intervals

[Dave Nereson]

>> 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