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
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