Conference napalm::commusic_v1

    !This program by Tom Janzen 12-89 calculates the Western doedaculpe
    !scale and prints it out in MIDI note numbers, frequencies, and note
    ! names.
    ! It is to run under VAX (tm) BASIC.
    ! The Western scale is a geometric series.
    ! An arithmetic series would not start with octaves as a division.
    ! you would simply count, 10 Hz, 20Hz, 30Hz, 40Hz, ....440Hz, 450Hz,
    !460Hz... etc. The effect would be that the intervals were too big in
    !the low register and very tiny in the middle and high register
    !(microtones).  I have played with prime-number frequency scale.
    ! It was interesting for 5 minutes.
    DECLARE REAL CONSTANT LOWC = 16.3516, A440 = 440.0
    DECLARE INTEGER CONSTANT NOTESPEROCTAVE = 12, TopnoteinOctave=11,TopOctave=7

    DECLARE REAL	PITCH
    DECLARE REAL	TemperedScale(0 TO TopOctave, 0 TO TopNoteInOctave)
    DECLARE STRING	NoteNames(0 TO TopNoteInOctave)
    
    RESTORE
    FOR Pitch = 0 to TopNoteInOctave
	READ NoteNames(Pitch)
    Next Pitch

    FOR Octave = 0 TO TopOctave STEP 1
	
	FOR Pitch = 0 TO NotesPerOctave-1 STEP 1
	    TemperedScale(TopOctave,Pitch)=LOWC*2**(((Octave*NotesPerOctave)+Pitch)/12)
            print "MIDI: ";Octave*NotesPerOctave + Pitch + 12;
	    PRINT "Pitch: ";NoteNames(Pitch);Octave;
	    print "Frequency: ";temperedscale(topoctave,pitch)
	NEXT Pitch
    NEXT Octave

DATA "C ", "C#", "D ", "D#", "E ", "F ", "F#", "G ", "G#", "A ", "A#", "B "
END PROGRAM 1

    The successive frequencies are
    
    	440 * 2**(1/12)    A
    	440 * 2**(2/12)    Bb
    	440 * 2**(3/12)    B
    	  ...
    	440 * 2**(11/12)   Ab
    	440 * 2**(12/12)   A'
    
    Please note that this is what's called an "even tempered" scale.
    There are other kinds of scales, e.g., the Pythagorean, where the
    frequency relationships are defined by ratios of small integers.
    Even temperament makes it possible to modulate into remote keys,
    without creating the need for lots of additional notes to "fill
    in the gaps".  It does so at the cost of some theoretical "sweetness"
    of the intervals.  Once you've been brought up on even tempered
    tuning (almost unavoidable these days), you may find, as I have,
    purer forms of tuning of only academic interest, not worth the trouble,
    and even unpleasantly "out of tune" (especially just or pythagorean
    7ths), ironic as that might be.  The slight "mistuning" of most
    of the even-tempered intervals results in a serendipitous "fattening"
    (chorus-like effect) of the harmonies. 
    
    This is an exponential (or logarithmic, depending on your perspective)
    progression.  It is based on the way the ear/brain system hears
    pitch.  Octaves are a factor of two apart.  This in itself argues
    very strongly for a nonlinear progression.  If the scale within
    octaves was linearly divided, one would expect it to be linear across
    octaves as well.  Otherwise you'd be constrained to one key.  Suppose
    I split the 440 Hz between A440 and A880 into 12 equal parts (why
    12?  Why not 8?  Only 8 notes in the diatonic scale, right?  But
    that's a different diversion).  Then each linear step is 36 2/3
    Hz.  So the B above A440 is 513.33... Hz, and the C above A440 is
    550 Hz.  But suppose I want to build a scale on B instead
    of A?  If I use the B at 513.33..., then the B an octave above is
    1016.66... (because the octave *has* to be twice the frequency;
    otherwise the ear will not perceive it as having the "same" pitch,
    just an octave higher).  So now I divide my B to B octave into 12
    parts, but the difference is now 513.33 Hz, not 440.  So the steps
    are now 42 7/9 Hz apart, rather than 36 2/3 Hz.  So the C (550 Hz)
    from the A scale is flat compared to the C (555 1/9 Hz) that the B
    scale wants.   And the problem just gets worse and worse and worse
    as I look at other scales I might want to build.
    
    Many physiological systems derive apparently linear perceptions
    from exponential physical relationships; loudness has a similar
    "nonlinear" dependency.
    
    What would a linear scale sound like?  It would sound weird.  It
    would sound "scrunched up" at the top; the bottom notes of the scale
    would seem too far apart and the top ones too close together.  And
    it would require an incredible proliferation of additional notes to
    accomodate modulations even to "neighboring" keys.                
    
    len.
    
    
    The exponential relationship between pitch and frequency is fortunate,
    as it provides some physical basis for harmony.  E.g., the fifth,
    at 2**(7/12) in the even tempered scale, is alm 

    As I started to say in my previous reply, before I got sidetracked,
    one other benefit (?) of the exponential progression of frequencies
    in scales is that it provides some physical basis for harmony.
    Why is an octave a nice sounding harmony?  It's been conjectured
    that it's because the harmonics of the two notes line up nicely;
    the harmonics of the octave correspond to every other harmonic of
    the unison.
    
    Similarly for the fifth, at 2**(7/12) relative frequency.  It turns
    out that 2**(7/12) is very very nearly 3/2, the pythagorean value
    of the fifth.  As a result, the harmonics of two notes a fifth apart
    also line up pretty nicely.  This would not be the case if the
    octave were linearly divided.
    
    len.
    

    re the last 2 - len, nice summary on the development of the even-
    tempered scale.  Tom, I'm somehow not surprised you've done computer
    experiments in this area.  Congratulations.
    
    Frohliche Neue Jahr, everybody !
    
    remember, Drink and Mow, Lose a Toe !
    
    karl

    	To carry on from where Len left off.
    
    	The even tempered scale has not always been in existence.  In fact,
    in Bach's time there were several tuning scales in use.  One of these
    tuning system had all octaves and fifths tuned perfect.  But of course
    this only works up to a point and the number of keys available to play
    in is restricted.
    
    	Another system often employed was to tune major thirds perfect.
    In fact the organ which Bach composed a lot of his works on was tuned
    to this system and in order to reproduce the effects and moods he was
    trying to convey it can be argued that one needs to tune the
    performance instrument accordingly. 
    
    	For example, on an instrument tuned as above, key signatures with
    no or only a few sharps or flats sound more in tune than even tempered
    scale (sweeter).  Key signatures with lots of sharps and flats sound
    more out of tune and discordant than even tempered.  Bach apparently
    used this in some of his religious works such as passion chorales and
    the first sections are often in keys with not many sharps or flats and
    then modulating to keys with many sharps and flats where he (Bach) was
    trying to convey feelings of pain and anguish.
    
    	There is an organ in a church near our office which is tuned to
    this system.  It is totally inappropriate to the needs of the church
    and is really of only academic interest, but to hear the difference the
    tuning can make to a particular piece of music is quite something.
    
    	I dont think there will be any moves away from the even tempered
    system of tuning that we have now, but it is interesting to note that
    we may not be hearing the works of some past composers as they were
    intended.
    
    David Bruce

    To elaborate a little further on Bruce's last remark - one inevitable
    consequence of even temperament is that all keys are rendered
    equivalent, in the sense that the pitch relationships (frequency ratios)
    between the notes of the scales are independent of key.  Even tempered
    keys differ only in absolute pitch, and which notes of the chromatic
    (12 tone) scale form the key's diatonic (8 tone) scale.  So the
    cost of being able to modulate to remote keys (without requiring
    dynamically retunable instruments, a possibility that computer based
    instruments create) is that these remote keys do not differ in
    any fundamental way from the closer keys - the notes are different,
    but the relationships among those notes (the basis for our perception
    of harmony) remain unchanged.                                
         
    len.
    

    
    ...which is why I've been trying to figure out how to build a Scalatron 
    keyboard cheaply, and with velocity and pressure sensing.
    
    [for those of you who aren't "into" such things, the Scalatron keyboard
    has many more notes per octave- and also has the interesting 
    property that a given chord (say, the minor seventh) always is 
    fingered the same way (for a given inversion).  No more screwing
    around counting half-steps, and no more major thirds that sound
    broken]

    I can't resist throwing in my 2 cents worth on this. I believe that
    what makes combinations of notes "chordant" is the degree to which
    their harmonics mesh. Dischord is marked by lots of beat frequencies
    among the notes' harmonics.
    
    If you run a simple organ-like patch through a fuzz box and play around
    with what notes fit together well, I think you'll find that the
    traditional, music theory of octaves, fifths and forths being "most
    pleasant" does in fact hold. I think the fuzz box accentuates the cross
    modulation that human hearing does automatically (perhaps subliminally)
    for us.
    
    I also think it's important to think of the equal-tempered scale as a
    human invention. It's an approximation of the "pure" intervals of
    fifths, forths... it gets pretty bad at thirds and sixths, and awful at
    sevenths. But we're all trained to hear these approximate intervals as
    "correct". It's just part of the game.
    
    I think I read somewhere, and I believe I have heard it happening:
    people playing intruments that require fine pitch adjustments (like
    violins, trombones, and of course, singing) will actually drift away
    from the equal-tempered pitches toward the "pure" intervals. I think
    barbershop is particularly marked by this effect. I also think that's
    why some people can't stand that style - it doesn't conform to the
    normal "correct" pitches of equal-tempering - and so it somehow sounds
    wrong.
    
    It would be a neat experiment to take an existing equal-temper
    composition and map it - not simply to a just-intoned scale - but
    rather, craft the exact frequency of each note in the context of the
    others at each point in time. Perhaps it would sound smoother. Perhaps
    it would sound stranger? Hmmmm... compute each note's pitch bend ...
    sounds like another project...

    re: .7
    
    I saw and heard the Scalatron at Uconn back in '75 or so. It ruined my
    hearing for several days - nothing sounded quite right. I think it had
    31 notes per octave, and more than one button for each note. The guy
    demoing it (forgot his name) played a few pieces that were beautiful,
    and then played a really harsh song - that turned out to be in our
    familiar equal-tempered scale. It sure was disorienting. He could
    program several different tunings into it.
    
    I think Roland has a keyboard out that has several different tunings
    built in, probably for educational use. I saw it around at least a year
    ago, don't recall it's model number. It was a stand alone unit, built
    in amplifier, made to look like a strangely fore-shortened baby grand.

    I have built a just synth (just took it apart for good) and used
    computers (including pdp11) to generate a variety of tunings, but just
    to do it for fun. 
    I think dissonances are most pleasant.  Especially the piano secondal
    chords entering under the strings in about the middle of Carter's
    second piano concerto.
    to each their own
    Tom

    Harry Partch did a lot with the idea that it is consonant if the
    harmonics match and dissonant otherwise.  He invented a scale to take
    advantage of this effect.  It had 43 steps per octave.  Among the
    things you CANT do with a scale like that is random transpositions.
    But there are other neat things you CAN do.

    
    I've been playing around with building a Scalatron-like MIDI controller
    for at least a year now (along with a garagefull of other ideas that
    don't get far).  
        
    Were there any published papers on the Scalatron design?  I know it had
    interesting fingering properties- all  similar chords were fingered the
    same way (i.e. the same relative  finger positions always made a major
    chord, a different set of  relative positions made a minor, etc.) and
    the fingering properties also automatically gave a correctly tempered
    intonation. (i.e. obeying the real integer ratios, not the
    approximations that a 12-tone scale forces on us) 	
        
    	-Bill

    Re: Note 2220.11 by KOBAL::DICKSON

>    Harry Partch did a lot with the idea that it is consonant if the
>    harmonics match and dissonant otherwise.  He invented a scale to take
>    advantage of this effect.  It had 43 steps per octave.  Among the
>    things you CANT do with a scale like that is random transpositions.
>    But there are other neat things you CAN do.

    Yes, saw a concert of his music once (which I think I mentioned some
    time back maybe).  WOW!  
    
    Maybe non-equal-temperament sounds "out of tune" to most people, but
    I've gotten rather used to it and find it a refreshing change of pace. 
    The Wendy Carlos' _Beauty_And_The_Beast_ album is all non-ET, as is
    nearly all Terry Riley (like _A_Rainbow_In_Curved_Air_, for example, 
    but not _In_C_).  Also all gamelan and Indian music.  Just remember
    that it's your own cultural habits that cause you to regard these
    tunings as "off" -- the tunings themselves are not to blame.  It's a
    habit you can break, though.
    
    Those of you who own Kurzweil 1000-PXs or similar can painlessly dial
    up a number of different tunings without having to plot out all those
    cents-corrections.  They do take a shortcut in that the retuning spans
    only one octave and is simply doubled or halved from there.  Thus there
    are no "stretch" tunings or tunings having other than 12 notes per
    octave.  But it is a nice and easy introduction to these tunings if
    you're interested.  
    
                                			Mike