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I have read the threads about CHAS, and I must admit I don't get it, even if I have a solid mathematical background.

Can someone explain it to me in plain terms? Or at least in mathematically unambiguous terms?

Would it be possible to tune CHAS with TuneLab?

Can you explain the bow and arrow part?

Chas is based on equal beating 12ths and 15ths.

The Chas paper (it's on the web) gives the equation; the semitone ratio comes out at 1.059486544. You can work out the offsets for Tunelab from that.

Nevertheless, Kees, the Chas paper gives the equation and the result. Some, if not all, versions of Tunelab allow for custom offsets. You are right that the differences will be small.

Well, TuneLab only takes offsets for the temperament octave and then only one type of octave tuning (e.g. 6:3) for the entire range below the temperament octave and one type of octave tuning (e.g. 4:1) for the entire range above the temperament octave. There is no way to make TuneLab use the same ratio between semitones for the entire range (if that is the point of CHAS?) and there is no way to choose compromises between octave types (for instance, halfway between 6:3 and 4:2 or equal beating 12ths and 15ths). Probably TuneLab could be used in a more manual fashion to do this.

Not just small, these "offsets" are completely overwhelmed by effects caused by inharmonicity, hence irrelevant to piano tuning.

Equal beating 12/15ths in piano tuning is explained properly here:
http://www.billbremmer.com/articles/aural_octave_tuning.pdf

Kees

(3-x)^(1/19)=(4+x)^(1/24)

Tunewerk: What this equation does to my understanding is simply define a new width for the semitone - the basis for a new width of ET rather than the 12th root of 2 - which could also be seen as a specific measure of theoretical stretch before inharmonicity, the mean between the pure octave fifth and the double octave.

Alfredo: Yes, as you say, the algorithm defines "a new width for the semitone", and to me it surely is "the basis for a new width of ET rather than the 12th root of 2", and it does stretch the scale before iH. Then, more than a "mean", the algorithm may be seen as the representation of many possible step widths, depending on the "s" value. This, in my view, describes the correct approach to the definition of the scale frequencies.

Tunewerk: It has only one solution for x, which makes this an equality or a point rather than an equation.

Alfredo: Yes, in my mind it makes a precise point. In 2006 Chas equation was plotted in MathLab, I seem to remember that 0.002125 was one out of perhaps 4 or 5 solutions. Should I/we check?

Tunewerk: I plotted out the curves, and there is one solution for 'x' in the real domain at approx. 0.00218. There is only one other solution, and it is imaginary.

This equality is then literally describing the same quantity x being taken away from the ratio 3/1 as is added to the ratio 4/1 to achieve a semitone size which will satisfy this requirement.

Alfredo: Yes, two "differences" from two partial matchings (3:1 and 4:1), chromatically (all across the scale) relative to two intervals now determine the scale incremental ratio, and consequently the scale frequencies. Using your words, 12 root of two achieves "a semitone size that will satisfy" the 2:1 ratio as referred to frequencies; CHAS equation, in its basic form, satisfies the 1:1 ratio referred to differences on two other ratios, in our case 3:1 and 4:1.

Tunewerk: It would not be the equal beating point theoretically (though in reality I doubt one could tell).

Alfredo: Good point. Perhaps saying "equal differing" would be more correct. I too could not tell whether, in my final tuning form, the beat rates for 12ths and 15ths were "perfectly" even. But that perception of mine took me to the idea of two equal differences for two intervals; in fact, within my tuning form I was looking for a "difference ratio" that could relate two intervals, and a for a single constant that I could use for modeling. [...] So, by managing "s" we can generate any incremental ratio and aim at any point. As I've mentioned, if I were to tune equal beating 12ths and 15ths there and then, I would not gain the final form I want.

Not just small, these "offsets" are completely overwhelmed by effects caused by inharmonicity, hence irrelevant to piano tuning.

Does it follow that equal beating 12/15ths cannot be tuned with an ETD like Tunelab without adjustments made aurally?

Reading the manual will show you how to do that.

Kees

It is not an equation, Withindale. It is an equality.

(3-x)^(1/19)=(4+x)^(1/24)

Tunewerk: What this equation ...

What Alfredo was describing (despite the rough math) is an entire concept of thinking about and understanding tuning that is not in the current lexicon. To put in CHAS offsets into a tuning machine is to not understand what he was saying.

Machines are part of the problem in the way they approach tuning - as a series of point solutions from a stretch algorithm, rather than an interrelated whole...

You can't put CHAS offsets into a machine and have it work, because the way the machine works, these offsets will be overwhelmed by the larger effects of inharmonicity. However, if you tune with ideas represented by CHAS (not exclusive to CHAS, by the way), they can be incorporated into the inharmonicity. In this way, the tuning can be completed to a higher level, where small changes are maintained because of aural cross-referencing, and not single variable output.

This is an extremely important point that most tuners (and Kees) are missing. These aren't offsets.