Give her a helping hand!

Give her a helping hand!
------------------------

(Or, how an EDO can arise by sequence, that is, repeating
a motif at another pitch.)

Early in January this year, I was writing a satirical song,
with the phrase: "Give her a helping hand!" to the tune of
the line: "That's what it's all about", following "(You do
the hokey-pokey and you turn around, and)" in "The
Hokey-Pokey".

FF:FG:A:Bb:

I wanted to repeat the phrase, based on a rising fourth,
another four times, each one tone higher (as one might
do, for example, in a musical), but ending on the octave
of the starting note.

The last repeat was problematical; if I kept the usual
tunings, it would have to be a semitone higher than the
previous repeat. But what if I tempered the intervals?

Here are the notes and step sizes I wanted; you can read
the melody from left to right, one line at a time:

....... tone ... tone ... smalltone
.... F ...... G ...... A ........... (Bb)*
tone
.... G ...... A ...... B ........... (C)*
tone
.... A ...... B ...... C# .......... (D)*
tone
.... B ...... C# ..... D# .......... (E)*
tone
.... C ...... D ...... E ........... F'
* = not the proper names!

In this tuning, C=C# and D=D#. Writing t for the tone
and s for the "smalltone" (not always a semitone),
2 = FF' = FGABCDEF' = s * t^6.

That is, this scale has seven notes in the octave, of
which 6 are tones and one is a smalltone.

For a leading tone effect, s<t, say s=t^r where 0<r<1;
in practice, say 1/6 < r < 5/6.

So 2 = t^(6+r), or t = 2^(1/(6+r)).

When r is a proper fraction, we have an EDO. For
example:

r . steps(s) steps(t) n-EDO smalltones/tone
1/6 ... 1 ..... 6 .... 37 ....... 6
1/5 ... 1 ..... 5 .... 31 ....... 5
1/4 ... 1 ..... 4 .... 25 ....... 4
1/3 ... 1 ..... 3 .... 19 ....... 3
2/5 ... 2 ..... 5 .... 32 ....... 2 1/2
1/2 ... 1 ..... 2 .... 13 ....... 2
3/5 ... 3 ..... 5 .... 33 ....... 1 2/3
2/3 ... 2 ..... 3 .... 20 ....... 1 1/2
3/4 ... 3 ..... 4 .... 27 ....... 1 1/3
4/5 ... 4 ..... 5 .... 34 ....... 1 1/4
5/6 ... 5 ..... 6 .... 41 ....... 1 1/5

(More generally, if r=p/q, then the n-EDO tuning, with
n=6q+p, has:
. 6 tones of q equal steps and
. 1 smalltone of p equal steps.)

So I could realise the required transpositions in EDOs of:
13, 19, 20, 25, 27, 31, 32, 33, 34, 37 or 41 steps/octave.

. 13-EDO gives 2 smalltones/tone.
. 19-EDO gives 3 smalltones/tone.
. 25-EDO gives 4 smalltones/tone.

---------------------
Yahya Abdal-Aziz
27 April 2006.
---------------------

Regards,
Yahya

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