Origin of scale

[BuyBuy]

So does anyone really know how our scale came to exist ?

I mean, when was it decided that an A sounds the way it does (44o Hz), that the scale was divided in 12 tones, and so forth ?

Were the modes used in Middle Ages music using the pitches that we use today ?

Anyone with insight on the subject ? Or maybe with a reference to a good web site about it ?

[Daevren]

A being 440 is irrelevant. A could be anything.

Modes were used as major and minor are used today at some point in history.

I am not a music historian. But its part physics. We are talking about sound waves. And there are always overtones in a sound. These overtones tell us what is constant and what is dissonant. The first overtone is the octave. So naturally an octave is the most constant interval, ignoring an unison. This makes alot of sense. The air waves are in sync with each other.

Then we have the tast of dividing the octave, how do we do that? It is interesting that several musical cultures descided to do it the same way independently.

The second overtone is the fifth, if you take the 440 note, which you call A, then the first overtone is the 880 freqeuncy. The second one is a freqeuncy 3/2 times higher. This is the fifth. This is 1320 Hz, which is an overtone of 660. So it makes sense that those notes are constant.

You can continue this, you get a major third, then an octave and a fifth again, I think, and then it gets a bit messy. You get something that approaches the Major 7th. Of course a major seventh isn't a nice constant.

You can also look down. 440 is the fifth of 293 Hz (D), or its octave 586,67

So we have the main note A(440) we have the octave A2(880), we have a fifth E (660), we have a major third(the actual overtone is closer to a major third than a minor third), 495(C#) we have a major seventh, 825(G#) and the fourth, because the A is an overtone of the fourth, 586,67(D).

So

A C# D E G# A

This is the pentatonic major scale. Invented by Pythagoras(or at least, he is credited for it, maybe one of his students). Also note that this scale is created by stacked fifths, the 3/2 interval, the second overtone. Its either done that way, or the way I did it, by measuring the overtones.

This scale was not only invented by Pythagoras, it is also used in Chinese music, in African music, in Indonesia, in Scottish and English folk music, all over the place.

The pentatonic scale is an universal truth, just like the Pythagorean triangle theorem. It was already there, a byproduct of the laws of the universe.

If you fiddle around with the pentatonic scale you will notice it is a very simple and melodic scale. You can end on any one and make it sound ok.

Now, for some reason the number 7 is a 'magical' number. I am not sure why, but for some reason it is best to divide the octave into 7 parts(so eight notes if you count the octave itself too, hence its called 'octave'). We added two notes, a major second inbetween the unison and the major third, where there was a gap, and in the other gap, inbetween the fifth and the major seventh, we added a major sixth.

I am not sure how they actually got those notes. But now we have the major scale. And to any human, using 7 notes plus the octave just sounds the most perfect and most natural.

But western music has a problem. Lets take the first note, 100 Hz.

So then the natural overtones are 200, 300, 400, 500. But if we are going to stack fifths then we get 100, 150, 225, 337,5 and then 506,25. Thats 6,25 Hz too much. Its 81/80 too much. This means you can only tune an instrument and get the perfect overtones build on only one frequency.

To tune a piano or any kind klavier/keyboard in a way all keys sound equal in tune they tuned the fifths slightly lower than dictated by the universe, and the thirds slightly bigger. This way you can make the piano sound in tune for any key. Otherwise you have to tune it to one key, lets say A major. This is where things like Well Tempered and Equal Tempered come from. This is why Bach composed a prelude and a fugue in every key, because which his new keyboard it was possible to sound in tune in every key.

There is another problem.

If you start stacking fifths on Gb then after 12 fifths you get a F#. That note is (3/2)^12 times higher. But if you start stacking octaves on Gb then after 7 of them you will get a Gb that is 2^7 times higher. Now (3/2)^12 is 129.75 and 7^2 is 128

This problem is alot more subtle and was solved in the same way.

The system we use in the west today does not use the 3/2 fractions we got from the overtones. No, we now divide the octave into 12 equal parts. So the twelfth root of two, which is 1,05946 and alot more numbers. Its kind of like a musica Pi.

With this number, we can make every key sound kind of in tune with the natural overtones. This number created the chromatic scale. Now, then you can pick any combinations of intervals and create new scales.

Look at this site for more info: https://www.phy.mtu.edu/~suits/scales.html

What did they do in other cultures?

Not every musical cultures uses chords and harmony like we do. Most of them don't have chords or keys.

Lets take south indian carnatic music, which I know some stuff about. They still use what is called the just scale. Their fifth is 3/2 and not 7 times the twelfth root of 2. The difference is 3/2 = 1.5000 and 1.49831.

Carnatic music is based on singing. There is no standard for notes like A = 440 and concert C is 273. A singer has a range and he needs to pick a base pitch which he can comfortably sing an octave and a half above and half an octave lower.

They don't have keys, there is only one key, and what the actual pitch is is different for every person. The instruments tune up with the singer, which is usually the solist.

So they have no use for a tempered scale. All their scales are just scales. Also, all their scales have a root and a perfect fifth. Diminished fifths and augmented fifths aren't used. Now they have a mathematical structure to fill in all the other notes. Like I said before, they also prefer 7 note scales, probably because of the way our brains work.

They have 72 seven note scales, they call them melakarta ragams. All have a perfect fifth on the base note. Thirthy-six of them have a perfect fourth, and the other 36 have augmented fourts.

So they have scales that mach ours. What they call  Sankarabharanam( or Dheera-Sankarabharanam) is the major scale. As you see, our name is a bit simpler. The Sankarabharanam ragam is a popular one.

Another aspect to their music style is the use of microtonal ornamentations. Because their music is more melodic and doesn't have a strict system of the notes to use they use alot of sounds inbetween their notes. This is very normal and very awkward to the average western ear. It is actually very colourful and melodic.

The same is done in the middle east and in a whole lot of other musical cultures. Even the blue note in the blues has this. This is why blues music is better played on guitar or sung instead of played on piano. On a guitar you can bend notes 'out of tune' to hit the awkward blue notes.

As for modes and major and minor. People started to change the modes at places so they could get better and more perfect cadences. So the modes kind of evolved into two subforms. Major and minor.

The reason that A is called 440 is because A had to be some note. It would have made no difference to have called it 441 or 430 or 457.

Well, this is alot of info. If I was incorrect or incomplete, please correct me.

[Daniel_piano]

This is what the musicologist Bob Fink has to say on this subject:
----------------------------------------------------------------------------------
 
    The process that historically forces the do, re, mi scale into existence is not hard to grasp.
    When a note is played, on a flute or by any instrument, the note is determined by how many vibrations per second it makes. Concert "A" for the orchestra is 44o vibes per second, for example.
    If you play or sing the octave to any note (the "same note, only higher"), then it has twice the vibes. The ratio between the notes, then, is 2 to 1 (or 2:1).
    Historically, the ear has preferred simple ratios as harmonious, and complex ratios have been avoided or considered noisy or dissonant (to be used only as an artistic contrast to harmoniousness -- as a sort of 'dueling tonalities').
    For example, two notes on the piano right next to each other have a complex ratio (play them together to hear this) and these kind of ratios cause "beats" -- a kind of repetitive "wow-wow-" effect, which is physically measured as unpleasant to the ear.
    You also need to know that whenever a single note, like "A," is played, we actually hear several notes at once, called overtones. They're very faint, but the different strengths of the mixture is mainly what tells us we are hearing a trumpet instead of a piano or a voice -- but all sounding what seems like the same single 'A' note.
    Now here are the grabbers:

    * The overtones of any one note all add up to its major chord, when played out loud rather than as overtones.
    * The most audible overtones of a tonic or keynote all have simple ratios, like 2:1 (octave), or 2:3 (fifth note of scale), and the 4th note of the scale, whose first different overtone is the given tonic, has a ratio of 3:4 . In fact these three notes are present in virtually every musical scale known on earth.
    * If you write out the overtones of these three notes and string out the three most audible ones of each within the span of an octave, you will get the major scale:
   
 Tonic C: Overtones are: C, G, E, and Bb (I've left out the additional octave overtones as redundant and too high and weak to be noticed within the framework of average human hearing.)
    Fifth G: Overtones are G, D, B, and F
    Fourth F: Overtones are F, C, A, and Eb.

    * If you substitute the three weakest ones (the 3rd, 6th and 7th notes of the scale) with another three notes (which includes the even weaker next overtones), and which are flatter, you get the minor scale. (The 6th note is strongest of the three because it forms no complex ratios with adjacent notes in the scale.).

    * If you leave these two -- the 3rd and 7th notes -- out altogether, you get what's called the 'Chinese scale' -- or the piano 'black notes' pentatonic 5-note scale -- found also in Africa, old Scottish and Irish folk music, and elsewhere.

    * Because these overtones are very weak, they were the last to come into the scale, and how to tune them was a matter of historic uncertainty -- and many people tuned them somewhere between minor and major (in the 'cracks' on the piano), producing what are known as 'blue' or neutral notes.

    * When you further consider the advent of harmony (in which there has been use of only the three chords of the tonic, dominant(5th) and subdominant (4th)) to harmonize all the 7 scale-notes in most of the folk melodies known, this further underscores that these three notes and their overtones were fundamental influences in the formation of the scale's notes. Even the names that evolved for them are perfect representations of their acoustic role, even though the names ('dominant' 'sub-dominant' & keynote/tonic) were also coined by people without acoustical knowledge.
    Now either all this is the greatest coincidence on earth -- that is, people who knew nothing of acoustics coming up with scales reflecting all these acoustic properties purely by chance -- or else, in fact, the ear was already able to discern the sounds as distinct between harmonious or dissonant because the ear could hear these acoustic properties without consciously knowing they existed. I have chosen to believe the latter, and in 1970 wrote The Origin of Music in order to demonstrate this idea more fully. Since that time there is confirming evidence in the 'babies experiment' (Trehub & Co.); the Kilmer et al, oldest song being in harmony and using the diatonic; and the oldest known instrument (Neanderthal flute).

    The glaring question in all this is the apparent absence of the full diatonic scale in so much of Asia and elsewhere. While the diatonic is found in much of the world, including Africa (Tracey; see also Nettl) and the near East, the Pentatonic scale predominates as well, or even exclusively, in a huge part of the non-European world (along with many other non-pentatonic nor diatonic scales). People raised on the music built by scales like these become used to them and the scales are entwined in the cultural matrix of the culture for many generations.
    In early music of Scotland, Ireland and the Orient one can often find the missing 3rd and 7th notes of the scale being used not as part of the official scale, but as passing notes or leading tones. That is, they are notes in the gaps that 'lead' 'to the fourth or 'pass over into' the octave. In different cultures the names for this are different, but have similar meaning. The Pien tones in Chinese pentatonic scales mean 'becoming' that is, a 7th 'becoming' the octave, in a sequence of melody or scale notes. The words are different, the concept and usage is similar. This is widely reported among musicologists and anthropologists.
    But in many places, not even these leading tones can be found. Sometimes they are used but 'banned' by tradition or religious authority. This dichotomy -- between official or religious systems of music and the actual practice among the common or pagan component of the population (who far less often could keep records or had notation as often did the heirarchical keepers of musical systems) -- is a dual history that has been recognized by numerous writers, exisiting in various ways in Europe as well as in the Orient and the Near East. [See Carl Engel, Music of the Most Ancient Nations, pp. 151-3; Curt Sachs, Rise of Music in the Ancient World pp. 116-118, 121; Fink, Origin of Music, pp 107-115, and many other sources).
    But still, even when we consider this, the pentatonic and many other scales still can be found in places with little or no reference to anything like the diatonic.
    Unfortunately, our life-spans and personal experience (especially in the recent past and earlier) rarely embraced enough time to see all aspects of the evolutionary world in motion. In a real sense, our experience is like a still-photo of a bird in flight. The bird (in the photo) could lead to the conclusion (based only on that one frame) that it is 'still,' but in reality, it flies. Taking evolutionary reality in general, we usually don't live long enough to see the next frame of it nor the previous frame -- except when histories give us a clue (in so far as they are accurate).
    But if we use our best efforts to fill in the frames, based on our research into the past, then we can rise above the 'still-life picture' limits of our personal experience and immediate surroundings -- and dimly see the forest, not just a few trees; see the whole, not just the part; see the motion, not just the fixed bird.
    Specifically, we don't consider the pentatonic scale as a contradiction to the acoustic, nearly-subliminal pressures that bring the diatonic scale into existence over long periods of time and across cultures, but instead, we see it as an earlier, or different but equal version, of the same acoustic influences and process.

    Now here are some known historic operating principles capable of being considered universal or nearly so:
    * The "tonic" (for lack of a better term), and its fourth and fifth, are almost everywhere used in scales, and their overtones are most frequently heard, although were not necessarily heard consciously as different notes.
    * In the course of making scales, Helmholtz has noted, I believe accurately, that most peoples have avoided notes in their scale which produce semitones as melodically dissonant. Semitones are admitted only apparently when justified by being melodic "leading tones" [whether moving melodically upward or downward] to one of the strongest notes (tonic, dominant or subdominant).
    * The overtones of a note, as listed above, become (generally) weaker or less audible from left to right
    * The octave to a note is usually considered or treated as "the SAME note, only higher."
   
    The role of harmony in a melodic musical culture is this: Harmony tends to overtly reveal between the notes of a melody what is hidden (in overtone relationships) by playing some of these overtone relationships out loud. This -- in the hands of a harmonicist/composer can either aid the perception of the connection between a melody's notes as they unfold through time, or even deliberately obscure those connections -- all to aesthetic effect. [To hear an example, click here.Another: Here.]
    The dissonances of harmony now become tolerable once a musician is concerned with the connections between notes of a melody. As said, a musician in an "earlier" stage, prior to such melodic practices, is concerned more with the aesthetics effects of single or pure tones, and so would avoid the dissonances of deliberate harmonies [with the possible exception of using octaves and fifths, which are assigned to different voice ranges as a form of "unison"].
    Therefore, we can so far explain from this why, in the limited data existing of the most ancient music, we likely will never find any deliberate harmonic practices existing unless and until the diatonic scale (or perhaps a pentatonic that includes occasional leading tones) has been established. (See Evidence.)
    Of course, as history unfolds, it never progresses in a neat & tidy manner to suit any theory of "stages." The stages can overlap or often be interrupted by a cessation of communication of, or inability to hand-down, past practices to a later community, and so the stages may end up beginning again in a later culture.
--------------------------------------------------------------------------------------------------

[magnus-y]

Ancient Geek modes had little to do with modern tempered scales. there is a long, convoluted history of the deriving of scales from modes. Essentially by the 15th c. notated music used the series of whole and 1/2 steps we use with "modes" characterized by different "finals." The major scale began to occur in music during and after Bach with the final 1/2 step evolving from cadential resulutions. Since pre- Common Practice  music was largely polyphonic interval sizes likely differed from modern equal-tempered scales. the modern temperament and scale usage solidified in the 18th-20th centuries largely due to the keyboard and minimization of harmonic dissonances.

[Daniel_piano]

Bob Fink wrote:
------------------------------------------------------------------
    Interesting to me is that the popular scale among Greeks was not the Lydian (like our major) but was the Dorian (e' down to e) -- and here's possibly why:
    In downward playing, the semitone-jumps of the Dorian are IDENTICAL to the semitone-jumps when we play our major scale upward:
Semitone jumps: .......................2 ....2 ..1...2 .. 2... 2....1
...of Major (upward): ................c ...d ...e -.f ...g ... a ...b -.c'
..of Dorian (downward): ...........e' ..d ...c -.b ..a ... g ...f..- e
    When Glareanus [16th century] measured usage of the medieval modes, he claimed the Ionian and Aeolian were most frequent, which are like our major/minor scales. It's interesting that, after the Greeks, when the modes came to be played upward, the popularity of the arrangement of semitone-jumps is what persisted (e.g., Downward Dorian in ancient Greece and upward Ionian in medieval times).
    That's probably because, when played according to the semitone-jumps, there is a halftone at the end of each scale that serves as a melodic "leading tone" to the final note (F to E in the downward Dorian, and B to C in the upward major [or "Lydian" in Greece; "Ionian" in Glareanus time].

     Basically, every note we think is a "single" note actually produces overtones, which can be heard, but not as different notes, but as a change in quality -- for example, allowing us to tell the difference between a piano and a trumpet even though each plays the same note. This is mostly due to the different strengths of the overtones of that note in the instrument.]
    A "natural 5th" is based on the first different overtone of any given note. For example, C's first different overtone is G, which when lowered one octave, is the natural "5th" of C. You can produce a series of such 5ths, or a "cycle of 5ths" as it has been called in music, such as C to G, then D (based on the preceding G), then A, based on the preceding D, etc. Such a series or cycle looks like this:
    C-g-d-a-e, etc.
    Simply put, starting on C, if you produce this series of perfect (natural) 5ths, eventually you come round again to the original note, but in the form of B#, which is an "enharmonic" name for C, the original starting note. But this later C (or B#) differs from the original C by an amount called a "Pythagorean comma."
    It's a natural "flaw" -- like the days of the year not fitting equally into one earth revolution round the sun, with a 1/4 day interval left over. You might call that a "Calendar Comma."
    While we cannot practically shorten each day by a tiny amount to make them fit the year by a whole number (365, instead of 365.25), in music we CAN correct by re-tuning the notes slightly off "natural." The way this is done on fixed-key instruments (like the harpsichord or piano) is to divide the octave into 12 equal semitones or half-tones.
    This process is called "Temperament," and as a result, the notes are no longer perfect or "natural" in relation to others. C to G is no longer a natural or perfect 5th, but very slightly out of tune. Also as a result, B# and Cb no longer appear to exist as separate notes in this tempered system. So you won't find music written in these "keys," as the enharmonic version of them (B instead of Cb, E instead of Fb, etc) is used.
    As a result, in several instances, two notes become one note (for ex, a Db becomes the same as C#, or a B# becomes a C, or a Cb becomes a B. "Naturally" speaking, each of these examples really are TWO different notes -- which are playable as different notes on violin fingerings or as sung by singers).
    Clearly, however, putting so many enharmonic notes into a fixed-key instrument like a piano (putting B# as well as C, or Cb as well as B, for example) would produce an unplayable cumbersome instrument.
    The motivation for tempering notes comes also from transposing music. It used to be that on fixed-key instruments that if you tuned the scale of C to perfect steps, then the scale, while sounding perfect in C, when transposed to other keys, would sound seriously out of tune because of the Pythagorean "error," or natural flaw. These other keys, for example, those requiring a Db, and NOT a C#, would be the cause of the mistuned sound. But when the 12 tones are tempered or equalized by dividing the "error" equally among all the 12 notes, then ALL the keys are equally "out of tune" by a very small amount, none of them so badly as to sound like a serious mistuning to most or average ears. [However, many singers still sing the true intervals despite what a piano plays as accompaniment. Indeed, some singers hate the piano because they find its temperament annoying to their sense of pitch.]

FURTHER SOURCES:
    As mathematics buff Harvey Reid (in a 1995 webpage) wrote: "Our ears actually prefer the Pythagorean intervals, and part of learning to be a musician is learning to accept the slightly sour tuning of well-tempered music. Tests that have been done on singers and players of instruments that can vary the pitch (such as violin and flute) show that the players and singers tend to sing the Pythagorean or sweeter notes whenever they can. More primitive ethnic musics from around the world generally do not use the well-tempered scale, and musicians run into intonation problems trying to play even Blues and Celtic music on modern instruments." (Not all ethnic music avoids equally-spaced notes -- especially when trying to suit convenience of the fingerholes on flutes, which are often spaced equidistantly, creating a "tempered" effect. See Sachs quote below)
    Ethnomusicologist Victor Grauer, in a march 3 letter to me, notes the same thing: "While tuning systems and scales were not investigated per se in Cantometrics, it did become apparent as we analyzed music from so many different types of society, that tunings based on more or less 'perfect' fourths, fifths, unisons and octaves could be found 'everywhere.' "And this came as something of a surprise, at least to me. While, indeed, there are certain primarily instrumental tunings which have anharmonic aspects (e.g., those based on equidistant intervals, as already mentioned), the vocal music which was the primary object of our study was with few exceptions something that could be notated (roughly to be sure) in terms of the usual scales."
    Notes in the Neanderthal Flute essay indicate similar penomena:
    Curt Sachs ["The Rise of Music in the Ancient World, East & West" N.Y.: W. W. Norton & Co., 1943] and Bruno Nettl, "Music in Primitive Culture" (Cambridge: Harvard University Press, 1956) and Marius Schneider, "Primitive Music," in "Ancient & Oriental Music," ed. By Egon Wellesz, Vol. I of The New Oxford History of Music London: Oxford University Press, 1957] all write of this general issue, each writing about a different culture::
    On p. 133, Sachs describes a phenomenon in which conflicting tendencies (toward and away from equal divisions of the scale) may be combined. "Singers do not pay much heed to this temperament." He adds one aria "in almost Western intervals alternates with orchestral ritornelli in Siamese tuning." That is, singers sang the unequal steps, but the instruments were tuned to the tempered or equal steps.
    Nettl confirms this idea. He writes: "...the instrumental scales rarely correspond exactly with the vocal scales occurring in the same tribe." (p. 60.)
    Schneider writes (p.14-15): "When the same song is performed simultaneously...by voices and instruments, the melody proceeds in two different tunings. The instruments...on their own scale, the voices in theirs..." He says we must suppose "that the vocal tone-system has been evolved in a natural and specifically musical fashion, whereas in the tuning of instruments...quite different principles were applied -- such as, for example, the breadth of the thumb as the standard for the space between flute holes," or such as when a need on the same instrument arises to transpose melodies into higher or lower keys, the notes are adjusted toward greater equality (tempered) so that each key will remain tolerable, if not perfect. Thus we have developed from this kind of typical behavior an expectation NOT to find perfect pitch tunings on an instrument.
    The operative word here, relative to criteria for what is "acceptable" is that these instruments are tolerated, but when perfect pitch is available (voice, strings) then musicians choose the perfect intervals. In practical terms of instrument-making, the tolerable amounts have been in the neighborhood of up to a Pythagorean comma
    These numerous observations can be taken as a signal they are a widespread practice, perhaps universal, and is evidence that none of it could occur so widely if there didn't exist some innate preference for perfect consonant intervals.
--------------------------------------------------------------------------------------------

[Brian Healey]

So does anyone really know how our scale came to exist ?

I made it up.

I was just killing time until Ricky Lake came on.