I looked for these in music books but they don’t answer these questions. It says we have a 7-note (diatonic) scale system which means we use 7 pitches of the octave, whereas in Africa and China for example the people specify only 5 pitches in the stretch of the octave. a) Why 7? b) Why 5? Even more puzzling is when you read on it says that actually there are 12 POSSIBLE notes in the octave, eg., the piano is set up so that within any octave and counting both black and white keys there are 12 pitches, but then it says we only use 7. c) Why do we only use 7? Except for Schoenberg et. al. who use all 12.
d)Where did this 12 come from? They never say. These numbers justs float in out of nowhere, 7,5,12, and there’s Debussy’s scale of 6.
Then you learn that there is this thing called the fifth, and all peoples recognize the fifth. For instance C and G. It makes the simplest ratio next to the octave. e)Where does this fit in in a 5 note scale? or a 6 note scale? Anyways, it is like C and G together in the 7 note system, or D and A for instance. The notes of the fifth are five pitches apart counting both ends, but not five whole steps apart. For it turns out that adjacent keys are defined as a half step apart, so from C to G we have for instance 8 half steps or three whole steps and one half step that go w w 1/2 w from C to G. From D to A it goes
w 1/2 w w, thus a different pattern but still 3 wholes and a half. I suppose this is consistent.Except now suppose I take a fifth from F#? The only reason I know this would be up to C# is by starting on C, which I know goes to G, and then going up half step by half step for each key, C# to G# and so on until I get to F# and see that it goes with C# as the fifth. f) But is there no way of knowing this other than by that process? g) and is there is a way of knowing exactly which five pitches are in that interval? I know it is probably considered irrelevant because the issue is the interval, not what is between, but somebody must know what these pitches would be for each inteval of the fifth.
h) I know there are 12 major diatonic keys but I think they never use some of them and/or they use only one name for some of them. Like I think I’ve heard of A flat but not G#, and afe they the same?
i - repries of a,c, and d:If nothing else I would really like most to know why did they pick 7 out of 12. Is it because this causes the different keys, since you can’t get seven evenly spaced things out of 12, and therefore you get 12 sets (keys)each of which has pitches in it that the other 11 don’t and thus more variety?
A great variety of scales have been used in the past and in different cultures; no single interval is common to all of them. In the 6th cent. B.C., Pythagoras defined the mathematical relationship of the perfect intervals (the octave, fourth, and fifth) and of the intervals between them (an interval being the difference in pitch between two tones). The Greek system was taken up by the Christian church, which adapted its note series to a number of modes used in medieval music.
The church modes, …became reduced in due course to the two characteristic scales of later Western music, the major and the minor. The major scale, called diatonic, has five whole tones (t) and two semitones (s) arranged thus: ttsttts (as in the white notes on the paino keyboard taken from one C to the next C); this scale, with certain modifications, became the basis of Western musical tonality until the end of the 19th cent. The dissemination and influence of the diatonic scale was therefore very great. The minor scale is based on tsttstt. This arrangement produces the lower third, sixth, and seventh degrees that are characteristic of the minor mode; the higher seventh degree, a semitone rather than a whole tone below the main note, or "tonic, is often borrowed from the major mode for use at cadences.
Akin to the modes, the concept of key was developed, whereby a home tone, or tonic, is the principal focus of a composition, and the various other tones assume importance according to their relationship to the tonic. The increasing complexity of instruments demanded more refined tuning systems. By J. S. Bach’s time equal temperament had become established. The resulting scale, called chromatic, consisted of 12 notes divided by semitone intervals (the white and black notes of the keyboard). Although the diatonic scale is basically heptatonic (seven-noted), music that is in a major or minor tonality usually employs the remaining five tones of the chromatic scale as auxiliary or ornamental tones. Music that employs them freely is said to be highly chromatic, while music that employs them sparingly is said to be diatonic.
12 comes about because it works. Really. If you work out what simple ratios between 1 and 2 are, they fall pretty close to powers of 2^(1/12). For example, 2^(7/12) = 1.4983 = 3/2 (approx.). 2^(5/12) = 1.3348 = 4/3 (approx.). 12 is the lowest number which works well. The next higher number of notes which works well is 19
The fractions corresponding to the 7-note scale, and the corresponding 12 note approximations, are
1 2^(0/12)
9/8 2^(2/12)
5/4 2^(4/12)
4/3 2^(5/12)
3/2 2^(7/12)
5/3 2^(9/12)
15/8 2^(11/12)
2 2^(12/12)
I’m sure you’ll get an expert along any minute who will be able to inform me that I’m woefully mistaken, but if dim memory serves, a fifth is just two notes whose frequency has some simple ratio like 3:2 or some such (I have no idea what the ratio is; an octave is 2:1, and that’s all I remember for sure).
There are some keys that are more COMMONLY used than others, so more pieces would have been written in A flat than in G#, but there’s no fundamental difference. As a WAG, I’d say that the reason for this is just that it’s easier to read the music if you have, say, 4 flats than it is with 8 sharps. After all, G# would have both a G and a G#, as well as an F, and that must be hard to keep track of after a while.
Minor hijack (pun unintentional): I think I’ve heard (but can’t cite) that at one point in the baroque era, people considered, say, C# and D flat to be different. That is, they had a concept of both, but the two were distinct. Is this true, and if so, how did it work? Was it only a matter of “color” and the two tones were actually the same, or were they actually different?
The different notes were actually different between sharps and flats (A# and Gb were slightly different pitches), back in the good old days when they got out the little wrench and retuned the instrument between every piece of music. For various reasons (possibly because retuning was a pain, or because a bunch of instruments got made that couldn’t be retuned easily), they quit doing that and picked one specific sequence of pitches, even going so far as to define the base “A=440” pitch that everybody the world over (as far as western culter was concerned) would use as the starting point for all instrument tuning.
Back in the day, however, it would probably have taken a true connoisseur of music to discern the difference between a A# scale and a Gb one.
Don’t take this as a hijacked thread, but I’ve often wondered about the 7 note and 12 note diatonic scale. When I first started learning (poorly) to play guitar, I wondered what good stuff we were missing in the key of H and K…
I heard that the first few bars of a Black Sabbath song (the name escapes me, not a fan) it goes “bow… DOW… bau…” is a combination of three cords which in medieval times which were never supposed to be played together under penalty of death. I suppose it was considered something satanic… It is a fairly creepy chord progression, what’s the deal with that?
Wow - lots to cover, no time to herd up some cites, but let’s play this out and see how it goes:
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Tedster - I am pretty sure you are referencing the beginning of the song “Iron Man” by the Sabs (great tune!). Anyway, through some combination of nature and nuture, humans (in general, it varies hugely across cultures) that pitches that are in certain mathematical relationship to one another are “good” and those that are in different relationships to one another are bad. Over time, the rules have been set in stone, and now everyone in modern society knows that certain tones and pitch combinations mean happy, sad, triumphant, etc. The notes Tommy Iommi hits in Iron Man = sad/scary/evil. So, yes, one could imagine that the old time monks might not have judged it to be good…
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Why 7? A perhaps more insightful way to frame the question is “Why 12 tones and then why 7 main ones in a scale?” As has been discussed above in the thread, the 12 is mathematical - if you play guitar and know what harmonics are (when you hold your finger over the 5th, 7th or 12th fret without pressing down all the way and pluck the string - like the beginning of “Roundabout” by Yes). You are causing the string to vibrate (this is an over-simplification, but humor me) at the 1/2 point (12th fret) 1/3 point (7th fret) and the 1/4 point (5th fret). What is a reasonably small number that contains 2, 3 and 4 as multipliers? 12. It’s that mathematically straightforward (I don’t know why 6 isn’t used, since it also is multiple for 12 - maybe early instruments weren’t precise enough to register a 6th harmonic.
Why 7 notes in a major scale? Well, IIRC, it’s has to do with Tedster’s question and my response. It is generally felt that different progressions produce different emotional and spiritual responses. So different Western cultures developed different ways to get from the 1st note (the Tonic) through the 3rd, 4th and 5th (the Dominant - the note most mathematically/vibrationally compatible with the tonic) back up to the tonic. To stay focused on the emotions/spirituality they wanted to achieve, the decided over time which of the 12 notes worked - in Western music, there were typically 7 of the twelve that you could use and stay consistently within your targeted emotional/spiritual response, and 5 that didn’t, or were used occasionally (and are referred to today as accidentals, I think). The Greeks captured most of this - they didn’t judge, they just took the 12 tones and said “if you start from tone 1, these notes are consistent with it and this is the Dorian mode” and so on until there were 12 (I think) modes. Over time, a few modes became the most popular; these ended up being name the Major and the Minor - although I think there are at least two minors - harmonic minor and relative minor, but my theory skills are completely shot.
So - math, practical capabilities of the instruments and genetic/cultural connotations of pitch relationshps - that’s why.
WordMan, it has nothing to do with 12 containing 2, 3, and 4 as multipliers, and everything to do with powers of the twelfth root of 2 being good approximations to ratios of small integers. As I mentioned above, a scale with 19 steps per octave also works well, even though 19 is prime. 19 works because powers of the nineteenth root of 2 are also good approximations to ratios of small integers.
ZenBeam,
First, thanks for the correction - it’s only been 15 years since I focused on this stuff - no surprise that I have forgotten some. Okay, lots.
Where does the “root of 2” come in? I feel like I am with you up to that point. I think you are saying that the basic rule of thumb is: take any number you are considering to represent the number of steps between octaves (5 steps, 12 steps, 34 steps, whatever). Then, consider each fraction (e.g., if your denominator is 12, consider 1/12, 2/12, 3/12 etc.), for each step in that scale. For each fraction - isn’t the question “Does a note at that point vibrate in a complementary way with the root note?” and the answer is based on whether the two notes vibrate together in a way that mathematically appears as a reasonable simple fraction? So a fifth sounds very complementary because its ratio with the root is 3/2, whereas the 7th note sounds less so because its ratio is 15/8. Right? So the question is what is the smallest number which has provides enough notes but the fractions of that number vibrate simply? Right?
That was definitely the point I was trying to make with my “multipliers of 12” argument, but I am hearing that while my intent may be okay, the argument ain’t makin’ it. So why roots of 2?
[added in preview to Wordman : The roots of 2 are used because you’re dividing up an octave (a ratio of 2:1)]
ZenBeam is correct. But there’s also an interesting point to consider about the guitar strings :
The twelve notes of the scale are only approximations to the real harmonic intervals. But when I pluck a string one-third of the way down, the ratio to that string when open will be exactly 3:2. So it won’t correspond with an evenly divided up 12-note scale which another instrument (say, a piano) might be using.
The variation is slight, true, but significant enough to make tuning considerations interesting. I don’t know enough about the history of it, but you could look up ‘even tempering’ or information on tuning to find out more.
Regarding the ‘Iron Man’ opening … I can’t recall the opening notes off the top of my head, but you’re probably referring to the use of the tritone, a very dissonant harmonic interval. If the name didn’t give it away, the tritone consists of three whole tones (so from C to F# is a tritone).
This was indeed referred to as ‘diabolis in musica’ (the devil in music) in Medieval times since it sounded so ugly. And I’m reasonably sure no one was ever put to death over it. While it undoubtedly evoked a certain sense of superstition, they probably just regarded you as a bad musician if you tried to use it.
Question for today : Why does the tritone, specifically, sound so bad?
If you consider what’s been here already – that lower harmonics are more pleasing to human ears, and then take a look at Zenbeam’s previously posted list for a little help, and you know a little math, you should be able to come up with the completely rational answer.
There’s a lot of good information above, (and some misinformation as well), but rather than try to condense it, I’ll start at the beginning.
Music deals with sound waves. I’m going to describe things with a vibrating string, but you could as easily describe a a bottle you blow over or other instrument you use to produce a sound.
If you take a string, say 2’ long, tied at both ends and pluck it, you get some note. The string vibrates at some frequency. If you cut that string exactly in half and pluck it, you will get another note, which is exactly double the first frequency. Those two notes sound good together. The second note is known as one octave above the first note.
Now if you take the original string and cut it exactly in thirds, and pluck it, you’ll get another note, which is exactly 3 times the frequency of the first. These two notes also sound good together. This new note is known as one fifth above the first note. (Actually, it’s an octave plus a fifth above, so let’s double the length of this new 1/3 string, to lower it one octave. Now it is one fifth above.)
Now, if you cut that new string in thirds again, you’ll get another note which is a fifth above the fifth above the original note. (again you’ll have to double it to lower an octave, so we don’t end up with micro strings) If you continue to do this you’ll get additional notes. After doing it 12 times, you’ll get a note which is extremely close to your original note. So close, that we define it as the same.
So that’s where the 12 notes come from. They come from the fact that a note played simultaneously with a note that is 3 times its frequency (or 3/2, or 3/4 or whatever) sound good together.
Now, those 12 notes in the order I called out look like this:
C G D A E B F# C# G# D# A# F (and above that another C)
You can rearrange them on a piano (or guitar or whatever) in order:
C C# D D# E F F# G G# A A# B (and again C)
When they’re in the piano order, it’s curious that we picked the seven we did, but when they’re in the order of fifths, it’s a lot clearer:
F is one fifth below C
C is the key
G is one fifth above C
D is two fifths above C
etc.
Why seven keys instead of six or eight? I dunno. Why one fifth below and 5 above? Well, sometimes you use other combos, these are called Modes, but that’s another lesson.
Also keys other than C are another lesson.
There’s another take on this subject here.
A couple of notes:
a) The main Iron Man chords don’t use tritones or anything funky at all. They’re pretty standard chords: (I think it’s in C which is a bit odd for guitar music, but irrelevant for this discussion. If it’s in something else, translate away)
C Eb Eb F F B Bb B Bb B Eb Eb F F
So, it’s stock I - iii - IV - VII stuff. Hell I’m a big New Order fan and a Blues fan to boot, so those are the only chords I know.
b) The reason the tritone is funky is that it is as far away as you can get from the root note. See my circle of fifths posting above. The Tritone for C is F#, which is as far away up as it is down from C. The fact that it sits right between the pleasant sounding fourth (a fifth down) and fifth (a fifth up) also makes it want desparately to lead to one of those two notes.
Err, that’s I - iii - IV - vii.
I’m probably just going to obfuscate things further, but here goes anyway. First, I think there would be seven classical modes, not twelve. You start with the seven note scale and then get a different mode by starting at each of the possible notes. Today we’re only familiar with two modes. If you take the scale consisting of all the white keys on a piano, you can start at C for a major scale or A for a minor scale. Starting at a different note would give you the Dorian mode, but we don’t use that much nowadays, or any of the others for that matter.
Now, as I understand it, the twelve notes of the chromatic scale were worked out by taking ratios of 3/2 then possiby dividing by two to bring the notes down to right octave. So, you start with C then multiply the frequency by 3/2 and you get G. Multiply the G by 3/2 and you get D. Divide the D’s frequency by two and you get the D what we call one full tone above C. OK, now do that a total of twelve times and you get to a note almost exactly 128 times above the original C, which equates to another C seven octaves up. It’s the first time you get to a frequency that is a power of two above the original. (128 = 2 ^ 7) (Does that seven bear any relation the seven notes in our scale? I have no friggin’ clue. Maybe.) Along the way, you’ve computed the values of the 12 notes of the chromatic scale C - C# - D - D# - E - F - F# - G - G# - A - A# - B (- C). If instead of multiplying by 3/2, you multiply by 2/3, you also get twelve notes, but they are called C - Db - D - Eb - E - F - Gb - G - Ab - A - Bb - B (- C). They have almost the same values either way you compute them … but not exactly the same values. This is where tempered tuning comes in.
Bach said it was a royal pain to retune instruments every time you want to change from playing sharps to playing flats. He said we should just make A# and Bb (for example), which were already almost the same note, exactly the same note. Using powers of the twelfth root of two is just a mathematical way of making this happen. Tempered tuning was apparently a bit of a hard sell because, for one thing, it screwed up the fundamental ratios. G is no longer exactly 3/2 the frequency of a C, but it’s pretty close. If you have a scientific calculator, you can figure it out by 2 ^ (7/12) = 1.498. (Bach would have had to figure that out using logarithms and God only knows how he tuned an instrument that way.) Anyway to prove his point that the ratios were more convenient, and close enough to the “true” ratios to sound just fine, Bach composed a series of works in each of the twelve keys, which he called “The Well Tempered Clavier”.
I didn’t even post this thread, but through putting out the misinformation and getting corrected, I learned a lot - thanks
ZenBeam - thanks for the math - once I understood the context behind it, it all made sense
panamajack - thanks for explaining the 2:1 ratio
Bill H - a great explanation - made everything easy to understand - I didn’t check some of the threads that other people published, but if any are like Bill H’s explanation, they would be worth checking out
Greg Charles - sorry for the mistake; you’re right on the number of modes
WordMan
Not according to Kyle Gann, the guy behind the website I referred to above. From another page on that site:
"If you are or were ever a college music student, you probably read, or were told, that Johann Sebastian Bach wrote his collection of preludes and fugues The Well-Tempered Clavier in all 24 major and minor keys in order to demonstrate equal tempered tuning.
If so, you were misinformed.
Bach did not use equal temperament. In fact, in his day there was no way to tune strings to equal temperament, because there were no devices to measure frequency. They had no scientific method to achieve real equal-ness; they could only approximate.
Bach was, however, interested in a tuning that would allow him the possibility of working in all 12 keys, that did not make certain triads off-limits. He was a master of counterpoint, and probably chafed and fumed when the music in his head demanded a triad on A-flat and the harpsichord in front of him couldn’t play it in tune. So he was glad to see tuners develop a tuning that, today, is known as well temperament. Back then, they did call it equal temperament - not because the 12 pitches were equally spaced, but because you could play equally well in all keys. Each key, however, was a little different, and Bach wrote The Well-Tempered Clavier in all 24 major and minor keys in order to capitalize on those differences, not because the differences didn’t exist.
In any case (according to Jorgensen), the error that Bach wrote the W.T.C. in order to take advantage of what we call equal temperament crept into the 1893 Grove Dictionary, and has since been uncritically taught as fact to millions of budding musicians. Lord knows how long it will take to get that error out of the universities. It’s still in all kinds of reference books."
I’m no music history expert, so I can’t vouch for this, but I thought it was interesting.
The notes Ab and G# are (almost) identical, so you can call the same note which ever you want. However, if you’re talking about a key, then Ab is preferrable. This is because the major scale goes like this:
Ab Bb C Db Eb F G (Ab)
If you instead did the same scale with sharps, it would be:
G# A# C C# D# F G (G#)
Now compare these two. The first has all the letters going in order, once each. With the second, there is no B or E, and C and G appear twice! Since music is written on a staff with a dot representing a letter, and you tell whether you should make it a sharp or flat by looking at the beginning of the line, you need a system where each letter is represented once.
Equal temperament is needed for keyed instruments. String (and brass, to a lesser extent) can adjust the tuning of a particular note on the fly. I have been told by a string musician that G# and Ab are indeed played slightly differently. String quartets and brass ensembles will always try for perfectly tuned harmony, rather than equal temperament. And barbershop quartets are known for striving for “ringing” harmony (perfectly tuned).
You can play with different tunings with the Java applet here: http://pages.globetrotter.net/roule/accord.htm