sorry, i think i pushed those digits out of whack



> It is really just counting, where the last item becomes the new first:
> 
> 12345
>         12345
> 123456789 

i.e., it's inclusive and 1-based


Richard Dobson wrote:
> On 15/01/2011 11:33, Chuckk Hubbard wrote:
>> Hi Richard.
>>
>> Great explanation. I'd just defend myself a little by saying I wasn't
>> conflating the two theories, but contrasting them. I am well aware
>> that they don't go together, but they're two different ways of naming
>> similar things. One of them makes perfect sense, and the other...
>> diminished sense. What kind of "theory" says that adding two 5ths
>> gives you a 9th?
> 
> Music theory, which like all theories is a systematic (and inevitably a 
> posteriori)  account of "best practice". In music theory, a "fifth" is 
> five scale steps (or the nominal 3/2 ratio), not 1/5!
> 
> It is really just counting, where the last item becomes the new first:
> 
> 12345
>     12345
> 123456789
> 
> I use this model all the time in flute teaching of the rhythmical 
> practice of rudiments - this is a "9-note group" where you start on 1 
> and stop on 9  (and resume from that same note as a new 1). the emphasis 
> here is on the rhythm - those 9 notes could be ~any~ nine notes - it is 
> a ~counting~ exercise.
> 
> If you want to express the adding of intervals  mathematically, you have 
> to be clear on what you are doing. When we ~count~ intervals, we count 
> them differently from  scale steps where the intervals vary (i.e. if 
> this is  a major scale, 1-2 is two semitones, 3-4 is just one). One very 
> ancient way of looking at it: The western major scale is made up of two 
> disjunct tetrachords, each with the same interval pattern: CDEF,GABC. 
> The ancient Greeks has their own way of expressing it mathematically, 
> which may come as a surprise: the standard expression of a scale was as 
> a descending form, and a note was either "long" or "short" in reference 
> to the length of the string. So their "low" notes are our "high" notes, 
> and v. versa.  So even with mathematical rigour, things can still be 
> described in many different ways. We just have to be consistent.
> 
> In term of frequencies, when we "add" intervals, we are actually 
> ~multiplying~ ratios: so 3/2 * 3/2 become 9/4 - which is thus the ratio 
> of a "pure" ninth (2.25). But as we are in daily life relating notes to 
> the five-line staff, or to the standard keyboard, we don't bother with 
> endelss discussion of rratios (which in any case will be wrong in terms 
> of equal temperament; the ET ninth is more like 2.45), we just count the 
> steps, and trust that everyone knows what we mean (which is why we learn 
> music theory). Simple!
> 
> PS: Richard Dawkins presents "the irrational" as the enemy of the 
> "rational". No mathematician could possibly agree - the irrationals are 
> just another class of number, after the rationals and the integers, and 
> their favourite class may well be the transcendentals. No class is the 
> enemy of another. Now if only Dawkins could get his maths right and be a 
> bit more transcendental...
> 
> 
> Richard Dobson
> 
> 
> 
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