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#1075257 - 08/25/04 07:58 PM INTERVALS -- Keeping a safe distance between you and your notes! (I)
Matt G. Offline
3000 Post Club Member

Registered: 06/22/01
Posts: 3789
Loc: Plainfield, IL
A musical expression consisting solely of a string of individual notes is referred to as monody. Monody was probably the first musical (as opposed to rhythmical) expression made by humankind, as it is the sound created by one person singing alone. Monody also carries the idea of melody, a monodic sequence of notes that is considered pleasant. In singing (or playing) a melody, one often jumps from one note to another, not a simple series of half steps. The distance between each successive note in a melody can be measured, and we call these measurements intervals.

Monody was fine for a long time, apparently, but it eventually gave rise to something new when a second singer or player came along and performed a different melody at the same time as the first. This is called polyphony (from Greek meaning "many sounds"), and nearly all of the music we enjoy today is polyphonic (not that monody doesn't still exist!). In a polyphonic musical expression, with two or more notes sounding simultaneously, we can also measure the relationship between these concurrent notes using the same "ruler" we used to measure jumps in a melody: intervals.

Let's look at the following example in a polyphonic style, where the basic intervals between the lower and upper notes are given:



You probably noticed that the odd-numbered intervals (unison being 1) are all in spaces on the staff, while the even-numbered intervals (octave being 8) are on the lines. This pattern will always hold true when the lower note is in a space on the staff; it is, naturally, reversed when the lower note is on a line. The interval holds true regardless of which note one picks as the starting point. One could say that C is a fifth above F, or that F is a fifth below C. The direction doesn't matter because the distance remains the same.

When the mathematician/musicians of ancient times considered the relationship between various notes, they found notes that had a very simple ratio (in terms of frequency or string length) between them, and others that had slightly more complex ratios. They referred to the first set of intervals as PERFECT, and the others as IMPERFECT.

The perfect intervals are unison, fourth, fifth and octave. The imperfect intervals are second, third, sixth and seventh.

In addition to their mathematical ratios being of different complexity, the other real difference between the two types is that perfect intervals have only one basic quality or "flavor," whereas the imperfect intervals have two basic qualities: major and minor. Why is that? Well, let's look at the intervals between the notes in a major scale and you'll see why. We'll just pick out some random intervals between notes in the C major scale:



The interval between C and E is comprised, essentially, of 4 half-steps. Now, let's look at another:



The interval between E and G is comprised of only 3 half-steps. Hmm, how do we reconcile that discrepancy? Well, the third between C and E is bigger, a.k.a. MAJOR. The third between E and G is smaller, thus MINOR. (Just so you know, these are exactly what major and minor mean in Latin!) What about the other imperfect intervals? How about seconds:



This second is comprised of 2 half-steps, whereas



this one is only 1 half-step. Here, then we have two types of seconds, major and minor. Sixths, you ask? How about



this one between D and B comprised of 9 half-steps, compared to the sixth between E and C



that is comprised of only 8 half-steps. Now, just for the sake of argument, compare the seventh between C and a B above made of 11 half-steps, with the seventh between D and and a C above made of 10 half-steps. Once again, here we have a major and a minor seventh.

(CONTINUED IN NEXT POST)
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#1075258 - 08/25/04 07:58 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
Matt G. Offline
3000 Post Club Member

Registered: 06/22/01
Posts: 3789
Loc: Plainfield, IL
_________________________
Sacred cows make the best hamburger. - Clemens

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#1075259 - 08/25/04 08:42 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
signa Offline
8000 Post Club Member

Registered: 06/06/04
Posts: 8483
Loc: Ohio, USA
although some of those i seem to recall, it is after all a great summarization of the subject, thorough and interesting.

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#1075260 - 08/25/04 11:31 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
TimR Offline
3000 Post Club Member

Registered: 08/17/04
Posts: 3201
Loc: Virginia, USA
Matt,

Excellent technical discussion.

But philosophically, I disagree with mixing the terms perfect and imperfect with major and minor.

Yes, I know, it is common practice and routinely considered correct. I still disagree.
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#1075261 - 08/26/04 05:11 AM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
cathys Offline
500 Post Club Member

Registered: 04/10/03
Posts: 917
Loc: Virginia
The new lesson is here
Thanks Matt!!

Cathy

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#1075262 - 08/30/04 10:52 AM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
Matt G. Offline
3000 Post Club Member

Registered: 06/22/01
Posts: 3789
Loc: Plainfield, IL
So, has everyone fully digested this lesson? Answers coming tomorrow!

P.S. - Next topic: THE "OTHER" MINOR SCALES -- As if one weren't bad enough! (II)
_________________________
Sacred cows make the best hamburger. - Clemens

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#1075263 - 08/30/04 12:16 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
Liesle Offline
Full Member

Registered: 06/25/04
Posts: 192
Loc: Southern Illinois
Matt - thank you. \:\)
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Liesle

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#1075264 - 09/01/04 07:09 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
Matt G. Offline
3000 Post Club Member

Registered: 06/22/01
Posts: 3789
Loc: Plainfield, IL
OK, in the first exercise, you were to provide the second note to form the interval. The blue notes are the ones you should have picked:



In the second exercise, I asked for the interval shown between the (black) notes, then provide the notes and interval for its inversion. The green notes shown are the inversion, and each interval is shown below the staff:



So, now that you have the answers I would have expected, what did you think? Was this a simple or difficult lesson? Any questions?
_________________________
Sacred cows make the best hamburger. - Clemens

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#1075265 - 09/01/04 08:56 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
cathys Offline
500 Post Club Member

Registered: 04/10/03
Posts: 917
Loc: Virginia
I liked it - I got everything right - (Oh, I can't tell a lie. I got a little messed up on the double augmented one in the second exercise; I had it that way to begin with but then started rethinking myself when I was counting steps and tripped and missed a few - I did have the notes right, though.)
Well I think this was about right for a Level I lesson.
I've filed the half step cheat sheets and the inversion charts until these things become second nature.

Again Matt THANK YOU!!

Cathy

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#1075266 - 09/02/04 02:40 AM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
RKVS1 Offline
3000 Post Club Member

Registered: 07/07/01
Posts: 3192
Loc: Topeka, Kansas
Happy to say I got them right the first time with the exception of example 1 #4, which I initially read as Ab instead of Gb, for some reason. And my initial burst of rage at the mixture of flats and sharps in example 2 #4 eventually subsided. \:\)

The rule of 9 is very handy and the inversion rules are easy to remember. I've used them before, but I'm glad you gave me an opportunity to check them out again with a possibility of feedback. (nice use of colors in the answers, btw .)

Just to rephrase, as a double-check of my understanding.
"Perfect intervals (IV, V and I ) Do NOT have minor designations, only diminished or double-diminished, while 2,3,6,7 use "minor" as the label for the first 1/2 step lowering. "

Is it incorrect or just sloppy to use the term "major 4th" or "major 5th"? i.e should the phrase "perfect 4th" or "perfect 5th" be used exclusively or is it just kind of a "shrug" ?

I'm assuming the term "Tri-Tone" is a one-of-a-kind usage, and nobody uses the phrase "diminished" or "augmented" tri-tone".


Matt, you do an excellent job of writing these lessons. They are concise, precise and comprehensive. (Well, the "not leaving anything important out" part, I'm taking mostly on faith. \:D )


Bob

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#1075267 - 09/02/04 04:01 AM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
TimR Offline
3000 Post Club Member

Registered: 08/17/04
Posts: 3201
Loc: Virginia, USA
Matt does indeed do an excellent job of presenting thess lessons and I hope my earlier comment about terminology was not taken as a criticism. It's just a reflection of my occasional pedantry.

Perfect intervals are not called major or minor. They predate the existence of the church modes and the "minor" distinction by a few hundreds or thousands of years. To my mind that is one reason not to use the term perfect. It does not add information the way adding major or minor does. Fifth is completely explanatory, third is not. But, sigh, that's just me. Another reason not to use perfect is that they're not. Perfect intervals would be beatless. Modern fourths and fifths are not, in equal temperament. Even if they were, on any but the limpest string the overtones would not be. However, ignore me and my obsessions; perfect fifth is considered correct, major fifth is definitely not, fifth by itself will be understood and I recommend it but most people are going to call it perfect fifth.
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#1075268 - 09/02/04 09:09 AM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
Matt G. Offline
3000 Post Club Member

Registered: 06/22/01
Posts: 3789
Loc: Plainfield, IL
WARNING: Really boring, pedantic technical discussion for anyone interested follows:

TimR is, in fact, quite correct in his assessment of using the term "perfect" as a descriptor for fourths and fifths. The perfect intervals defined so very long ago have very simple ratios (1:1 for unison, 2:1 for an octave, 3:2 for the fifth, 4:3 for the fourth) and using this mathematically pure tuning would be (at least theoretically) beatless. In this just intonation[/b] scenario, the fourth and fifth are the perfect internal intervals and (again theoretically) reside outside the realm of half-step counting and the assigning of numerical names to intervals. It was only after the development of the early 6- and 7-note scales that these intervals were given their scale positions and numerical names. Even so, there was really no need to describe any fourths or fifths -- at the time, all fourths and fifths in use were the same, identical to the perfect intervals defined previously.

This system survived quite well for centuries, until the rising acceptance of the leading tone and the prototype of the major scale came into existence. Many theoretical debates swirled around the use of the "raised" seventh scale degree, and there were holdouts insisting that it was an artificial note well into the Baroque! (As an historical footnote, this explains the still-extant use of the letter B[/b] in German notation for what we consider Bb, and using H[/b] for what we call a B natural.)

Part of this may be attributable to the use of the circle of fifths as a tuning method. Starting from a known pitch, one could (relatively) easily tune up and down in fifths, then tune all of the octaves to correspond. However, if one does the mathematical calculations, the major seventh above the base tuning note is discernably out of tune. At the time, this was considered a flaw not of tuning, but of the theoretical role of the raised seventh. (Using the circle of fifths, note that it takes 5 fifths upwards to get to the major seventh, whereas the minor seventh is only 2 fifths downward.) Later, when it was found more pleasing to tune this major seventh by calculating its position as the major third above the dominant, a whole new world of musical possibilities opened up. Unfortunately, with those possibilities came a problem -- the tritone.

The poor tritone didn't quite fit into the highly-structured world of the fourths and fifths that were in common use. The interval between scale degree 4 and this "raised" scale degree 7, in written form, looked like a fourth or a fifth (depending on the inversion) but was not a perfect interval. Some theoreticians agreed to refer to the tritone as an "augmented" fourth or a "diminished" fifth, while others felt this was a needless, overprecise nicety, preferring to call it simply a tritone, since the distance between was always the same value, regardless of inversion.

Those who adopted the augmented fourth and diminished fifth terminologies later found it helpful (if not a bit redundant) to refer to the unmodified, original intervals as "perfect" in an attempt to differentiate them from these newfangled interlopers. But, for those who considered the tritone an entity altogether separate from fourths and fifths, tagging a fourth or fifth as "perfect" bordered on the heretical.

In this discussion, we've adopted the "perfect" moniker for fourths and fifths just as a demonstration that the original perfect intervals were never ambiguous in terms of major or minor. In many ways the terminology is incorrect, but as tonal, chromatic and, later, atonal music flourished along with equal temperament, the rational objection to use of "perfect" fourths and fifths becomes less meaningful.

The purists among music theoreticians can still insist, and rightly so, that a fourth is a fourth, dagnabbit!, and a fifth is just a fifth needing no modifier. Whichever method one subscribes to, it is considered a central part of music theory to understand both viewpoints intimately.
_________________________
Sacred cows make the best hamburger. - Clemens

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#1075269 - 09/02/04 11:31 PM Re: INTERVALS -- Keeping a safe distance between you and your notes! (I)
TimR Offline
3000 Post Club Member

Registered: 08/17/04
Posts: 3201
Loc: Virginia, USA
For my part, when singing in my church choir, I expect to recognize and sightread a fourth or fifth without difficulty. (Yes, there really is direct practical application to everything Matt says.)

But the sinister tritone - Agggh! It's only half an octave, it should not be that hard. But I am not alone in missing the tritone interval almost every time. It's almost as if it is different on some physiological level.

Thanks again for the lesson Matt, much appreciated.
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gotta go practice

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