Cents: was help with bad tuning...Ric B. Eastern Scale?

[Stéphane Collin]

Hello Ric.

----- Original Message ----- 
From: "Richard Brekne" <Richard.Brekne@grieg.uib.no>

>
> Grin.... Sorry, that should read 7 in all.  Made the same mistake as Don 
> did with his 1300 cents.
>
> 6 steps doesnt finish the octave. 6 Different notes yes... but 7 to 
> complete the octave
>

I don't know what is a step.  From your posts, I guess it is the interval 
between two adjacent notes in a diatonic scale.  From other posts, I guess 
it is equal to two half steps, one half step being equal to one semi-tone. 
Sure, both suspected definitions don't match one with the other, because, as 
Dave said, the interval between two adjacent notes in a diatonic (may I add 
heptaphonic ?) scale is not necessarily equal to two semi-tones.
So, in our westernly standard scale figured by the white keys of a piano 
(which is, when played from C to B, a so called major scale, or ionian 
mode), there are 7 different notes (which figure 7 particular sounds with 
each a different pitch, knowing that all pitches who have their frequencies 
in a ratio of positive or negative powers of 2 bear the same note name) (by 
the way, did I understand these are your steps ?), defined by the very ratio 
of their frequency and that of a reference note giving it's name to part of 
the scale name.  The choice of those ratios, who uniquely defines the very 
scale, proves to be purely cultural, and in reasoned nomenclature gives the 
other part of the scale name.  Only in equally tempered scales do those 
ratios coincide with some whole multiples of the twelfth root of two, 
because historically our western culture found convenient to theoretically 
divide the octave (the interval between two pitches whose frequencies ratio 
is 2) in 12 (logarithmically) equal intervals (this was a result of building 
the same scale on different notes of the first reference one, and trying to 
make them all fit together).  This value (without unit) twelfth root of two 
is what we should call an equally tempered semi-tone (as other differently 
tempered semi-tones occur in other places).  The pattern of intervals 
(expressed in semi-tones) which defines the western standard major scale is 
: 2,4,5,7,9,11 (this means, D is 2 semi-tones away from C, the reference ; E 
is 4 semi-tones away from C, F is 5 semi-tones away from C, etc.).  You can 
also say, like Dave did, that D is a whole tone (2 semi-tones) away from C, 
and E is a whole tone from D, and F is a semi-tone from E, and G a whole 
tone from F.  Note that 6 intervals define 7 notes (hence the 6-7 
confusion).  Note again that two notes which form an octave bear the same 
note name, and that 7 (not 8) notes define our standard scale (hence the 7-8 
confusion).  The process of building a similar scale (that is, replicating 
the same pattern of intervals) on all the notes of the reference scale lead 
to use some pitches which are not in the original reference scale.  But 
because of the convenient division of the octave in 12 equal semi-tones and 
because of the artificially simplified pattern of intervals in the standard 
major scale, all notes generated by this process happen to share only 12 
different pitches.  Again, this is only true thanks to the simplification, 
unique to the western music of the second half of the second millenium.  But 
this leads to another confusion about the term "note" , which was uptill now 
one of the 7 possible pitches in a standard scale.  We use to call note any 
of the 12 pitches possible in an octave, regardless to their meaning in a 
particular scale.  Even more, we sometimes distinguish between two pitches 
who have their frequencies in a ratio entire multiple of 2 saying they are 
different notes (while in a scale, they bear the same note name).  Anyway, 
the term cent was more clearly defined as 1/100 of a semi-tone, and, being a 
ratio of a ratio between values of the same physical unit (Hertz), it has no 
physical unit.

Stéphane Collin