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Thread: Help a beginner out! Basic intervals qn

  1. #1
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    Help a beginner out! Basic intervals qn

    Hi guys,

    I've been trying to teach myself music theory from scratch, and I have a really basic question I want answered before I continue.

    Why is a fifth called a fifth? The name would seem to assume you have already invented a 7-note diatonic scale, so that a perfect fifth represents a difference of 5 "letters", ie. C-G. If you used a scale with less notes, the name wouldn't make sense. Is this right, or is there a reason for it being called a fifth, independent of how many notes there are in the scale?

    A related question, concerning interval classes: Why are some intervals known as "major" and others "perfect"? I understand that "minor" means one semitone less than a major interval, and "diminished" means one semitone less than a perfect interval, but I don't know what the difference is between "major" and "perfect". Also, is there a difference between "minor" and "diminished" intervals other than their relation to major and perfect intervals. In other words, instead of having minor 2nd's, 3rd's, 6th's & 7th's, could we describe them all as diminished (or vice versa for 4'th's, 5th's & 8ve's), or is there a reason to distinguish them?

    I hope these questions make sense to everyone. Your help would be appreciated

  2. #2
    Registered User Malcolm's Avatar
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    Quote Originally Posted by Jaffui View Post
    Hi guys,

    I've been trying to teach myself music theory from scratch, and I have a really basic question I want answered before I continue.

    Why is a fifth called a fifth? The name would seem to assume you have already invented a 7-note diatonic scale, so that a perfect fifth represents a difference of 5 "letters", ie. C-G. If you used a scale with less notes, the name wouldn't make sense. Is this right, or is there a reason for it being called a fifth, independent of how many notes there are in the scale?
    Why is a fifth called a fifth? Because it is the 5th note in the scale. Yes some scales have 8 notes, some have 5, but, most that you and I will be using have 7 notes. Now to the other question -- a 5 note major pentatonic scale is made of the 1-2-3-5-6 intervals of the parent's (1's) major 7 note scale. There is a chart below that should clear this up.
    A related question, concerning interval classes: Why are some intervals known as "major" and others "perfect"? I understand that "minor" means one semitone less than a major interval, and "diminished" means one semitone less than a perfect interval, but I don't know what the difference is between "major" and "perfect".
    I found all that major, minor, perfect 5, etc. to be confusing and not a big help so I just skipped over it. I'll let the other guys give you the reasons to know this. We have to name "stuff" so we know exactly what we are talking about, I found those names too confusing. And understand their functions using other means. But, do not let me keep you from knowing this. It can't hurt.
    Also, is there a difference between "minor" and "diminished" intervals other than their relation to major and perfect intervals. In other words, instead of having minor 2nd's, 3rd's, 6th's & 7th's, could we describe them all as diminished (or vice versa for 4'th's, 5th's & 8ve's), or is there a reason to distinguish them?
    The difference between minor and diminished - good question. A diminished chord is also minor, however, a minor chord is not necessarily diminished. If the 5th is flatted your moving toward diminished. Minor usually refers to the 3rd interval. The C major chord is made of the 1, 3 & 5 interval of the C major scale, or C, E, G. The Cm chord is made of the 1, b3 & 5 interval of the C major scale C, Eb, G. A major chord will have a natural 3rd interval and a minor chord will have a flatted third interval - always. Diminished, refers to the 5th interval. If it is a b5 or flatted 5 the chord made from it is said to be diminished, i.e. 1-b3-b5. Diminished chord are also minor so both a b3 and a b5 will appear in a diminished chord. If you raised the 5, i.e. #5 you augmented the chord. Copy the following chart, it will come in handy. http://www.smithfowler.org/music/Chord_Formulas.htm Those 9's, 11's and 13's are from the same scale. Their just notes in the next octave.

    The following charts may help.
    The following are generic numbers, i.e. the C major chord is made from the R-3-5 intervals of the C major scale. R = the 1 note. The F major chord is made from the R-3-5 intervals of the F major scale. Major chords have the same generic structure you just change the specific notes of the scale to get the specific chord or scale you want.

    Basic Chords
    • Major Triad = R-3-5
    • Minor Triad = R-b3-5
    • Diminished Chord = R-b3-b5
    7th Chords
    • Maj7 = R-3-5-7
    • Minor 7 = R-b3-5-b7
    • Dominant 7 = R-3-5-b7
    • ½ diminished = R-b3-b5-b7
    • Full diminished = R-b3-b5-bb7

    Scales
    • Major Scale = R-2-3-4-5-6-7 Home base
    • Major Pentatonic = R-2-3-5-6 Leave out the 4 & 7
    • Natural Minor Scale = R-2-b3-4-5-b6-b7 Major scale with the 3, 6 & 7 flatted.
    • Minor Pentatonic = R-b3-4-5-b7 Leave out the 2 & 6.
    • Blues = R-b3-4-b5-5-b7 Minor pentatonic with the blue note b5 added.
    • Harmonic Minor Scale = R-2-b3-4-5-b6-7 Natural minor with a natural 7.
    • Melodic Minor Scale = R-2-b3-4-5-6-7 Major scale with a b3.
    All the fog went away when I realized the structure of chords and scales are generic.
    All we need is a generic box pattern. Place the root and let the pattern bring the correct notes
    to our finger tips.
    Code:
    Major Scale Box Pattern
    
    E|---7---|--R(8)-|-------|---2---| 1st string
    B|-------|---5---|-------|---6---|
    G|---2---|-------|---3---|---4---| 
    D|---6---|-------|---7---|--R(8)-|
    A|---3---|---4---|-------|---5---|
    E|-------|---R---|-------|---2---| 6th string

    Here is a very good paper on music theory. http://www.billygreen.pwp.blueyonder...20Advanced.pdf Page 11 will have the notes in the major scales. Kinda important. C major scale has no flats or sharps, E major scale has a F#, G#, C# and D# in it so you have to take all that into account. Copy that and keep it handy. The first 20 pages take you through the basics. Expect to spend several months on those 20 pages. Then and only then venture into the next 60 pages.

    I also found this site to be helpful. Want to know the chords made from a specific scale - look here until you understand how to stack 3rds. Something you need to know, but, later. Right now just use these charts. http://www.guitar-chords.org.uk/chords-key-c.html And then of course ask specific questions here.

    Have fun.
    Last edited by Malcolm; 04-29-2012 at 12:47 AM.

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    A related question, concerning interval classes: Why are some intervals known as "major" and others "perfect"? I understand that "minor" means one semitone less than a major interval, and "diminished" means one semitone less than a perfect interval, but I don't know what the difference is between "major" and "perfect".
    The answer to this question lays in history and in the concept of consonance and dissonance. In the past (I believe before common practice period-before A.D. 1600) the 4th and 5th intervals were considered as perfect because their sound was the most pleasant to the ear, from the ones available.
    I understand your question because we always try to find the meaning of names in some kind of scientific background when, at least in this case, it is in music history. Obviously, there are scientific reasons on why these intervals were more pleasant than the others. If you are interested in that, try to read something about harmonic series and how is that applied to an instrument (http://en.wikipedia.org/wiki/Harmonic_series_(music).

    About the difference between majors and perfects, all have different nş of semitones between notes that form the interval. This means (once again we need harmonic series) that each of these intervals have their own frequency ratios. See http://en.wikipedia.org/wiki/Interva...nant_intervals and http://en.wikipedia.org/wiki/Interval_ratio. Basically, the simpler the ratio the more consonant (more pleasant or perfect) are the intervals.

  4. #4
    Registered User Malcolm's Avatar
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    Quote Originally Posted by rbarata View Post
    The answer to this question lays in history and in the concept of consonance and dissonance. In the past (I believe before common practice period-before A.D. 1600) the 4th and 5th intervals were considered as perfect because their sound was the most pleasant to the ear, from the ones available.
    Good explanation.

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    Good explanation.
    Thank you...as long as Jaffui finds it useful.

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    Just to add a few personal thoughts to what’s been said above -

    - it’s probably simplest to think of intervals (eg 5th) in terms of scales, and most musicians probably do think of it that way, but it’s not actually essential to have a scale in order to have the concept of an “interval”. An interval is just a musical separation or distance between any two notes. So whether you have a scale or not, a 5th for example is a distance of seven semitones. And you can, and everyone does, describe intervals that way in terms of distance in semitones or whole tones.

    Perfect intervals are those which are not normally thought of as becoming either major or minor sounds when they are raised or lowered by a semi-tone. So, eg, a 5th comes a flat-5th rather than a minor 5th.

    As rbarata says, much of this, and a great many things in music, are really of historic origin from a time when classical or pre-classical musicians preferred to describe all sorts of musical ideas in terms of subjective listening, or even in terms of religious ideas such as a “tritone”. So historically, a “perfect” interval was one that was regarded as sounding “perfectly consonant”, ie not at all dissonant or uncomfortable sounding … very subjective. As always, Wiki to the rescue -

    http://en.wikipedia.org/wiki/Interval_(music)
     
    To digress for a moment - what we now call “science” was really unknown until about 1800 to 1850 onwards. Before that time, up until about the time of Galileo c.1600, people tried to understand and describe things in philosophical terms which were strongly influenced by religious belief. That began to change when early philosophers such as Galileo began to realise that philosophy based on word arguments was, and is, in fact incapable of accurately describing the real universe, and that instead what was required was experimental observation and strict mathematical calculation

    … the rest is history (and also a source of much lingering angst amongst philosophers and philosophy students who are of course highly offended when scientist like Stephen Hawking say “philosophy is dead” … he means that, compared to science, philosophy is dead as a method of ever truly discovering and explaining real events in a real universe).

    The point being - a lot of the musical terms which we still use today, come from a time before modern science. So they don’t always make a great deal of sense in objective or logical scientific terms.

    On a more practical note - I found that practicing intervals was one of the best things I ever decided to do. That has given me a much better familiarity with finding my way around the fretboard in real time playing. By which I mean, when practicing I constantly find myself using visual recognition of interval shapes on the fretboard to locate the next note that I want to play, whether that note is a scale tone or a chord tone, or a voice-leading note, or a chromatic approach note etc.

    And the way I got into learning intervals and realising how useful that was, was simply that I wanted to learn how some guitarists used arpeggios more than others. So I began practicing a whole load of arpeggio patterns from a book (Guthrie Govan Creative Guitar vol-1), and then later I began to think about those arpeggio intervals when practicing scales and chords. And then later still, when practicing improvising either with scales or with methods targeting chord tones (eg triads and triad extensions ... see the two books by Garison Fewell) I began to find that visual recognition of those interval shapes was highly effective when trying to locate the next note that I wanted to target (eg in "voice leading").

    OK, that was digression I know, but I hope it was useful digression in saying why I think it’s so useful to learn/memorise interval shapes visually. And last thing, the reason I keep stressing “visually” is that most people think it sounds more erudite and correct if they say you should learn the intervals by their sound rather than by sight. But I found that visual recognition is actually far more important. So I’d recommend practicing that (eg starting from the patterns in Govan, then proceeding to improv. using the books by Garison Fewell, for example).
    Last edited by Crossroads; 04-29-2012 at 08:13 AM.

  7. #7
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    In case it's not clear from the above or the wiki article (all correct!):

    Intervals are "ordinal" numbers - ie counts rather than measurements. Or rather, they're counted first (in note letters, 1st, 2nd, 3rd, etc), then measured afterwards (quantity of half-steps).

    PERFECT INTERVALS
    Unison (same pitch)
    4th (5 half-steps)
    5th (7 half-steps)
    Octave (12 half-steps)

    So-called because they come in one standard size, perceived as strongly consonant. They were also the basic scale divisions in ancient modal systems.

    Other intervals come in TWO common sizes, MAJOR (larger) or MINOR (smaller).

    2nds (1 or 2 half-steps)
    3rds (3 or 4 half-steps)
    6ths (8 or 9 half-steps)
    7ths (10 or 11 half-steps)

    NB: the initial note count is still critical. Not every interval that measures (eg) 3 half-steps is going to be a minor 3rd - see below.

    When a PERFECT or MAJOR interval is ENLARGED by a half-step it's AUGMENTED;
    When a PERFECT or MINOR interval is REDUCED by a half-step it's DIMINISHED.

    Eg, you'll notice that none of the above intervals measures 6 half-steps. A 6-half-step interval would be either a "diminished 5th" or "augmented 4th".
    B-F is a diminished 5th, because it's a half-step smaller than the perfect 5ths B-F#, or Bb-F.
    F-B, meanwhile, is an augmented 4th, because it's a half-step bigger than the perfect 4ths F#-B or F-Bb.
    6 half-steps is often known as a "tritone" (3 tones, ie 6 semitones).

    B-F and F-B are INVERSIONS of each other.

    When a perfect interval is inverted it becomes another perfect one.
    When a major interval is inverted it becomes a minor one (and vice versa).
    When an augmented interval is inverted it becomes a diminished one (and vice versa).
    When you add the numbers they come to 9. (Eg, an inverted major 3rd becomes a minor 6th.)


    All the above interval types occur in the MAJOR scale (and its modes). A few others occur in HARMONIC and MELODIC MINOR.
    Eg, the A harmonic minor scale contains an "augmented 2nd" (F-G#) and its inversion a "diminished 7th" (G#-F) (which is where "dim7" chords come from ).
    It also contains an augmented 5th (C-G#) and its inversion a diminished 4th (G#-C).
    These intervals (like some others) are "enharmonic" with (sound the same as) more common ones, but shouldn't be confused with them. Eg, a diminished 7th is 9 half-steps, so sounds like a major 6th. G#-F sounds like Ab-F. But the note count is different (7 notes or 6), and this usually matters in a musical context.
    Sometimes, however, the ambiguity of enharmonic intervals is musically useful, eg the tritone. (The fact that G#-D sounds the same as Ab-D means a Bb7 chord can often stand for an E7 chord.)


    As Ian says, you don't need a scale to understand intervals. Music actually begins with intervals, and scales and chords derive from them. Eg, a "major" scale and "major" chord are defined by (named after) the 3rd interval from the root; likewise a minor scale and minor chord. (Although a perfect 5th is assumed in all cases.)

    Eg, some people say that a "major" scale is so-called because it contains major and perfect intervals only. Or - worse - that "major" intervals are so-called because they come from a "major scale". But that doesn't work for minor scales, which contain the same perfect intervals, plus a mix of major and minor intervals. The 3rd alone is what distinguishes them.

    Augmented and diminished triads are named after their 5ths. "Maj7" and "dim7" chords are named after their 7ths. In each case, it's the most distinctive interval that the chord is named after (they include other intervals too of course).
    Chord intervals are always measured from the root. Eg, the idea that a major chord is a minor 3rd stacked on top of a major 3rd - while true - is unhelpful. It's a major 3rd and a perfect 5th, both measured from the root; the interval between 3rd and 5th is irrelevant.

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    Thanks for the help everyone. It does seem to me that some of these definitions and meanings are just conventional, relics of the history of musical discovery. The problem is that when you go to a book of the fundamentals of music theory, they often don't go into the history of how these conventions came about, they just describe the system as we have it today. I think it makes music theory one of the hardest subjects I've ever tried to teach myself. Perhaps a really good book on the history of music theory would be helpful, but I haven't found one so far.

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    Thanks for the help everyone. It does seem to me that some of these definitions and meanings are just conventional, relics of the history of musical discovery. The problem is that when you go to a book of the fundamentals of music theory, they often don't go into the history of how these conventions came about, they just describe the system as we have it today. I think it makes music theory one of the hardest subjects I've ever tried to teach myself. Perhaps a really good book on the history of music theory would be helpful, but I haven't found one so far.
    Exactly!!! I also had, and sometimes still have, this difficultie. In fact I have the oppinion that music history (and theory history) is fundamental to theory students. At least makes our life easier.

    About the history book...I don't think you need one. The internet has plenty of good articles about it.

  10. #10
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    Quote Originally Posted by Jaffui View Post
    Thanks for the help everyone. It does seem to me that some of these definitions and meanings are just conventional, relics of the history of musical discovery. The problem is that when you go to a book of the fundamentals of music theory, they often don't go into the history of how these conventions came about, they just describe the system as we have it today. I think it makes music theory one of the hardest subjects I've ever tried to teach myself. Perhaps a really good book on the history of music theory would be helpful, but I haven't found one so far.
    The best view of theory, IMO, is it's like the grammar of a language.

    It's often referred back to the so-called "Common Practice Period" (CPP), which is a term covering the Baroque, Classical and Romantic eras of European music (approx 1600-1900).
    That's where all our contemporary music theory comes from, and the term is significant: "common practice" means just that. It was the kind of things that most composers chose to do, most of the time. IOW, the theory followed the practice, not vice versa.

    The grammar of a language is an attempt to discern rules in the way people speak. The people who speak the language don't learn it from reading books, they pick it up by ear as infants; probably not even conscious of the rules they're following. But rules of grammar that can be written in books matter for any foreigner wanting to learn the language.
    Same (more or less) with music. For most of us, as adults, music is a "foreign language". Not very foreign - we know instinctively when music sounds right or wrong (because we've heard it all our lives). But to actually become a musician - rather than just enjoy it as a listener - learning by ear alone is tricky and time-consuming. A "book of grammar" can therefore be handy, so we know the right order to put notes in, so that they "sound right".
    Also, just like a language, music can have dialects, accents and slang. Jazz and rock are like "local dialect", or "street slang" versions of "proper" (classical) music. They borrow some of the old rules and treat them quite liberally, but do have distinctive and important rules of their own (by which we identify them, of course).

    Naturally the Baroque composers didn't just invent everything from scratch - any more than dialect speakers invent their own language wholesale. They built on the previous Renaissance era, which itself grew out of the Medieval period. But the CPP was significantly different from previous periods, in that it represented the establishment of major and minor keys (12 of each) as the basis of all composition. The other phrase for the basic concept of the period is the "major-minor key system", or "functional harmony".
    We take the idea of major and minor keys for granted now, but it didn't exist before the Renaissance; it was essentially all modes before then. The notion of "key" (and "tonality") steadily grew through the Renaissance, refining and developing certain tendencies that had become increasingly common in the late modal period.

    But - to get back on topic! - the old ideas that the new CPP system inherited included the terminology of intervals, and of concepts like the "dominant", meaning originally the "reciting tone" of the old plainchant era - which was not always the 5th note of the scale, but was always distinct from the "finalis" or "final", the equivalent of our modern "tonic" or "keynote.

    As rbarata says, there are plenty of websites with overviews of music history - although I don't know any with a comprehensive history of how theory itself developed (eg when specific terms would have been first used).
    Here's one I like:
    http://www.midicode.com/tunings/index.shtml
    It's mainly about tuning and scales, but is good on the science and history of intervals.

    A good (very readable) book on music history - focussed on a handful of crucial inventions - is this:
    http://www.amazon.co.uk/Big-Bangs-Fi...5722725&sr=8-1
    http://www.howardgoodall.co.uk/presenting/bigtext.htm
    It was a TV series too, and you can find excerts on youtube -
    http://www.youtube.com/watch?v=XgQptKXzcTY
    along with other excellent programmes Goodall's done on aspects of music itself.
    http://www.youtube.com/watch?v=PnbOWi6f_IM
    Last edited by JonR; 04-29-2012 at 07:12 PM.

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    When I started learning this stuff, my goal was to find out why we have the peculiar arrangement of white and black keys on a piano. I thought if I understood that it would help answer a lot of more general questions.

    Now, I think I understand why we have 12 notes in an octave (at least, according to this explanation: http://thinkzone.wlonk.com/Music/12Tone.htm) but that doesn't in itself explain the particular layout of white and black keys. Why, for instance, don't we just have 12 notes going from A to L? The reason for that, as far as I can tell, is that that arrangement must have been invented after the 7-note scale was already prevalent, so that when we came to add more notes they had to be slotted in between the 7 we already had.

    I think the problem I had with the interval names is that they already assume a certain number of notes, but then my books talk about building the diatonic scale by stacking intervals on top of each other, and it felt a bit like a chicken-and-egg problem. We must have named the intervals after we had a 7 note scale.

    Anyway, I guess the names aren't that important. If my understanding of the arrangement of piano keys is basically correct, I think that understanding is good enough for now, and I can start thinking about moving into deeper waters

  12. #12
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    Quote Originally Posted by Jaffui View Post
    When I started learning this stuff, my goal was to find out why we have the peculiar arrangement of white and black keys on a piano. I thought if I understood that it would help answer a lot of more general questions.

    Now, I think I understand why we have 12 notes in an octave (at least, according to this explanation: http://thinkzone.wlonk.com/Music/12Tone.htm) but that doesn't in itself explain the particular layout of white and black keys. Why, for instance, don't we just have 12 notes going from A to L? The reason for that, as far as I can tell, is that that arrangement must have been invented after the 7-note scale was already prevalent, so that when we came to add more notes they had to be slotted in between the 7 we already had.
    Precisely .

    That site is a little misleading - not wrong, just doesn't tell the whole story - which I guess no website could anyway!.

    Here's my take on the history, as I understand it from the various bits and pieces I've read over the last 40 years or so. (I didn't study this at college, I'm just some guy on the internet, so don't take my word...)

    Our system of scale division goes way back to ancient Greece, where the only mathematical relationship they cared about was the ratios of 4th and 5th (not 3rds or all the rest). You may have heard of Pythagoras... he's the guy supposedly responsible for spotting that those nice sounds (octave, 4th and 5th) were produced by simple ratios; not of frequency, which they couldn't measure then, but of weights of metal bars, or lengths of string or pipe.
    He noticed that a ratio of 2:1 produced the octave, and 3:2 produced the perfect 5th. ("Octave" is from Latin, not Greek, for "8th", and that term probably dates from the middle ages, much later, when they wrote everything in Latin.)
    Because science, philosophy and religion in ancient Greece were basically all the same thing, he deduced from this that God must be a mathematician (nice to know, seeing as Pythagoras was one too... ); they also believed that the planets moved in exact circles in similarly simple proportional orbits, and actually made heavenly music while doing so: "harmony of the spheres". (I wonder what drugs they had in those days...)

    The Greek system was based on "tetrachords", which means "4-string" and refers to the lyre: a small hand-held harp tuned to a 4-note scale. The outer notes (as I understand it) would be a perfect 4th - Pythagoras's 4:3 ratio, an inverted 3:2. The inner 2 notes were found by ear, and could be set to various different points - because no one position for the smaller intervals sounded as "perfect" as the 4th, but different ones had different effects. Some would have matched our whole and half-steps (approximately), some would have been quarter-tones.
    Adding two tetrachords together (each with varying inner notes) produced the legendary Greek modal system. Ta-daaa... IOW, they all had 7 notes, even back then. (4+4 = octave, but of course the outer ones are the same note.)
    The names (dorian, lydian etc) referred to Greek tribes or regions, because the sounds of each one supposedly expressed the character of the region. If this sounds bizarre, imagine calling the blues scale "Mississippi Delta Mode", or phrygian dominant "Andalusian mode" (you might need to look those up), and you can see how it might make sense.

    Much later - about 600 AD - the Catholic church wanted to standardize their worship music, to help consolidate their power, and drew on a contemporary Roman writer named Boethius. He mistranslated some old Greek texts (either accidentally or deliberately), and the old Greek mode names got switched around so they referred to different scale structures. (A possible source of misunderstanding was that the Greeks spelled their scales from the top down.)
    The Greeks had more than 7 modes (thanks to quarter tones), but the Church chose just four: dorian, phrygian, lydian and mixolydian (as I say, Greek names, but not the same scales the Greeks would have known by those names). For nearly 1000 years, European church music was based on just these four - which initially did not transpose, so there was no dorian but D dorian, no phrygian but E phrygian, etc. (Although AFAIK they didn't use letters; they might have used Greek terms, alpha beta, etc., or possibly numbers.)
    They did make use of what were called "plagal" modes, which were the same four with a different range. Eg "hypodorian" ran from A-A instead of D-D ("Hypo" means "beneath") - but D was still the keynote, so it wasn't like Aeolian.
    "Gregorian chant" (named after Pope Gregory, who got the whole show on the road) is the expression of this early modal system.

    They used no harmony at all to begin with. Singers would sing either in unison or octaves. Eventually someone thought other intervals might be cool, but even then they only went for the perfect ones - 4ths and 5ths. (3rds and all the rest were considered "dissonant".) This was known as "organum".
    You might get some idea of why they were so strict when you remember the huge cavernous cathedral spaces they sang in. Voices in perfect intervals would blend into an ethereal, heavenly sound, almost sounding as one, but with a richer texcture than unisons or octaves. Other intervals were more difficult to tune intuitively by ear, so risked veering off into more impure sounds.
    Very slowly (over a few centuries), 3rds and 6ths became acceptable; but one interval - the tritone - was carefully avoided. "Diabolus in musica" they called it: not (IMO) because they believed it was genuinely evil, but because it sounded so unpleasant. Every time the harmonies risked encountering one - eg if one voice stepped up from E to F while another stepped down from C to B - they would alter either F or B to make either a perfect 4th or 5th. This is how F# and Bb were invented.
    It's important to realise they only did this at those rare occasions where there was a danger of a tritone. Otherwise they were quite happy with the notes ABCDEFG (in whatever mode).

    By 1000 AD, a system of "hexachords" had developed, which were 6-note scales in which one note (usually the 7th) could be altered (minor or major in our terms) for different effects. This meant that one could now transpose modes, to a limited extent, because the 7th could also be the 4th of another mode (if I've got this right ). There were "hard" scales - which had the raised note - and "soft" scales, which had the lowered one. The "H" symbol for hard eventually became our "natural" sign; but in Germanic Europe it's still the letter they use for what we call B natural. What they call "B" is the old "soft" B that we now call Bb. (Of course, our flat symbol is derived from the lower case "b" .)

    The guy who invented this hexachord system - or maybe just wrote up what was becoming common practice - was Guido D'Arezzo. He also invented the first type of music notation which didn't depend on you having heard the tune before. Previous notation systems - "neumes" - were rather like tab for voice: reminders of roughly how a tune went. You needed to be taught the tune by ear first, and then the neumes just helped guide your memory.
    Guido's system was truly revolutionary. He also invented a system of using parts of the hand to mean different notes, and pointing to them to teach singers (some of whom who might well have been illiterate). He supposedly astonished the Pope by teaching him to sing a tune he (the Pope) had never heard before. Naturally this meant new music could be taught accurately over big geographical distances, without the kind of "chinese whispers" errors that aural teaching naturally suffered from.
    http://en.wikipedia.org/wiki/Guido_of_Arezzo
    http://en.wikipedia.org/wiki/Guidonian_hand

    [to be continued]

  13. #13
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    [Part 2]

    Notation - though Guido could never have imagined this - also led to the incredible explosion of classical music, beginning with the Renaissance around 1400. It led to the cult of the composer. Before notation, a composer was a minor figure, more like a choir leader or arranger. He had to teach all his singers or musicians by singing or playing the tune himself. (This is probably one reason why music developed so slowly - it had to be kept simple enough to teach and learn by ear, and rules had to be kept strict so as to be easily understood.)
    With notation, he didn't even have to be a performer. He didn't have to even meet his musicians. He could give the music to a messenger on a horse, and musicians 100s of miles away could play it perfectly.
    Moreover, he could write for many more instruments together than were possible before, and much more lengthy and complicated pieces than before. Without notation, every line of music would need to be held in the composer's head simultaneously. Now each line could be written out, and others imagined - and then written out too.
    So the concept of the "orchestra" started to take shape: the bizarre notion of many instruments of different kinds playing together. Trumpets playing alongside violins (or rather viols)? whatever next!

    We're so used to this concept now that it doesn't seem odd. But it's unnaturalness survives in the notion of "transposing" instruments. Trumpets and similar instruments play in different keys from instruments like strings or keyboards. Even today, when an arranger writes for trumpet and wants the trumpet player to play the sound of a "C", he must write a "D" on the trumpeter's music.

    BTW, Guido also invented the notion of memorising a scale as "do re mi fa so la ti do" - although in his day "do" was "ut". The syllables are the first of each line of a famous hymn of the time which - usefully - started on a different scale note on each line. (see abovelinks)

    IOW, by 1000 AD, the idea of what we'd call a major scale as a primary scale was already (apparently) in place - although they didn't use such a term, and had no concept of "key" as we know it. Bizarrely (given this "ut re mi" idea), it would be 500 years before Ionian mode was accepted into the modal system.


    In 1547 a music theorist named Glareanus finally included the modes Ionian and Aeolian in a written treatise. Again this wasn't some bright idea he suddenly had: it would have been something that was increasingly part of the music practice of the day, such that it was due for official recognition. (Previously the church had regarded Ionian mode - our "major scale" - as somewhat vulgar, a folk scale beloved of those disreputable troubadours...might encourage... dancing [shudder], or other dangerously pleasurable activities... )

    That was really the beginning of the end of the modal era, because Ionian quickly took over as the scale of preference for the new concept of "harmony" - involving those previously unacceptable intervals of 3rds and 6ths as the new "sweet" sounds, and building the kinds of harmony that we would now call "chords" - although it was still some time before such things would be recognisable (to us) in music.

    As I said before, by about 1600 the "key system" was pretty much fully in place.
    They were still struggling with "Just Intonation", however: a system of scales built on pure intervals. Although in theory 12 keys were now available, JI meant that only a few keys could be in tune at any one time. The more you moved away from the key you'd tuned to, the more out of tune it became.
    The standard key was of course C major - being the relative Ionian mode of the old medieval ABCDEFG notes. (Ie, despite all the modernisation of the system, they still kept the old note allocations, pitch names and scale structure.)
    So, C major would sound fine. But when you went to D major - OK you could easily raise F and C to F# and C#; but the interval D-E was a different kind of whole step from C-D (the equivalent interval in key of C). This meant different keys truly had different characters - subtle differences but real ones.
    Singers and string players (and horn players to some extent) could adjust their intonation as they played of course, so it didn't trouble them too much. But keyboard players were screwed. Inventors tried to get round it with nightmare keyboards containing 19 notes per octave (ie two different keys for F# and Gb, etc). With hindsight it's amazing that equal temperament (ET) took so long to become acceptable. Although people lke Galileo had proposed it centuries before, it wasn't fully established - officially - until the early 20th century, although people had really been using it for some time before then.
    Bach had taken a major step in that direction with his "well tempered keyboard" - a system close to but not identical to equal temperament, but which still allowed all 24 major and minor keys to be played (the remote ones would sound different from ones close to C, but not different enough to be "bad").
    The truth is, of course, that ET is strictly speaking "out of tune". We tune all our instruments carefully and precisely to digital tuners, but we end up with something fundamentally out of tune in pure terms. Luckily most people's ears have a threshold of tolerance. A few cents out either way, and we tend not to notice (the clashes are really only in the upper overtones of the notes, very faint).


    Quote Originally Posted by Jaffui View Post
    I think the problem I had with the interval names is that they already assume a certain number of notes, but then my books talk about building the diatonic scale by stacking intervals on top of each other, and it felt a bit like a chicken-and-egg problem. We must have named the intervals after we had a 7 note scale.
    Again, yes.
    Quote Originally Posted by Jaffui View Post
    Anyway, I guess the names aren't that important. If my understanding of the arrangement of piano keys is basically correct, I think that understanding is good enough for now, and I can start thinking about moving into deeper waters
    Yes. If the above makes your head explode, feel free to ignore it (or is it already too late for that...?)
    Last edited by JonR; 04-30-2012 at 10:56 AM.

  14. #14
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    Quote Originally Posted by Jaffui View Post
    Hi guys,

    I've been trying to teach myself music theory from scratch, and I have a really basic question I want answered before I continue.

    Why is a fifth called a fifth? The name would seem to assume you have already invented a 7-note diatonic scale, so that a perfect fifth represents a difference of 5 "letters", ie. C-G. If you used a scale with less notes, the name wouldn't make sense. Is this right, or is there a reason for it being called a fifth, independent of how many notes there are in the scale?
    I'll add my newbie answer. What works the best for me is that a fifth is five different consecutive letters.

    FIfths can only be Augemented, 8 semitones, Perfect, 7 semis or Diminished, 6 semis.

    So if you take any one note and count up in semitones from the root you will find all of your 5ths.

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    Quote Originally Posted by rockabilly View Post
    I'll add my newbie answer. What works the best for me is that a fifth is five different consecutive letters.
    FIfths can only be Augemented, 8 semitones, Perfect, 7 semis or Diminished, 6 semis.

    So if you take any one note and count up in semitones from the root you will find all of your 5ths.
    Just in case it's not clear - instead of counting semi-tones, it's much more useful to recognise intervals by the geometric shapes they make on the fretboard ....

    ... the most obvious and useful 5th shape is - from your first note, go to the next higher string and go up a whole step - that shape is the most common 5th interval, and it's the one you see in all "power chords".

    But also - from your first note, go to the next lower string at the same fret position, that is also a 5th, but now it's a power chord inversion, ie the 5th is now in the base relative to your first note (as always, when doing that between the G string and the B string, you need to go up one extra fret on the B-string, because the B string is tuned one fret lower than the other strings).

    So in general - if you don’t already know what all the standard shapes look like for 3rds, 7ths, octaves and 5th's, then it's a huge help to learn those. Eg get a book which shows those shapes clearly ... or, you can always work these things out for yourself of course (but it's usually easier to start with the patterns in a book).

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