intervals

[Clyde Hollinger]

OK, since Dave did such a good job with this, I will not be redundant.  Those I
told I would write an explanation, read Dave's stuff.

Regards, Clyde

Dave Nereson wrote:

> >> 2. Like it or not, we are tuners... and our profession historically has
> had more to do with these intervals than not. And isn't it really only in
> the relatively narrow ET world that these intervals are wholly
> irrelevant?
> 4.  In fact one could go so far as to say "we" invented those diverse
> intervals.... all those hundreds of years back through music history.
> Why then should we be so willing to forget such an easy part of our
> heritage?<<
>
>     Yes!  I agree.  When teaching beginning tuning, I spoze it's alright to
> refer to all the black keys as sharps, to minimize confusion.  But for
> deeper understanding, a tuner should be aware that when tuning a Major
> third, s/he's tuning an interval that's vibrating in a ratio of roughly 5 :
> 4 and if it gets expanded enough, will approach 4 : 3 and start sounding
> more like a fourth.
>
> (The rest of this is for those who would like a little music theory about
> intervals;  skip if you know this stuff. . . .  It may seem off topic, but
> it's really not.)
>
>      But when it does become a 4th, you can't spell it with C and E anymore;
> you have to use C and F, even though E# is "the same as"  F natural.  (It's
> really not, but in ET it is).  Counting notes, or tones, from C up to F, and
> with the first one as "one", and not "zero" (since the tonic or fundamental
> of the scale is considered "one", not "zero"), it's four notes (white ones,
> or letters of the alphabet) from C up to F.  So C up to F is a fourth, no
> matter how many sharps or flats are in front of those letters.  And C up to
> E is a third, because C-D-E is three notes, or letters of the alphabet, in
> order.
>     Intervals can be Perfect, Major, minor, diminished, Augmented, doubly
> diminished, or doubly augmented, the last two being rare.
>     And intervals can be defined by how wide they are in number of steps and
> half-steps.  A whole step, or whole tone, is from one letter of the alphabet
> to the next, except B & C and E & F, which, we know, are only a half-step,
> or half-tone apart.  B to C is two letters, so it's a second, but it's a
> minor 2nd, the smallest interval of our western diatonic scale. Same with E
> & F.  All the other adjacent letters (A & B,  C & D,  D & E,  F & G,  G & A)
> are all whole steps, so they're all Major 2nds.  So A up to B is a Major
> 2nd,  A to Bb (flat) is a minor second,  A to Bbb (double flat) would be a
> "diminished second" and is an absurd interval because it's not an interval
> anymore-- it's a unison.  But a unison would be two A's or two B's, not an A
> and a B, which has to be called some kind of 2nd.  A to A is of course a
> unison.  A to A# would be an "augmented unison", another absurd interval
> because if one of the A's is sharped, it's not a unison any more.  But
> because of the way it's spelled, it can't be called a minor 2nd, because two
> A's are used, rather than A and B.
>     About the only intervals never used in tuning, or at least very seldom,
> are minor and Major 2nds, the Major 7th, and the tritone, or augmented
> 4th/diminished 5th, because they beat so fast, it's hard to use them for
> "smoothness-of-progression tests".
>
>     The most commonly used intervals used in music are:  m2,  M2,  m3,  M3,
> P4,  A4,  D5,  P5,  A5,  m6,  M6,  D7,  m7, M7, and P8 (omitting intervals
> wider than an octave for now).    [m - minor;  M - Major;  d - diminished;
> A - augmented;       P- Perfect]
>
>     Only Unisons (U), sometimes called Primes, 4ths, 5ths, and Octaves (8)
> can be Perfect.
>     Only 2nds, 3rds, 6ths, and 7ths can be Major or minor.
>     Any interval can be diminished or augmented or doubly diminished or
> augmented, but in some cases "absurd" intervals are produced.
>     To use the example of the 3rd and 4th:  C up to E is a Major 3rd.  C up
> to Eb (flat) is a minor 3rd.  C# to Eb is a diminished 3rd  because the
> minor 3rd was made even smaller (narrower, or diminished) by sharping the C.
> It now sounds like a 2nd but has to be called a 3rd if it's spelled with C
> and E.  C up to E# or Cb up to E would be Augmented thirds because a Major
> third was made even wider by sharping the top note or flatting the bottom
> one.  They sound like 4ths but have to be called 3rds because of the letters
> they're spelled with.  C to E# sounds like C to F, but the first is an
> Augmented 3rd and the second a Perfect 4th.  Intervals that sound the same
> are "enharmonic".  If C to F is a Perfect 4th (P4), then C to Fb is a
> diminished 4th, C to F# is an Augmented 4th, and C flat to F# would be a
> "doubly Augmented 4th" because it's made another half-step wider by flatting
> the C, but still spelled with C and F (C-D-E-F;  1-2-3-4).  It sounds like a
> 5th, but to be correctly spelled according to the rules of music theory, it
> has to be spelled with C and G  (C-D-E-F-G; 1-2-3-4-5) as Cb to Gb, or with
> B and F (B-C-D-E-F; 1-2-3-4-5) as B and F#.
>     C up to C is an octave.  C up to Cb would be a "diminished octave" (an
> octave made smaller by a half-step -- another "absurd" interval) but if
> spelled as C up to B natural, you'd call it a Major 7th, even though they
> sound the same.  C to Bb is the minor 7th;  C to Bbb (double flat) the
> diminished 7th (although many times on sheet music, especially pop music,
> it's written as 'A'), which sounds the same as the Major 6th  (C up to A).
> C to A# would be an Augmented 6th, which sounds the same as a minor 7th.
> But C to A is 6 letters of the scale and C up to B is 7.   If C up to C is
> an octave, then Cb up to C, or C up to C# would be an "Augmented octave",
> another absurd interval, but C up to Dflat would be a minor 9th, C to D
> natural the Major 9th, and C to D# an Augmented 9th, sounding the same as a
> minor 10th, but the 10th has to be spelled with C and E.  C4 up to Eb 5
> would be a minor 10th.  C4 up to E5 a Major 10th.  And so on.  A flat up to
> C is a Major 3rd, but if it's spelled G# to C, it's a diminished 4th;  if
> spelled G# to B#, it's a Major 3rd, and if spelled A flat to B#, it's a
> doubly Augmented 2nd.  They're all enharmonic (sound the same but spelled
> differently).
>
>     So it's all in the spelling, either as done here, or on music paper when
> placing notes on the staff.
>     But, tuner-to-tuner, I guess we have license to call them all sharps,
> since it's a tradition, makes it easier, and we "know what we mean".
> Sincerely,  David Nereson, RPT