Fwd: [tuning] Re: Marchetto

For crying out loud, stupid Yahoo messed up my tables again. Here:

Dave, Marchetto seems to take the odd numbers from 1 to 9, where the
first part is the first diesis. So, the diagram should be:

| 0-1 | 1-2 | 2-3 | 3-4 | 4-5 | 5-6 | 6-7 | 7-8 | 8-9 |
|--1--| |-----3-----| |-----5-----|
|-----2-----| |-----4-----|

I interpret this to mean:

Type of interval Size in dieses Ratio Cents
-----------------------------------------------------
whole tone 5 dieses 9:8 204
chromatic semitone 4 dieses 27:25 133
diatonic semitone 3 dieses 15:14 119
enharmonic semitone 2 dieses 25:24 70

x......X...x....X.....x.....X.x..X.......x........X (ninths of a tone)
0......1........2...........3....4................5 (dieses)
0......35......70..........119..133...............204 cents

Oz.

> Oz.
>
> On Aug 16, 2008, at 7:01 AM, Dave Keenan wrote:
>
>> I think that Marchetto was one confused dude, mathematically.
>>
>> When I read this:
>>
>> "We acknowledge therefore that the parts of itself will have to be
>> inequalities, so that 1 is the first part; from 1 to 3, the second;
>> from the 3 to 5, the third; from 5 to 7, the fourth; from 7 to 9, the
>> fifth; and such 5th part is the fifth odd number of the total 9."
>>
>> I get this:
>>
>> | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Marchetto ninths of a tone
>> |-1-| |-----3-----| |-----5-----| Marchetto "parts" (later >> dieses)
>> |-----2-----| |-----4-----| " " " "
>>
>> When he says the 5th part is the fifth odd number, that implies that
>> _all_ the parts were odd numbers. In what sense can a part _be_ an >> odd
>> number. At least in this interpretation we have an obvious answer: >> by
>> it's size in ninth-tones. None of the other interpretations I've seen
>> have such an obvious sense in which all the parts _are_ odd numbers.
>>
>> However I completely fail to match that up with this table:
>>
>> Type of interval Size in dieses Ratio Cents
>> -----------------------------------------------------
>> whole tone 5 dieses 9/8 204
>> chromatic semitone 4 dieses -- --
>> diatonic semitone 3 dieses 17/16 105
>> enharmonic semitone 2 dieses 18/17 99
>>
>> It seems extremely odd that he should have
>>
>> diatonic_semitone + enharmonic_semitone = whole tone
>>
>> instead of the usual
>>
>> chromatic_semitone + diatonic_semitone = whole tone
>>
>> The fact that he's talking about singers makes me think "5-limit" if
>> there was any harmony involved.
>>
>> -- Dave Keenan
>