 The Clarinet BBoard
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Author: clairannette
Date: 2005-11-29 21:11
Today I had a student ask me "how come there is no note between E and F or B and C?" and I am lost on how to answer. All I could come up with on the spot was telling her that it just is, and to look at how a piano is arranged.
But here's the question for you folks: why is there really no note in between? Does it have more to do with music theory or music history? What can I tell my student to clarify this? Is it just a "why is the sky blue" kind of question?? Any help would be appreciated! Thanks!
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Author: sfalexi
Date: 2005-11-29 21:16
I could make an educated guess (which in my mind forms relationships between the formation of a major scale, solfeggio, and scale notes), but I'll wait for someone that's actually studied up on the material. But I do wonder if my thoughts are on the right track . . . .
US Army Japan Band
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Author: Hank Lehrer
Date: 2005-11-29 21:33
Hi Clarinette,
As far as the sky being blue, that's all about the scattering of different light waves as well as which ones we then see. You can find the answer for this question as well as many other cool scientific facts at http://www.sciencemadesimple.com/index.html
I'll let GBK explain about the "apparent" missing 1/2 step between E and F and B and C on the piano keyboard. He loves to share that kind of info.
HRL
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Author: Liquorice
Date: 2005-11-29 21:34
In German Bflat= B, and B=H. So at least in German there is a H between B and C, which gives the highly ingenious ascending order of ABHCDEFG. Try explaining that to a student!
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Author: Don Berger
Date: 2005-11-29 21:53
I believe? that I learned there were two "tetrachords" in each octave, 2 "fullsteps" plus a 1/2 step. Will look forward to a better description. Don
Thanx, Mark, Don
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Author: tictactux ★2017
Date: 2005-11-29 22:13
...but in "normal" circumstances, "B" and "H" are mutually exclusive (eg. I haven't seen a scale with both).
What I have more problems with is when to use "G#" and when to use "Ab" (although I faintly remember we more often use the # ('-is') than the 'b'. "Do you want me to play a G# or an Ab?")
--
Ben
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Author: GBK
Date: 2005-11-29 22:18
In short:
In Western music, a half-step (semi tone) is one half of a whole tone.
Some music historians credit Pythagoras, in the 6th century B.C. by use of a single vibrating string called a "monochord, for establishing the mathematical ratios of the scale which has been the basis of Western music.
The phenomenon of half-steps occurs because people began singing modes before they began writing them down.
Later, as diatonic fretted instruments were built, the distance between E and F, and B and C, were made proportionately smaller than the other frets.
As the keyboard was developed, these intervals remained as all the keys were built to be a half step apart.
Basic piano fundamentals can be found here:
http://www.martymethod.com/basiclesson.shtml ...GBK
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Author: ohsuzan
Date: 2005-11-29 22:35
Digging back through the memory banks here . . .
It's got something to do with scale construction and the medieval modes -- probably more history than theory, because AFAIK, the scales came first, and the theory came after.
Once upon a time, there were numerous scales in common use, all of which had characteristic patterns of whole and half tones. Can't remember them all now -- Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Ionian -- and maybe a couple more.
You can hear the modal scales by starting on the particular white piano key that will get you, diatonically, the particular pattern of whole and half steps characteristic of each modal scale. Dorian, for example, is constructed diatonically from the keyboard "D". Aeolian is from the keyboard "A", Mixolydian is from the "G", and Ionian is from "C". (I remember those, because I played dulcimer many years ago.)
What we have left in common use today are only the Aeolian (our natural minor) and the Ionian (our major). The piano keyboard is arranged in such a way as to make the C major scale normative -- and all the other major scales take off from that. But no matter what key, major scales still preserve the characteristic whole-whole-half/whole-whole-half pattern of the Ionian mode. Same for the (natural) minors -- always the same pattern
And yes, it is two tetrachords joined together. But I don't know how the modes and the tetrachord idea fit together -- or if they do.
Perhaps the use of whole and half tones intermixed has something to do, also, with concepts of movement and rest (stasis). I recall that certain modes were always associated with certain emotions, or moods, as well -- and what we now call the minor was actually considered a happy mode.
Here's a site I just found that tells more:
http://www.banjolin.supanet.com/modesandscales.htm
Think I'll go read it now.
Susan
Post Edited (2005-11-29 22:38)
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Author: Shorthand
Date: 2005-11-29 23:19
Misnomers plenty to go around. I wouldn't skirt around the issue, though, its time to explain that much of our music is the way it is because that's how it got that way. "Its arbitrary" is a perfectly good answer, and kids will be fine with it in most cases - they understand the need to settle on a standard and move on.
I'd try to explain that there are really 12 notes in an "octave" which is just when one tone is double the frequency of the one an "octave" below it. We call it octave because we have an 8 note (7 unique, but 8 intervals) scale.
In the west, our fundamental, chromatic scale was indeed identified by Pythagoras, and has 11 unique notes, but should be considered a 12 note scale (the math gets easier that way).
In China, they traditionally use a pentatonic scale, which is a 6 note scale - half the intervals of ours. (I think there is some unevenness in that, though. It is also closely related to Native American Scales AFAIK.)
There are a number of different 8 note modes on top of the 12 note scale:
http://www.ericweisstein.com/encyclopedias/music/topics/ScalesandModes.html
If you're going to cram 8 notes on top of 12, you're always going to get 6 whole steps and 2 halfs (unless you do something silly and have bigger intervals between the notes.) The major scale is just what we settled on as sounding the most "consonant" here in the west. In general notes with simple ratios (low denominators) of frequency (1:3 or 1:4 etc.) sound more consonant than notes with more complex ratios, but this rule isn't hard and fast so there is a little juggling over time, and some periods/cultures like dissonance more than others.
Traditional Indian classical music is based on a 24 note octave, I believe. Other musical traditions only think of Bass-range notes as musical, higher range stuff just sounds like screaming to them.
Post Edited (2005-11-29 23:20)
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Author: chuck
Date: 2005-11-30 04:00
Why not show your student how a major scale is constructed? Why is there no note between D and Eb, and A and Bb, in a Bb major scale? You could then lead into a session on the construction of a Harmonic minor scale and/or a whole tone scale. Harmony 1. Chuck
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Author: EEBaum
Date: 2005-11-30 06:21
Regarding posts above...
An octave is an octave because of how we define intervals... unisons, seconds, thirds, etc. We count, with the note we start on as 1. Later comes sixth, seventh, eighth ("octave"). Since the eighth is the same note name as the note you start on, the term "octave" is also used to denote how many notes the scale has before you get to that note again.
Our major and minor scales have 7 notes per octave.
The chromatic scale has 12 unique notes per octave.
The pentatonic has 5 notes per octave.
Octatonic scales are seen in contemporary music, with 8 notes per octave.
Whole-tone scales have 6 notes per octave.
I wouldn't call blues or harmonic minor scales silly.
Indian classical music has more of what we would call quarter tones, but I think calling it a 24-note octave might be a stretch. Careful listening reveals that it is closer to our system of music than one would immediately guess. The music is just more highly focused on development of scales than harmony.
I never learned tetrachords, and think they may serve to confuse in the long run.
While part of it is "because it was always that way," the major scale (and therefore all the modes as displaced relatives of the major scale) can be derived from the harmonic series of any given pitch. For example, pretend that you call 100 Hz a C. 200 Hz is again a C, 300 a G, 400 a C. 500 = E (touch flat from E.T.), 600 = G, 700 = Bb (flat from E.T.)...
Then... (with many of them way out of whack from Equal Temperament, hence the abundance of tuning systems before Bach's time)
800 = C
900 = D
1000 = E (touch flat from E.T.)
1100 = F (ish)
1200 = G
1300 = A (ish)
1400 = Bb (flat from E.T.)
1500 = B
1600 = C
The "flatness" of E and Bb's natural harmonic tendencies explain why minor thirds and sevenths sound in tune when equal-tempered instruments (e.g. clarinet) bring their pitch down. The sound is more consonant when the overtones are allowed to align.
-Alex
www.mostlydifferent.com
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Author: Bassie
Date: 2005-11-30 08:10
This is an evil subject. Here's my take:
Let's start with the simplest scale - call it 'A'. Make it a minor scale, and call the notes 'ABCDEFG' - and back to A, which sounds like the place you started (physically, because it's twice the frequency).
Turns out if you start on 'C', with those notes, you get what we call a 'major' scale.
NOW: let's start somewhere like 'B'. The scale sounds odd to our modern Western ears (unless we've been singing Walton recently) - unless we change some of the notes. We need a note between C and D, between D and E, etc.
Start somewhere else - again, new notes are needed. But after you've started everywhere you can think, you'll find you can get away with 12 notes to play all the scales you can think of: A, A# .... G#.
And there's nothing between E and F and between B and C.
WHY is there nothing between E and F and between B and C? Because you started with an uneven set of notes as the definition of 'A minor'. Turns out those notes fit a pattern of frequency ratios. We call them 'intervals': a minor third, a fourth, a fifth etc. - but in fact they're frequencies in simple ratio.
The scale of A minor is not an even scale. That's why.
Post Edited (2005-11-30 08:42)
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Author: Bassie
Date: 2005-11-30 08:41
Above message edited for silly mistake...
</blush>
Post Edited (2005-11-30 08:43)
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Author: tictactux ★2017
Date: 2005-11-30 08:45
Attachment: minor.jpg (7k)
> The scale of A minor is not an even scale. That's why.
But the perception of "evenness" is a convention. Or a 7-note octave is a convention, dragging a lot of history behind.
FWIW, I always get hungry when I hear "minor" (see .jpg)
--
Ben
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Author: Bassie
Date: 2005-11-30 13:18
tixtactux -
I meant 'uneven' in a physical sense - the gaps between the notes in frequency terms are uneven - i.e., on a string the frets would be unevenly spaced. And they're spaced s.t. if you fill in only 5 extra notes you get an (almost) even scale.
But 'why are there seven notes in a scale' is a truly excellent question. I have no idea. I know that some cultures have 5 and at some point in the twentieth century people in the West started thinking 12 was a better number - but why? Can you enlighten any?
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Author: Bassie
Date: 2005-11-30 13:39
I feel like I'm on the verge of learning something important here... are you saying (GBK, EEBaum, etc.) that if you take a string fretted at an octave (halfway) and eight equal intervals (halfway, halfway and halfway again, something I imagine Pythagoras could have come up with) and fiddle with it (no pun intended) until it 'sounds' in tune, you end up with a seven note scale with two semitones?
(Bit of a red herring, this... more below.)
Post Edited (2005-12-01 10:33)
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Author: Markael
Date: 2005-11-30 14:07
Wow.
What a crazy topic.
At first glace the student's question seemed (to me) silly, but it really is deep, because it implies or suggests many questions.
As I see it, the fundamental issue underlying all this has to do with unraveling what there is about our approach to music that stems A) from convention, and B) from the fundamental nature of music itself.
It is, of course, arbitrary that we decide to give "F" its own name, with the alternate name of E#, whereas the black key notes on a keyboard must derive their names from their white neighbors.
Octaves, however, are not arbitrary, just as years and days on the calendar are not arbitrary. A pure octave is defined by vibrations which the ear can detect.
The basic phenomena of major chords and seventh chords (dominant seventh type) are not arbitrary either, because the consonant sound they produce comes from sympathetic overtones.
Once you have established octaves and major chords, you have to fill in the blanks, and set up the chromatic scale and other scales. That's when you run into a problem vaguely similar to the one you have when dividing the year into months.
Months are based on moons, but when you divide the year into moons the math doesn't work out right without some adjustment. It is arbitrary that we have assigned thirty-one days to January and twenty-eight (or twenty-nine) to February, but the need to make some kind of arbitrary adjustment is based on the way things are.
Similarly, you can create a scale starting with a perfectly tuned C major chord, that is, perfectly tuned to the overtone series with no audible beats. From there you can construct a scale, and you will end up with some chords that sound terribly out of tune.
Thus, various forms of temperament have evolved. Equal temperament is now the norm.
Equally spaced half steps is a convention. In my opinion, however, dividing of music into half steps is a little more than convention; it's kind of the way things are, even though some forms of world music have traditionally played "in the cracks."
Also, I think the arrangement of the keys on a piano keyboard is a marvel.
Post Edited (2005-11-30 14:10)
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Author: Gordon (NZ)
Date: 2005-11-30 14:13
My simplistic take:
Forget the names for a while.
There are 12 basic steps (half tones) to get from one note to the same note an octave higher. These are represented by all the white notes going up an octave on the piano.
Then the NAMING of those 11 notes involved in those 12 intervals is just a convention we settled on a long time ago...... partly because if we used white notes for all of them, the piano would be far too wide. (Well, not really, but it makes sense and sounds good!)
So the question is the same as asking "Why is there no other note between F# and G, or F and F#", or "why is there no whole number between 4 and 5?" There just isn't!
Well, I tried. You'd need a degree to comprehend most of this thread, let alone pass it on to a young student!
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Author: Shorthand
Date: 2005-11-30 15:28
Attachment: Fun_With_Scales.pdf (14k)
You Swiss and your Chocolate!
Here's an excellent link:
http://www.phy.mtu.edu/~suits/scales.html
I had the good/bad fortune of understanding exponents when I learned the nature of a scale. As a result, I learned the equi-tempered first and didn't understand the true nature of a just scale until just now when I looked it up above.
The ET scale frequencies can be generated by the following formula:
f = 440*2^(x/12)
where x is the number of half-steps away from 440 A.
This has the benefit that any given interval always brings up the same ratio of frequencies regardless of where you start, making it possible to play any key on a given instrument. The downside is that the ratio isn't simple by any stretch, in fact its irrational, but it is close to what you would get with the correct, just intonation. The just scale is an excerpt from the harmonic sequence (1/2, 1/3, 1/4, etc.), whereas an ET scale is a geometric sequence (1/2, 1/4, 1/8 or 1/3, 1/9, 1/29, etc.) - they'll never quite jive.
In general, to keep costs in check, everyone except choirs technically uses equi-tempered tuning. I have heard of choirmasters having paino's retuned into a certain key (or perhaps a compromise of 2 or 3 keys) for a certain concert.
Similarly, I get the feeling that good musicians unconsciously tweak their intonation closer to the just scale for whatever key they're in.
As an academic exercise, I looked and saw how hard it would be to wedge the important parts of a just scale into a 15 note Equi Tempered monster. It wasn't. I think any number of steps divisble by three will get you close enough. (16 was much harder to pull off.) 9 was a squeeze but it worked.
See attachment.
Post Edited (2005-11-30 15:36)
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Author: ohsuzan
Date: 2005-11-30 15:58
Markael says, "Also, I think the arrangement of the keys on a piano keyboard is a marvel."
True enough. But have you ever had the sense that this same marvelous arrangement also *limits* the way we think about music?
I sometimes think the keyboard is a tyrant.
Susan
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Author: redwine
Date: 2005-11-30 16:00
Hello,
To be fair to your student, you should mention that there are notes between half steps. We in western music don't use all of them, however. Also, there would be an infinite number of them, so they could not be counted. Of course, in jazz we do use some of them by bending pitches. Sometimes these are called "blue notes". When we play Gershwin's Rhapsody In Blue solo, we are also playing notes between the half steps, but there are no keys to identify each note as such. Western music theory has conveniently divided the notes into the 12 that we use. This is how we explain what it is that we do.
Ben Redwine, DMA
owner, RJ Music Group
Assistant Professor, The Catholic University of America
Selmer Paris artist
www.rjmusicgroup.com
www.redwinejazz.com
www.reedwizard.com
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Author: Shorthand
Date: 2005-11-30 16:01
One more thing.
The physics of instruments evokes the harmonic sequence. (This is pretty much true of any harmonic oscillator.)
Slapping a geometric sequence (based on the 2 harmonic) on top of this is probably the most fundamental compromise in Clarinet design - and I guess the fundamental breakthough in both the Albert and Boehm clarinets.
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Author: Don Poulsen
Date: 2005-11-30 19:20
Gordon's answer is a good one in my opinion, but I can' t resist putting it in my own words.
In what we call an "octave" we (in western music) use twelve different tones. The "distance" between each of those tones and the next highest is the same. It's just the way they are named that is weird.
I now refer you to the above posts to see why the notes becamed named the way they did.
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Author: EEBaum
Date: 2005-12-01 07:09
Shorthand wrote:
>>Similarly, I get the feeling that good musicians unconsciously tweak their intonation closer to the just scale for whatever key they're in.
Unconsciously? We do it consciously in my ensembles. Anything else sounds out of tune.
If you want to go WAY in depth into how to "ideally" divide the octave for proper harmonic placement of pitches, look into the notational systems devised by Ben Johnston. IIRC, he indicated a few dozen pitches per octave. Granted, the net effect is, in my opinion, not much different than a group of seasoned musicians that adjusts notes based on their placement in the harmonic structure, and a whole lot more difficult to play.
-Alex
www.mostlydifferent.com
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Author: allencole
Date: 2005-12-01 08:01
Here's what I tell my students:
The major scale is the yardstick by which we measure pitch in western music. Be grateful for the two half-steps because they are what tell our ears where we are in the scale.
If it were all whole-tones or all half-tones, we'd get lost. If it were whole-half-whole-half or half-whole-half-whole (as in diminished scales) we'd also get lost.
We could divide our octave into 6 whole steps with half-steps between each. If we did, our piano keyboards would be completely symmetrical and middle C would be VERY hard to find. Also, our scales as we use them would not have a unique letter name for each note, and therefore would not rise and fall in a smooth diagonal on the staff. I'm trying to imagine what key signatures would be possible under this system, but my brain can't handle it. <g>
I generally play them examples to illustrate this, and they end up being glad for the two half-steps...and less prone to look this diatonic gift-horse in the mouth.
Allen Cole
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Author: Gordon (NZ)
Date: 2005-12-01 10:18
Shorthand wrote:
"...In general, to keep costs in check, everyone except choirs technically uses equi-tempered tuning...."
The harp is an exception.
There are 7 pedals, one for each note in the scale.
To sharpen all the F strings to F#, you raise the F pedal. To flatten all the G strings to Gb, you lower the G pedal.
And the result is that F# is not the same as Gb. The same phenomenon occurs for all sharpenings and flattenings.
Shorthand wrote:
"...Similarly, I get the feeling that good musicians unconsciously tweak their intonation closer to the just scale for whatever key they're in...."
I am sure that the intonationally aware instrumentalist player who often plays with keyboard accompaniment, will get so used to adjusting to a well-tempered scale, that when he joins a chamber music group that excludes keyboards, he will probably play out of tune to the the just scale that such a group probably tends towards.
And vice versa.
Post Edited (2005-12-01 10:23)
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Author: Bassie
Date: 2005-12-01 10:32
Okay, having read around this for a while, it seems to me that there are seven notes in an octave for the following reason...
The Ancient Greeks built music round a thing called a 'tetrachord': two strings tuned a fourth apart (that's a ratio of 4:3 in frequency) and two strings inbetween.
Now take the top string, and add another fourth above this, and then another two strings to fill the gap. That gives two tetrachords and seven notes. The top note is now almost exactly a tone shy of the octave, so there's no point adding any more notes. Tune the strings in the middle of each tetrachord to intervals of tone-semitone-tone and you're there: a seven tone (minor) scale with five tone-sized intervals and two semitones.
Post Edited (2005-12-01 10:39)
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Author: Gordon (NZ)
Date: 2005-12-01 10:37
Shorthand wrote:
"....As an academic exercise, I looked and saw how hard it would be to wedge the important parts of a just scale into a 15 note Equi Tempered monster. It wasn't. I think any number of steps divisible by three will get you close enough...."
A local acquaintance spent much of his life experimenting with dividing octaves into different numbers of intervals. He also adapted instruments to play these, e.g. a guitar with all the frets removed and replaced with more, and and a modified autoharp. His preference was for an octave divided into 15. He wrote, played, and recorded such music.
I listened to some. It had a most refreshing quality - quite beautiful. For well-known tunes adapted to this scale, a preconditioned part of my mind would tell me that a melody note was out of tune, but immediately afterwards, the harmony would tell me that it was correct. A weird feeling. I really enjoyed this voyage into the unfamiliar.
There were also some very interesting dissonances available, well outside the scope of the scales we are used to.
This may well be the music of the future. A shame that all this guy's work will probably never be recognised. It was shunned by local university music professors. It seems they were just too entrenched in their traditional outlook.
That would render all our woodwind instruments obsolete, and Boehm would have to start again!
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Author: tictactux ★2017
Date: 2005-12-01 10:44
> The harp is an exception.
> There are 7 pedals, one for each note in the scale.
Gordon,
AFAICR Spock's Vulcan harp had no pedals (but a strange yet refreshing sound nonetheless)...
--
Ben
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