That's only true if we want the scale to "fit" within an octave. Some cultures didn't care about that.

>
> 3. The psychological perception of music is highly
> complex, and the harmonic series seems to have a
> vanishingly small role to play.

Or could it be, it never really had all that much of a role to play to begin with ;-)

Research has shown
> that even the octave is not as simple as you'd
> think. Subjects seemed to prefer the sound of an
> octave tuned slightly wide, to one tuned to a
> perfect 2:1.

I'd need to see that study. Because Piano has stretched octaves, that preference may have developed as part of that conditioning.


If that's true, it doesn't suggest
> math will help us much in explaining far more
> complex musical phenomena.
> Other musical cultures enjoy scales we'd consider
> to be badly "out of tune", such as Indonesion
> pelogs, pentatonics with no relationship to our
> divisions of the octave.

Exactly. So one of the pitfalls of these conversations is, someone is trying to "justify" the Major scale, or Tonality, or whatever, because they're starting with the assumption (certainly a self-centered one) that it's somehow "better" or "more perfect" whether they say that directly or not (it's also a conditioned response).

Steve
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