Equal beating well temperaments, reply to John

This post is a reply to one long ago of John Chalmers of Tue, 4 Apr 95.
See also the ones before that of 31 March '95 and 3 April '95.

I was calculating some equal beating temperaments and found that John
had done so too:

[snip..]
> I derived them as approximations to
> Equal Beating tunings, a type described by Rudolf Rach in Music
> Perception 1(3): 308-322 (1984). Rasch examined tunings in which
> intervals and their inversions had the same beat rate (e.g. 1/4 comma
> in which the major third and minor 6th have 0 beats). In 1991 I
> decided to examine other temperaments in which pairs of non-
> inversionally related intervals had the same beat rate. I was able
> to find solutions for 9 such tunings. I also found fractional comma
> approximations for them as well.

I think only 2 are correct.

> E.B. Intervals Fifths Approximation
> 1. 4th = min 6th 695.80957 2/7-comma
> 2. 4th = Maj 6th 696.29584 5/19-comma
> 3. min 6th= Maj 6th 696.0063 5/18, 13/47
> 4. 5th = Maj 3rd 695.63044 5/17, 2/7
> 5. 5th = min 3rd 695.80957 2/7
> 6. Maj 3rd = min 3rd 695.72794 11/38, 9/31
> 7. min 3rd = min 6th 698.8780 1/7, 65/454
> 8. 4th = Maj 3rd 697.47472 5/24, 1/5
> 9. 5th = min 6th 697.0389 8/35, 3/13


1. 4th = min 6th 695.80957 not ok, 695.163
2. 4th = Maj 6th 696.29584 ok
3. min 6th= Maj 6th 696.0063 not ok, not computed
4. 5th = Maj 3rd 695.63044 not ok, 697.817
5. 5th = min 3rd 695.80957 not ok, 696.895 appr. 1/4-comma
6. Maj 3rd = min 3rd 695.72794 almost ok, not computed
7. min 3rd = min 6th 698.8780 ok
8. 4th = Maj 3rd 697.47472 not ok, 697.710 appr. 2/11 pyth. comma
9. 5th = min 6th 697.0389 not ok, 697.599 appr. 3/16 pyth. comma

These are the other ones I did.

E.B. Intervals Fifths Approximation
5/4 = twice 3/2 698.590 1/7 pyth. comma
3/2 = twice 5/4 697.278 1/5 comma
6/5 = 3/2 696.895 1/4 comma
6/5 = twice 3/2 698.048 2 schisma
3/2 = twice 6/5 696.022 5/18 comma
3/2 = 7/4 697.282 1/5 pyth. comma
5/4 = 3/2 other one 694.273 5/14 comma
6/5 = 3/2 other one 689.734 too extreme

Computing these fifths involves finding the root of a polynomial of the
fourth order or higher.
To see beat rates in Scala, use SHOW DISTANCE.

Manuel Op de Coul coul@ezh.nl

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