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why is the major scale constructed like it is?

Posted by expuddle 
Re: why is the major scale constructed like it is?
February 06, 2013 01:50PM
oliTUTilo Wrote:
-------------------------------------------------------

> 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.

Not nothing. But a whole lot less than most people assume.

Steve
ttw
Re: why is the major scale constructed like it is?
February 06, 2013 05:40PM
Not all western music can be described through major or minor or modal organization.

[www.youtube.com]
ttw
Re: why is the major scale constructed like it is?
February 06, 2013 05:55PM
The Persian dastgah "rast" corresponds to the notes of the major scale. The music doesn't sound similar to western music though.

[www.youtube.com]
oliTUTilo
Re: why is the major scale constructed like it is?
February 06, 2013 06:26PM
JonR Wrote:
-------------------------------------------------------
> 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.
>

William Sethares argues that the mathematics of the overtones (not necessarily harmonics) in an instrument can be used to understand the scales that evolved for use with those instruments. Thus Indonesian and Thai scales, which are very different from the Western diatonic or chromatic scales, can be understood as particular to the instruments used alongside them. You can find an overview concerning his work here:
[sethares.engr.wisc.edu]

But I recommend picking up his book Tuning, Timbre, Spectrum, Scale.

He also argues that given either an instrument's overtone series or a scale one can come up with a corresponding scale or overtone series to best complement it. Check out the first five sound examples of an harmonic instrument but with a slightly stretched overtone series here:
[sethares.engr.wisc.edu]
The first example compares two "octaves" on this instrument. The instrument's overtones lie at the frequencies n^(log(2.1)/log(2)), so that the 2*nth overtone is 2.1 larger than the nth (instead of twice the size), the 3*nth is 3.24 times the nth, etc. The first octave heard is for notes an exact octave apart (frequency ratio of 2). The second octave heard is for an accordingly stretched octave (2.1 base frequency ratio).
The next four examples compare the instrument either with or without stretched overtones playing a piece of music that is either in the 12-TET scale or in a 12-TET scale stretched accordingly. So instead of notes based on 2^(n/12) frequencies, we'd have (2^(n/12))^(log(2.1)/log(2)) for the base frequencies of the stretched scale.

Sethares' concern is with a particular type of consonance. While the song examples are interesting, it really seems like the octave's integrity depends on the actual overtones of the instrument. His book (or the accompanying CD) has lots of sound examples of various overtone series and scales, including those of Indonesia and Thailand.
yessir
Re: why is the major scale constructed like it is?
June 29, 2013 06:11PM
The major scale is simply 3 major triads (e.g., C, F, and G).

Not all civilizations discovered the wheel at the same time (or even close).

Tom
Re: why is the major scale constructed like it is?
July 01, 2013 09:58PM
The major scale predates triads of all qualities.
Re: why is the major scale constructed like it is?
July 02, 2013 01:24PM
LuxembourgianSixth Wrote:
-------------------------------------------------------
> The major scale predates triads of all qualities.

Well, not "officially"...

SCALES (or modes) predate triads. But the use of triads (and even 7th chords) was in force well before what we consider the "Major" scale (i.e., not the Ionian, or Lascivious modes, etc.).

At any rate, the previous poster's comment is not very informative.

Steve
ttw
Re: why is the major scale constructed like it is?
July 05, 2013 05:44PM
yessir Wrote:
-------------------------------------------------------
> The major scale is simply 3 major triads (e.g., C,
> F, and G).
>
> Not all civilizations discovered the wheel at the
> same time (or even close).
>
> Tom


Or two major and one minor triad: G, dm, G or two minor triads and a major triad am, dm, G or a major triad and a dominant ninth C, G9....
Michael Lusk
Re: why is the major scale constructed like it is?
July 20, 2013 12:48AM
I disagree with Steve: I think the construction of the major scale does have to do with the overtone series and is not just a matter of habit.

Before I explain why, I should say that 9/10ths of the difficulty of this subject, or any music theory subject, is not because of the concepts themselves, but because the terminology is so messed up - and one big source of confusion is between what is meant by scales and what is meant by modes. A scale is simply the division of an octave into a series of pitches. A mode is based on a scale, but implies a central pitch and harmony, and places that central pitch in different intervallic relationships with the other pitches. A given scale may form the basis for multiple modes. The most famous and widely used scale in Western and other music is the diatonic scale, and it forms the basis of the so-called church modes: the Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian modes. The 'major scale' is not really a scale, it's a mode, and should be called 'the Ionian mode of the diatonic scale'.

So expuddle's question can be broken down into two: 1) why has the diatonic scale become the standard scale; and 2) why of all the diatonic modes, has the Ionian become the standard mode?

The diatonic scale became the standard because of all possible seven-note scales, it has the most internal harmonies. All but one of the 7 triadic chords formed from a diatonic scale contain a perfect fifth and a minor or major third. Now, why the human ear perceives intervals from low in the overtones series, that is, octaves, fifths and thirds and their inversions as harmonious and other intervals as dissonant is a good question, and I don't know the answer. I don't think anyone does. But it's universally true, and we just have to accept it. Triads built from the whole-tone scale that expuddle proposes, by way of contrast, are all augmented - containing no perfect fifths - which people find unpleasant.

The Ionian mode became standard because of all the diatonic modes, it alone contains a V7 chord containing a leading tone and a tritone, and so is the most strongly tonal. It became the most frequently used mode when music decisively shifted to tonality in the Baroque period.
Re: why is the major scale constructed like it is?
July 20, 2013 07:30AM
Michael Lusk Wrote:
-------------------------------------------------------
> I disagree with Steve: ...


I was going to write a lengthy reply, but I don't know where to start! - You're mixing up an awful lot of things there.

Seriously, I'd listen to Steve. The guy's a university professor for goodness sake, he clearly knows what he's talking about.

Check out this textbook on Four Part Harmony.
Re: why is the major scale constructed like it is?
July 20, 2013 06:42PM
Michael Lusk Wrote:
A scale is simply the division of an
> octave into a series of pitches.


Scale comes from the Italian (or Latin?) "scala" which is "ladder" and "to climb" is implied (sorry, my Italian is not good).

A "scale" is a series of pitches in ascending order. It's got NOTHING to do with dividing an octave. There are many non-western european scales that don't use the octave as a boundary.


A mode is based
> on a scale, but implies a central pitch and
> harmony, and places that central pitch in
> different intervallic relationships with the other
> pitches.


This is only true for "rotational modes" - modes that are "derived" from a so-called "parent" scale. While this is the way the 7 modern modes are often taught and understood. At any rate, they are "modes based on a scale". However, the Modes proper were not "rotationally related" in that way. And not all modes are.


A given scale may form the basis for
> multiple modes. The most famous and widely used
> scale in Western and other music is the diatonic
> scale, and it forms the basis of the so-called
> church modes: the Ionian, Dorian, Phrygian,
> Lydian, Mixolydian, Aeolian, and Locrian modes.
> The 'major scale' is not really a scale, it's a
> mode, and should be called 'the Ionian mode of the
> diatonic scale'.

Firstly, the modes pre-date Major and minor. Secondly, the Greek modes pre-date all of them, but if we want to use familiar terminology, we can look at the REAL "so-called church modes" which are, in reality, the 8 Ecclesiastical Modes. These are Dorian, Phrygian, Lydian, and Mixolydian and their "Hypo-" versions. Ionian and Aeolian didn't come until much later, and it wasn't until later that they became Major and Minor.

And "The Major Scale" is not a mode. The pitch content of a mode, listed in ascending order, is a scale. We also define it more specifically as a mode to differentiate between it and other modes and scales with similar note content. But there is a distinction between The Major Scale and The Ionian Mode. You clearly don't understand this distinction.


>
> The diatonic scale became the standard because of
> all possible seven-note scales, it has the most
> internal harmonies.

No, that's not why. The diatonic scales (7 note scales in Greek ('diatonic' is Greek for 'through the tones') and Ecclesiastical Modes) PREDATE HARMONY.

Harmony was not a deciding factor in the note choice of scales or modes because the concept of harmony did not yet exist.


All but one of the 7 triadic
> chords formed from a diatonic scale contain a
> perfect fifth and a minor or major third. Now, why
> the human ear perceives intervals from low in the
> overtones series, that is, octaves, fifths and
> thirds and their inversions as harmonious and
> other intervals as dissonant is a good question,
> and I don't know the answer.

That's right. You don't. You've been led astray my friend. But don't feel bad, many (too many) people have. It seems SO believable doesn't it? But it has nothing to do with the actual historical evolution of modes and scales.



I don't think anyone
> does. But it's universally true, and we just have
> to accept it.


No we don't. We can THINK instead of just believe some gobbledy-gook someone told us becuase, "hey, that sounds like a good reason".


>
> The Ionian mode became standard because of all the
> diatonic modes, it alone contains a V7 chord
> containing a leading tone and a tritone, and so is
> the most strongly tonal. It became the most
> frequently used mode when music decisively shifted
> to tonality in the Baroque period.

This is partly true. Composers did in fact begin to favor the Ionian and Aeolian modes from the Renaissance, and there was a gradual "absorption" of Mixolydian and Lydian into Ionian, and Dorian and Phrygian into Aeolian (though elements of Dorian lasted into the Common Practice Era). In essence, the desire for a "functionally progressing" harmonic practice ended up favoring what are called "ascending progressions" and those chords that were unique in defining a Tonality. But it's this distinction between The Major Scale in Tonality versus the Ionian Mode in Modality - and while there is certainly some transition from one into the other, music theorists distinguish between Ionian and Major for good reason: though the pitch content and "key note" are the same, the WAY in which they're used is quite different, and conflating the two is considered the mark of the ill-informed.

Steve
Re: why is the major scale constructed like it is?
July 21, 2013 04:40AM
Michael Lusk Wrote:
-------------------------------------------------------
> I disagree with Steve: I think the construction of
> the major scale does have to do with the overtone
> series and is not just a matter of habit.

Not wishing to disagree with steve - and he may shoot me down in flames anyway :-) - but there is a connection between the harmonic series, certain basic scale divisions, and ideas of consonance.

It's just not a defining connection, in terms of music theory and practice, which is what I think he is saying.

Eg, the "perfect" intervals align with very simple frequency ratios. Octave = 2:1, 5th = 3:2, 4th (inverted 5th) = 4:3. As I understand it, Pythagorean intonation and scale structure begin from these ratios.
We can go further and invoke the factor of 5, which gives us a major 3rd of 5:4, a minor 3rd of 6:5, and their inversions.
However, scales based on these ratios - using factors of 2, 3 and 5 to calculate other pitches - soon hit problems, due to "commas" (Pythagorean or syntonic).
Our modern 12-note octave division into 12 equal semitones is an artificial compromise - a "temperament" - enabling every key to be "near enough" in tune, although none of the frequency ratios are pure. (5ths and 4ths are closest, at 2 cents out.)

When it comes to chords, the major triad - whose pure ratios are 4:5:6 - corresponds to octaves of the lower harmonics of the root.
IOW, the root of the chord has an acoustic property. The chord sounds consonant, probably (or at least in part) because all its pitches "belong" to one single lower fundamental. Eg, a pure A major triad can be formed with frequencies of 440 (A), 550 (C#) and 660 (E), all of which are harmonics of 110, a lower A note.
These are not quite in tune with the equal tempered frequencies we use (C# = 554, E = 658), but arguably "close enough".

But a crucial point is that music theory (and our whole musical culture) is not interested in this acoustic science. The argument about WHY certain sounds we use "sound good" is largely irrelevant. Much of the reason, in any case, is simply familiarity (over centuries).
Moreover - as steve hints - many musical cultures favour sounds we'd regard as "dissonant" or even "discordant" (ie unpleasantly dissonant - seeing as we actually enjoy certain kinds of dissonance). Do they enjoy dissonance more than we do? Or do they hear (tune into) different aspects of musical sound to us?

Looking at a music like Blues is salutary. Blues singers and improvisers habitually aim for notes that are not part of our diatonic system, and don't align with any simple frequency ratio. (Some theorists point to "7-limit" tuning, invoking ratios using a factor of 7, but these don't convince me; blues is just too variable to pin down to any system like that. In blues I've analysed, I've not seen convincing evidence of 7-limit pitching.)

This site has a pretty good survey of the history of scale development and the math behind it (such as it is)- as well as the story of modes:
[www.midicode.com]



Edited 3 time(s). Last edit at 07/21/2013 04:43AM by JonR. (view changes)
Re: why is the major scale constructed like it is?
July 21, 2013 05:41AM
This is something I wrote some time ago for another forum. It's not strictly relevant to the title of this thread, but it might provide some interesting background material.
Obviously it's not intended to be exhaustive

The Ecclesiastical Modes

Although the origins of the modal system go back to the ancient Greeks, the concept of mode as a function of scale and final originated in the 10th and 11th centuries, when an attempt was made to organise the ("Gregorian") chants of the Roman church according to the categories of ancient Greek music theory. This classification and categorisation was done, in part, to aid the memorisation of the melodies.

Specifically, antiphons (short syllabic chants) were compared with psalm tones (fixed melodies to which Psalm verses were sung) to see how the interval was filled in between their ending note (the finalis or "final", similar to what we now call the tonic) and the pitch corresponding to the psalm tone's reciting tone (the "tuba" or "tenor", similar to what we now call the dominant), which was normally a fifth above.

There are four ways a fifth can be filled in using the diatonic pitch set and its arrangement of tones (T) and semitones (S):

1) TSTT
e.g. the white notes descending from A to D

2) STTT
e.g. the white notes descending from B to E

3) TTTS
e.g. the white notes descending from C to F

4) TTST
e.g. the white notes descending from D to G

The ending notes, D,E,F,G were called "the four finals" and each was named according to their Greek ordinal numbers; protus, deuterus, tritus and tetrardus respectively. (It must be remembered however that the notes are only an abstract convenience and do not actually refer to literal pitches).

The range of the chants was also considered.
Those with the final at the bottom of their range (usually extending to an octave above it) were said to be "authentic", while those that extended lower than their finals so that the final occurred in the middle of their range (usually from a fourth below to a fifth above), were called "plagal".

Thus, each of the four finals governed two modes:
1. protus authenticus TSTTTST
2. protus plagalis TSTTSTT
3. deuterus authenticus STTTSTT
4. deuterus plagalis STTSTTT
5. tritus authenticus TTTSTTS
6. tritus plagalis TTSTTTS
7. tetrardus authenticus TTSTTST
8. tetrardus plagalis TSTTTST (NB: This has the same order of intervals as protus authenticus, but they have different finals).

Originally, these 8 modes were reckoned as 4 pairs (there is a fable that St. Ambrose made the authentic modes in the 4th century and St. Gregory made the plagal ones in the 6th century).

With authentic modes, the tuba/tenor lies a fifth above the final. However, where the tuba/tenor would fall on B, it was later changed to C.
With plagal modes, the tuba/tenor lies a third below that of its authentic counterpart.

Later theorists assigned the modes different names adopted from late Greek sources (although the Greek usage was different and the nomenclature was technically incorrect):

                   Range  Dominant  Final
1. Dorian           D-D       A       D
2. Hypodorian       A-A       F       D
3. Phrygian         E-E       C       E
4. Hypophrygian     B-B       A       E
5. Lydian           F-F       C       F
6. Hypolydian       C-C       A       F
7. Mixolydian       G-G       D       G
8. Hypomixolydian   D-D       C       G
The Greek prefix "hypo" is roughly synonymous with the word "plagal".

After this system had been perfected, it began to serve not only as a description of existing music, but as a prescriptive guide to new compositions. (Though modal theory was not extended to the analysis of polyphonic music until the late 15th century where the tenor line was usually used as the primary reference point).

Centuries later (around 1547), a humanist called Glareanus recognised 4 additional modes which came to be knows as Ionian and Aeolian, and their plagal forms. The Ionian mode has its final on C, and the Aeolian on A (the Locrian and Hypolocrian modes, with B as a final, barely existed).

Ionian and Aeolian modes were not necessary however and existed in practice long before they were given specific names. Singers often used a Bb to avoid the augmented fourth from F to B, even though this wasn't always specified in the notation. Since the 11th century, the use of the Lydian mode with a Bb provided the "Ionian" mode (which corresponds to what we now call "major"), and the Dorian mode with Bb provided the "Aeolian" mode (which corresponds to what we now call "minor").

With the rise of harmony, a leading-note became a necessity, and the "Ionian Mode" effectively became one of the favourite modes. - Both this and the "Aeolian mode" were more suitable for harmony.

In addition, more notes began to be chromatically altered. In the Mixolydian mode for example, the 7th was often sharpened to provide a leading-note (thus making it identical to the Ionian Mode). The ancient modes gradually disappeared until only the "major" and "minor" modes remained.

Between around 1450 and 1650, there was a gradual change from "modal" to "tonal" thinking, the latter based on triadic harmony and the diatonic circle of fifths.

Check out this textbook on Four Part Harmony.
Dan McCaughern
Re: why is the major scale constructed like it is?
July 23, 2013 02:40AM
It may have happened like this;
Someone blew into a wind instrument or bonked a piece of metal (sword maybe) and realised that there was another note present which was quite loud ( 5th).
"Hello!" Said that person with a quizzical look on his/her face. "I wonder what would happen if I made an instrument tuned to that other note" He/she did and low and behold this gave the desired pitch- plus the overtone of this, which, of course would have been a 5th above this.
He/she did this again twice and the pentatonic scale was born. Doing it again twice more would produce the Mixolydian mode. Melodies using would tend to resolve on to the Ionian mode.( major scale) This is how it happened......maybe.
Re: why is the major scale constructed like it is?
July 23, 2013 06:58AM
Dan McCaughern Wrote:
-------------------------------------------------------

> "Hello!" Said that person with a quizzical look on
> his/her face. "I wonder what would happen if I
> made an instrument tuned to that other note"

But he or she wouldn't have said "I wonder what would happen if I made an instrument tuned to that note". He or she would have said, "I wonder if any currently existing instruments already have that note, and if so, where can I pick one up cheaply?" ;-)

The first musical instrument was the human voice and it already had every conceivable pitch (within vocal range) at its disposal. People didn't have to wait for this guy and his discovery of overtones in order to make music using pitches that sounded good in relation to each other. If some of their preferred pitch relationships coincided with lower members of the as-yet-undiscovered harmonic series, it's probably just because simple pitch relationships are easier to recognise, remember and produce to order than complex ones - rather than them being unconsciously influenced by the vanishingly faint overtones produced by their vocal cords.

Overtones aren't necessary in pitch relationships. If two pure audio sine waves (i.e., NO overtones present) a minor 3rd apart are sounded, we still hear a minor 3rd - the interval still has all the qualities of a minor 3rd. Tone-wise it sounds like sh**, of course, as it's just boring sine waves without overtones, but pitch-wise or pitch relationship-wise, it's unaffected by the lack of overtones.



Edited 2 time(s). Last edit at 07/23/2013 07:28AM by Fretsource. (view changes)
Re: why is the major scale constructed like it is?
July 23, 2013 08:00AM
It's worth pointing out (again) that the overtone series was not discovered and described until the 18th century.

The idea that our major scale (or even our diatonic pitch set) was generated by successive fifths away from a central tone is an after-the-fact rationalisation. That's not the way it happened historically.

Check out this textbook on Four Part Harmony.
Re: why is the major scale constructed like it is?
July 23, 2013 08:33AM
JumpingJackFlash Wrote:
-------------------------------------------------------
> It's worth pointing out (again) that the overtone
> series was not discovered and described until the
> 18th century.
>
> The idea that our major scale (or even our
> diatonic pitch set) was generated by successive
> fifths away from a central tone is an
> after-the-fact rationalisation. That's not the way
> it happened historically.

Well, the original division of the octave using 4ths and 5ths dates back at least to Ancient Greece and their tetrachords. Unless this is all lies:
[www.midicode.com]

Pythagoras didn't need to know about frequency or overtones to observe that pitches in ratios of 2:1 and 3:2 sounded consonant, blended smoothly. He just had to (a) observe that they did, and (b) decide that that observation was important, had meaning. (For him, the sweet blending of simple ratios proved God was a mathematician.)
It's therefore not a "rationalisation" to attribute the effect to the overtone series; it's an explanation. (Pythagoras's concept of God and the harmony of the spheres was a "rationalisation" of his observations.)

Perhaps not a complete explanation, however, because - as Fretsource says - pitches with no overtones will still blend when in simple ratios.
My own explanation (rationalisation? hypothesis?) for that is that we intuit pitches in simple ratios as being overtones of a single lower pitch.
Harmonic intervals don't exist in nature, any more than sine waves do; but single pitches (with their overtones) often do. So if we hear two pitches of (say) 550 and 660 - whether generated by a musical instrument or as sine waves - we assume (subconsciously) they are overtones of a single natural pitch of 110. That's the reason for the sensation of "blending" of those two pitches, the sense that they "belong together".
The lower the greatest common divisor (GCD) of two pitches - the "acoustic root" of the interval - the more likely that frequency is to be beneath our threshold of pitch perception (around 20 cps). This seems to me to align with the lesser consonance (increasing dissonance) of pitches in more complex ratios, where the GCD will be a smaller figure. If the pitch that two frequencies are overtones of is too low for us to hear, we won't recognise those two frequencies as belonging to any natural sound; that would, in theory, result in a perception of dissonance (lack of relation).



Edited 3 time(s). Last edit at 07/23/2013 08:54AM by JonR. (view changes)
Re: why is the major scale constructed like it is?
July 23, 2013 09:31AM
JonR Wrote:

> Well, the original division of the octave using
> 4ths and 5ths dates back at least to Ancient
> Greece and their tetrachords. Unless this is all
> lies:
> [www.midicode.com]


No, what I meant was when some people assume that our diatonic pitch set arose because someone started stacking fifths together.

One theory that is commonly banded around is that you start on any note, add three fifths above it and three fifths below; so for example starting on D, going up you get A, E and B, and going down you get G, C and F. Put them all together in order and you get DEFGABC.

But this is almost certainly not what happened. The diatonic pitch set has been in use as far back into recorded history as it is possible to go, long before the ancient Greeks came along.

And the stuff about Pythagorous was legend even to the Greeks. None of his writings have survived, and all the stories about the anvils and so on are almost certainly apocryphal. The first recorded mention of it didn't occur until several years after his death.

Check out this textbook on Four Part Harmony.
Re: why is the major scale constructed like it is?
July 23, 2013 11:31AM
JumpingJackFlash Wrote:
-------------------------------------------------------
> JonR Wrote:
>
> > Well, the original division of the octave using
> > 4ths and 5ths dates back at least to Ancient
> > Greece and their tetrachords. Unless this is
> all
> > lies:
> > [www.midicode.com]
>
>
> No, what I meant was when some people assume that
> our diatonic pitch set arose because someone
> started stacking fifths together.
>
> One theory that is commonly banded around is that
> you start on any note, add three fifths above it
> and three fifths below; so for example starting on
> D, going up you get A, E and B, and going down you
> get G, C and F. Put them all together in order and
> you get DEFGABC.

Yes, I've heard that. Do you know where that "theory" started from?

This suggests something very similar dates from the Renaissance:
[www.midicode.com]
That site links the 5ths idea with Pythagoras, but I guess you're saying that's mistaken? That it isn't as old as that?

> But this is almost certainly not what happened.
> The diatonic pitch set has been in use as far back
> into recorded history as it is possible to go,
> long before the ancient Greeks came along.

Do you have a reference for that? As I understand it, the ancient Greek system differed from our diatonic pitch set, in that while it roughly aligned with our 7-note system, they sometimes made use of quarter-tones, not just tones and semitones. (I've forgotten where I read that.)

> And the stuff about Pythagorous was legend even to
> the Greeks. None of his writings have survived,
> and all the stories about the anvils and so on are
> almost certainly apocryphal. The first recorded
> mention of it didn't occur until several years
> after his death.

Yes, I'm aware of that.
Re: why is the major scale constructed like it is?
July 23, 2013 11:52AM
JonR Wrote:
>
> This suggests something very similar dates from
> the Renaissance:
> [www.midicode.com]
> That site links the 5ths idea with Pythagoras, but
> I guess you're saying that's mistaken? That it
> isn't as old as that?


I wouldn't necessarily trust that site.
It talks about the monochord as though it was something that Pythagorous had access to. He didn't. The monochord was first theorised in a treatise completed in 391 (AD) by Aurelius Augustinus (St. Augustine).


>>As I understand
> it, the ancient Greek system differed from our
> diatonic pitch set, in that while it roughly
> aligned with our 7-note system, they sometimes
> made use of quarter-tones, not just tones and
> semitones. (I've forgotten where I read that.)

It's true, Greek theorists describe an "enharmonic" genus which could involve quarter tones, but I'm not sure this was used much in practice.

Check out this textbook on Four Part Harmony.
Re: why is the major scale constructed like it is?
July 23, 2013 05:46PM
Dan McCaughern Wrote:
-------------------------------------------------------
> It may have happened like this;
> Someone blew into a wind instrument or bonked a
> piece of metal (sword maybe) and realised that
> there was another note present which was quite
> loud ( 5th).

The problem with this is that things like a sword weren't necessarily designed to produce harmonic overtones. They may have indeed produced overtones which may have been inharmonic or harmonic. A great example is the casting of bells (like church bells). Over the years, they have sought better ways to cast the bells to make a "pure" tone - that is, one with harmonic overtones. This tells us that, in the past, they had anything but (and if they did, it would have been purely by accident, and non-reproducable at that time).

The Harmonic Series in Acoustics happens the way it happens becuase we're dealing with PHYSICAL media - metal bars, columns of air, strings, etc. I think this is the reason why people have a hard time separating the two. But for a physical thing to vibrate with harmonic overtones, it would have to be designed that way to begin with. And in the real world, often the practicality of making something strong, durable, artistic, etc. overrides the need for making it vibrate with harmonic partials.


> "Hello!" Said that person with a quizzical look on
> his/her face. "I wonder what would happen if I
> made an instrument tuned to that other note"
> He/she did and low and behold this gave the
> desired pitch- plus the overtone of this, which,
> of course would have been a 5th above this.

Unfortunately, there are no historical examples that bear this out. Most instruments that play a 2nd pitch play a tone a semi-tone or tone (approximately) higher than the "tuned" note. And instruments with the ability to play diatonic scales (or other NON HARMONIC SERIES BASED scales in other cultures) are not pentatonic - they're tones and semitones.

>maybe.

Or maybe not. Evidence points to a non-overtone series-based evolution of scales.

The development of harmony seems to be "consonance-based" rather than "overtone-based" but there are of course natural parallels between the two.

At any rate, if harmony was at all at least initially harmonic series-based, it comparatively quickly moved away from overtones and simply to artistic choice.

Steve
PaulG
Re: why is the major scale constructed like it is?
July 29, 2013 07:25PM
This discussion goes back and forth in a way reminiscent of the difference between science and engineering, with SteveI as the engineer and JonR as the scientist - they are both right - in their context. Science discovers laws by observation. Engineers, due to physical and practical limitations, get "close enough". I believe Pythagoras' discovery and projection of the fifths gave rise to the scales and modes as they evolved, from a practical point of view of trying to simplify and codify. Attempts to build machines that reproduced those scales in every key became unwieldily, so the engineer in Bach came to the rescue with the well-tempered scale, which was close enough. And, though we became accustomed to it, this harmonic scale, and its 7-note modes, is not as pleasing as it could be, because of the compromises.

Now the 5-limit of ratios (also referred to as Ptolemaic) made additional pleasing ratios. The 7-limit makes chords that are even richer. While 7-limit is not practical for fixed-scale instruments, they are quite possible for non-fretted instruments and slide instruments and bendable instruments.

And then there are artists. The "blues" note, a flatted minor 7th (>30 cents flat), is also the basis for barbershop harmony, which I sing. The major-minor 7th chord is the most stimulating sound, with its dissonant tritone (3-m7) accompanying a consonant major 5th (3:2). (The third being the flat 5:4 ratio, too, so the tritone is 7:5 = 1.4 instead of 1.41414....). The leading tone JonR mentioned above is the third of the VMm7, which resolves to the I chord, of which VII is the 3 of the previous chord. As JonR pointed out, the 3 of the major triad, to some ears, sounds sharp when played on a well-tempered instrument. And, finally, the major triad produces the most overtones (above the sung notes) that match and "ring", especially with a bright vowel. Mm7 chords add that interesting sound and its leading tone tension. 6ths and 9ths, with their 2nd intervals, add more pleasing dissonance for interest, too.

So how much of this discussion is about scales and how much is about chords? The original question was about "most pleasing" scale. It may be the most pleasing of fixed scales (to western ears), but not necessarily providing the most pleasing chord. While we notate our barbershop music on a standard grand staff (scale), we sing chords and the "ear" adjusts to harmonics. In the next chord, we may need to adjust our pitch to fit the new harmonics. We eschew fixed scale instruments, because they are "out of tune", so we sing a cappella to match the harmonics.

Context IS important! Is it science, engineering or artistry?

Paul
Re: why is the major scale constructed like it is?
July 30, 2013 02:13AM
PaulG Wrote:
-------------------------------------------------------

> And then there are artists. The "blues" note, a
> flatted minor 7th (>30 cents flat), is also the
> basis for barbershop harmony, which I sing. The
> major-minor 7th chord is the most stimulating
> sound, with its dissonant tritone (3-m7)
> accompanying a consonant major 5th (3:2).

"Perfect" 5th, please! ;-)

> So how much of this discussion is about scales and
> how much is about chords?

Good question...

> The original question
> was about "most pleasing" scale. It may be the
> most pleasing of fixed scales (to western ears),
> but not necessarily providing the most pleasing
> chord.

Well, IMO, that depends on who's listening.
Personally I've never had any issue with the equal tempered scale - or chords harmonised from it; it all sounds fine to me.
Whenever I've heard music played in just intonation, either I hear no difference, or I think the JI version sounds - very slightly - more "out of tune". (Of course it isn't, so I guess I just mean "less pleasing" ;).)

I'd say there are two incarnations, as it were, of what you call the "VMm7" chord (dominant 7th). There's the "pure" one you're talking about - ratios 4:5:6:7 - and there's the equal tempered one, where the interval between 5th and 7th (a minor 3rd) is closer to 5:6 than 6:7. I.e., in pure ratio terms, the chord is more like 20:25:30:36 - ie a lot less consonant, in theory anyway. (And of course in equal temperament, the 7th is even sharper than that.)
In tonal music, the whole point of the dom7 is tension. It isn't supposed to be the pure barbershop entity.

IOW, there's nothing wrong with the ET V7 chord. It does its job fine. Tuning the 7th flat to make a pure 7:4 with the root is beside the point. (Unless you're singing barbershop, of course...;-))

> While we notate our barbershop music on a
> standard grand staff (scale), we sing chords and
> the "ear" adjusts to harmonics. In the next chord,
> we may need to adjust our pitch to fit the new
> harmonics. We eschew fixed scale instruments,
> because they are "out of tune", so we sing a
> cappella to match the harmonics.

Right - that's the luxury of unaccompanied vocals.
Of course, you're not consciously "matching harmonics" - you're finding pitches that blend pleasingly. Do you know of scientific research that has confirmed exactly what frequencies barbershop quartets actually produce?

> Context IS important! Is it science, engineering
> or artistry?

Well, as you're saying, it's all three depending on your point of view. ;-)

I'm much more on what you call steve's "engineering" side than it might appear. The science interests me, but only out of idle curiosity. Connections obviously exist between music and acoustic physics, but they're far more complicated than a simple reduction to harmonics and overtones. And they don't explain music anyway.
Music is - obviously - an artform, a cultural construct - "engineered" according to cultural requirements and habits. Acoustic science can't explain music any more than the physics of light can explain painting.
Re: why is the major scale constructed like it is?
August 03, 2013 07:52AM
JonR Wrote:
Acoustic science can't
> explain music any more than the physics of light
> can explain painting.

I've always wanted to join an artists' forum to see if they have discussions about how the primary and secondary colors evolved from the rainbow, or how painters use an exact shade of purple because the exact ratios of the light waves of red and blue combine to make a "pure" color. Is there an "in tune" purple that matches the exact center frequency of "violet" on the rainbow, and are all others "out of tune". Did painters prefer a whiter shade of pale through the Renaissance but gravitate towards darker shades of pale in the Classical period?

Steve
Re: why is the major scale constructed like it is?
August 03, 2013 11:10AM
Nice analogy, but not quite exact.
After all, a slight difference in audio frequencies (in harmonies) makes a much more noticeable difference to the sound than a slight difference in light frequencies. The difference between being "in tune" and "out of tune" - according to any cultural criteria you care to name - is a matter of tiny differences (in Hz or cents). A very slight deviation from consonance produces a strong dissonance, which reduces as the pitches move further apart - especially with two pitches that start out as unison. (This is regardless of what value a particular culture might attach to such experiences.)
When two colours go together, one of them can change very slightly without suddently becoming "dissonant" with the other. With two shades of the same colour, the difference becomes gradually more noticeable as they get further apart - no sense of sudden dissonance with a minimal change.
The main difference in experience, of course, is that two pitches heard together become one sound. Two colours seen together don't become one.
OTOH, when you mix colours (in paint or in light), they become a different single colour, in which the originals are no longer visible. Two mixed pitches produce a single sound, but the individual pitches are still discernible.

If our perception of colour really did behave the same way as sound, you can bet artists (some of them anyway) would discuss light frequencies as obsessively as some musicians discuss audio frequencies. Abstract painters would learn about colour theory and argue about whether to apply it as in the books or just follow their eyes. ;-)
As it is, there are theories of colour dissonance, of complementary and opposite colours, as displayed on a colour wheel (which some have tried - in vain - to compare to the circle of 5ths). Artists certainly do learn about that sort of thing, although the education - in my experience - stops short of electromagnetic and optical science, just as music theory stops short of acoustic science. (I did an art degree, btw.)

BTW, I did once know a harmonica player who had a degree in optical science - and went to work in art restoration, where such things do matter... ;-). In a sense, I guess, that's comparable to sound engineers, who work with EQ and understand frequency much more scientifically than the average musician.

Of course, when it comes to more direct art/music links, we can rule out those synaesthetes who link rainbow colours with specific pitches or keys (because such associations are entirely subjective).
Ty
Re: why is the major scale constructed like it is?
August 17, 2013 01:35AM
I think SteveL and others are arguing at cross-purposes here.

To the question, "why is the major scale constructed like it is?" Steve has offered a perfectly sensible historical argument, which essentially says the major scale is a matter of fashion. To him, the major scale is not much different than a tweed jacket, it's just a matter of style.

But I believe this begs the question. The heart of the matter is why such a preference is running 400 years and counting?

Steve's ostentation doesn't answer that question any better than the arguments from acoustics do.

Looking for an acoustical, objective basis for aesthetic preferences in music seems to me a very honorable and intelligent thing to do. Folks are looking for the frame that undergirds what we like in music. The eminent theorist, Heinrich Schenker, has ideas in stark contradistinction from SteveL, and for those looking for powerful arguments in support of an acoustical basis for the predominance of the major scale, I highly recommend reading his text, Harmony.
Ty
Re: why is the major scale constructed like it is?
August 17, 2013 01:48AM
stevel Wrote:
-------------------------------------------------------
> JonR Wrote:
> Acoustic science can't
> > explain music any more than the physics of
> light
> > can explain painting.
>
> I've always wanted to join an artists' forum to
> see if they have discussions about how the primary
> and secondary colors evolved from the rainbow, or
> how painters use an exact shade of purple because
> the exact ratios of the light waves of red and
> blue combine to make a "pure" color. Is there an
> "in tune" purple that matches the exact center
> frequency of "violet" on the rainbow, and are all
> others "out of tune". Did painters prefer a whiter
> shade of pale through the Renaissance but
> gravitate towards darker shades of pale in the
> Classical period?
>
> Steve

To both Steve and JonR, it's not about "explaining," it's about understanding the relationship between nature and art. To what extent are human creations artificial, to what extent are they in accord with nature? How does this dynamic relate to aesthetic beauty, style, pleastantness, fashion, etc.?

DaVinci is a great example of someone who used science to broaden and deepen his craft. He is precisely the type of man who might study acoustics and its relationship to music making were he comparably musical or alive today.
Ty
Re: why is the major scale constructed like it is?
August 17, 2013 01:50AM
Steve, an honest question for you:

Is the prohibition of consecutive 5ths and 8ves in the 4-part harmonic texture of CPE music simply a matter of style, or is there an acoustical reason for it? Or both?



Edited 1 time(s). Last edit at 08/17/2013 01:52AM by Ty. (view changes)
Re: why is the major scale constructed like it is?
August 17, 2013 03:15PM
Both, but primarily style.

To expand:

The purpose of counterpoint (from which that style of writing evolved) was to maintain the independence of the voices. When you use parallel intervals of any type for too long, one voice becomes dependent on the other - it gets "subsumed" into it. This is especially true of 5ths, and even more so for Octaves (and obviously, unisons!).

In essence, nothing destroys independence of the voices like parallel 8ves or 5ths.

Even in multi-voice textures, these things tend to "stand out" (not to our modern corrupted ears, but to them) especially in combination with the upper voice, or in the outer voices, etc.

I also believe that there's an element of "anti-old-fahioned-ism" going on: parallelism was THE primary element of parallel organum and is seen in stuff like Faux Bourdon, etc. What that tells me is, that at some point, some young brats didn't want to compose like their old man, so they avoided anything that sounded like that - just like a young punk rocker today would not (necessarily) choose to play a Chuck Berry style riff - it sounds "too 50s". Now, we're talking centuries here, but the idea is a reasonable one.

But - if this is where you're going - the acoustics have never changed. But our tastes do. Changes like, a preference for parallelism, changing to a preference for anti-parallelism, then back to parallelism (middle ages, to CPP, to modern) is simply a matter of style. The acoustic properties aren't changing. So stylistic conventions often happen *in spite of* acoustic principles. That's why it's art. That doesn't mean that sometimes the art agrees with the science or is inspired by it, but it means that as far as composers writing music (at least until the modern era when people got all enchanted by this stuff) is concerned, the science aspect is far removed from their thought process.

Steve
Re: why is the major scale constructed like it is?
August 18, 2013 08:40PM
Ty Wrote:

>
> DaVinci is a great example of someone who used
> science to broaden and deepen his craft. He is
> precisely the type of man who might study
> acoustics and its relationship to music making
> were he comparably musical or alive today.

However, many people find DaVinci's work to be too "academic" - in much the same way as people see Bach's music. Although it's a Romantic ideal, people tend - still - to favor more "emotion" in their art. La Giaconda's appeal seems to be less about it's *supposed* architectural positioning and more about the "enigmatic smile" which has nothing to do (that I've ever heard) with any scientific or physical elements - it's purely the emotional aspect of that smile which "connects" with people.

There is plenty of music out there written *very specifically* using acoustical principles in a far more obvious manner than other styles of music. It just does not connect with people in any significant way that would lead one to believe that what little intentional acoustic elements included in more popular music have any impact on a listener - it's ultimately leaning towards the "art" side.

Steve
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