Yahoo Tuning Groups Ultimate Backup tuning Beats analysis of meantones etc.

I am very pleased to see that an interest in the analysis of beat frequencies is developing on this list.

To my mind this could develop into a very useful way to examine tunings, on the assumption that the "allusive harmonics" do actually beat.

I would be very grateful if someone could post all the relevant links to the various methods of calculation which are being proposed,
so that I can explore any new (to me) ideas.

Until recently, members of the various tuning lists have seemed less than enthusiastic to even entertain the idea that the beat rates between
frequencies are significant, unless they are at integer frequency ratios.

I have experimented with many different mathematical models of this phenomena over the years,
and have often felt as though I was shouted down by the JI mob, whenever I insisted that harmonics "should" beat.

Let's really explore what is happening with beating from every direction,
for I am convinced that this is the most potentially productive direction for future microtuning research.

Charles Lucy - lucy@harmonics.com
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🔗Tom Dent <stringph@gmail.com>

Certainly for Baroque/Classical repertoire on a normal keyboard, you
can't do without beating...

I use a comma-based approach which may not be exact in getting
arithmetically proportional beat rates but seems to work within a tiny
fraction of a cent.

The fifth is then flat by some fraction F of a comma, whereas the
major third is sharp (or flat!) by a fraction T of a comma. The beat
note for the 5th is the 3rd harmonic of the root, that for the 3rd is
the 5th harmonic. The relative beat rates then go as 3F : 5T.

With the minor third as well, its beat note is an octave below the
15th harmonic of the root therefore if it is out of tune by a fraction
t of a comma, the resulting beat ratios are 3F : 5T : 15t/2.

So for 5/17 comma meantone, F is 5/17, T is 3/17 (flat), t is 2/17
(sharp) therefore the ratios are

15/17 : 15/17 : 15/17 or 1 : 1 : 1 !

For 1/5 comma T is 1/5, t is 2/5 therefore we get

3/5 : 1 : 3 or 3 : 5 : 15

For 5/29 comma T is 9/29, t is 14/29 therefore we get

15/29 : 45/29 : 105/29 or 1 : 3 : 7

etc. etc. etc.

The (tiny) corrections to this come from the fact that when you divide
up the comma it is a geometric division, whereas you would need to
divide up a frequency difference arithmetically to get exact integer
frequency ratios.

~~~T~~~

--- In tuning@yahoogroups.com, Charles Lucy <lucy@h...> wrote:
> I am very pleased to see that an interest in the analysis of beat
> frequencies is developing on this list.
>
> To my mind this could develop into a very useful way to examine
> tunings, on the assumption that the "allusive harmonics" do actually
> beat.
>
> I would be very grateful if someone could post all the relevant links
> to the various methods of calculation which are being proposed,
> so that I can explore any new (to me) ideas.
>
> Until recently, members of the various tuning lists have seemed less
> than enthusiastic to even entertain the idea that the beat rates between
> frequencies are significant, unless they are at integer frequency
> ratios.
>
> I have experimented with many different mathematical models of this
> phenomena over the years,
> and have often felt as though I was shouted down by the JI mob,
> whenever I insisted that harmonics "should" beat.
>
> Let's really explore what is happening with beating from every
> direction,
> for I am convinced that this is the most potentially productive
> direction for future microtuning research.
>
>
> Charles Lucy - lucy@h...

🔗Ozan Yarman <ozanyarman@superonline.com>

I agree with you Charles. I also feel that brats come into play in music, maybe even to the detriment of Just Intonation. That is probably why meantone temperaments with proportional beats are so much appealing to me.

My proposal for ascertaining the concordance of a chord was this:

----------------------

Assuming that z>x...

if `z` ( i.e. 1.5) becomes `z+n`

then `x` (i.e. 1.25) becomes x+(x/z times n)

For example, if one modifies the pure fifth so that the new value is 732 cents, then to preserve the consonance of the chord, one needs to modify the third by this amount:

386.31+(30.04 * 386.31/701.96)= 386+16.5= 402.84 cents

-----------------------------------------------------

According to Paul Erlich, the explanation is that the error of the minor third should be distributed between the major third and the fifth in an optimal manner. Yet, he may have a different proposal in mind.

My approach produces near-proportional beating:

If one modifies the pure third so that the new value is 392 cents (+5.686 cents), then to preserve the consonance of the chord, one needs to modify the fifth by this amount:

701.955+(5.686 times 701.955 / 386.314)= 701.955+10.332= 712.288 cents

However, a 392 cents major third from the base frequence of 260 Hz has the beat rate:

2^ (392/1200) * 260 = 326.07 Hz

(326.07 * 4) - (260 * 5) = 1304.28-1300 bps

which makes 4.28 bps or

1 over 4.28 = one beat every 0.233645 seconds.

In order to achieve the same beat rate for the fifth in a major triad, I have to temper 3/2 so that the result is:

(f*2) - (260 * 3) = 4.28 bps

thence,

2f - 780 = 4.28

2f = 780 + 4.28

f= 784.28 / 2

f= 392.14 Hz

The relative frequency of the fifth from the base is:

392.14 / 260 Hz = 1.5082307692307692307692307692308

The size in cents is:

1200 / (log 2) * log 1.5082307692307692307692307692308 = 3986.3137 * 0.1784678 = ~711.429 cents

The equal beating tempered triad has the frequencies:

260 Hz
312.68 Hz
392.14 Hz

and the cent values are:

0
392
711.429

In contrast, my solution resulted in 712.288 cents.The difference is less than a cent: 0.859

Cordially,
Ozan

From: Charles Lucy
To: tuning@yahoogroups.com
Sent: 07 Eylül 2005 Çarşamba 16:46
Subject: [tuning] Beats analysis of meantones etc. - links please.

I am very pleased to see that an interest in the analysis of beat
frequencies is developing on this list.

To my mind this could develop into a very useful way to examine
tunings, on the assumption that the "allusive harmonics" do actually
beat.

I would be very grateful if someone could post all the relevant links
to the various methods of calculation which are being proposed,
so that I can explore any new (to me) ideas.

Until recently, members of the various tuning lists have seemed less
than enthusiastic to even entertain the idea that the beat rates between
frequencies are significant, unless they are at integer frequency
ratios.

I have experimented with many different mathematical models of this
phenomena over the years,
and have often felt as though I was shouted down by the JI mob,
whenever I insisted that harmonics "should" beat.

Let's really explore what is happening with beating from every
direction,
for I am convinced that this is the most potentially productive
direction for future microtuning research.

Beats analysis of meantones etc. - links please.

🔗Charles Lucy <lucy@harmonics.com>

🔗Tom Dent <stringph@gmail.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗George D. Secor <gdsecor@yahoo.com>