The HP Original for sale

In the RF spectrum, do the HPs stand on the extreme left, or are they simply (politically) blase?

More precisely, they were accurate to a rational (fractional) scale, not to a well-tempered scale. So you lost all the single-digit Hz variations. Also, the lower octaves were again all divided exactly by two, so there was no "stretch tuning" the pianos are tuned.

Pianists differ in the amount of stretch tuning they prefer, but a full semitone distributed between the 87 steps is usual, and some prefer up to two full semitones, meaning that the top octave is numerically 12% high (a full tone sharp) in frequency compared to the bottom octave.

There is a physical justification for this. The Mersenne equation does not account for string inharmonicity, usually attributed to "end effects" - the the fixed ends of the string acting as bending columns with Bessel behaviour. This results in increasing harmonics of a string being increasingly sharp. A well-tempered tuning can tune the fundamentals only (which is numerically double) or it can bend towards harmonising the second or third harmonic - which results in stretch tuning.

Needless to say, simplistic digital dividers like the top octave generators do not produce any of these effects!

Clifford Heath.

This one's on the extreme left, yes.

The top octave was likely tuned to 12th-root-of-two equal temperament, same as most modern non-piano polyphonic instruments (and synthesizers)

Yeah, but I don't think that's the reason they sounded "thin", it was mainly because linearly mixing simple square waves of different frequencies but the same phase relationships just sounds bad.

Tonewheel organs just generated top ocatave sine waves too, but sounded great because you could mix in different harmonics from wheels of various sizes to do basic additive synthesis.

Then push the result through an overdriven tube amplifier and a rotating speaker.

Still popular in all styles of music:

Yes. Yesss....

No - they were integer approximations of the required dividers. The most widely used chips audibly diverged from equal temperament. You can do the sums yourself:

It does. But all the ratios in stretch tuning and equal temperament that are not rational numbers generate a lot of low frequency beats that create a kind of vibrancy that you just don't get with simple fractions.

I love my rigol DG1022, ~1 milli-Hz to 20 MHz. (Sine). It has two channels with synchronous phase control for each. (Well there's an align phase button you have to press.)

George H.

Yeah, as the 12th root of 2 is irrational you would have to truncate the real number somewhere into a rational approximation, and apply something like a Stern-Brocot tree to find the best approximation with the granularity you have available.

I guess the top-octave chips of the time were using counters that didn't have many "bits", so it sounded off.

"DCOs" in synthesizers from the 1980s had counters with more resolution; my Roland Juno (circa 1986) uses Intel 8253 16 bit PITs under uP control to divide an 8 MHz crystal clock and sync the analog sawtooth generators to that.

The Farfisa organ sounded OK in moderation:

It's trivial to find rational approximations using continued fractions. Here's the output of my program, given the twelfth root of 2:

$ ~/bin/continued_fraction 1.059463095 Usage: ratvalue [number [maxnumerator]] number defaults to PI, maxnumerator to 500 Rational approximation to 1.059463095000000 1 / 1 = 1.000000000000000 with error 0.059463095000000 10 / 9 = 1.111111111111111 with error 0.051648016111111 11 / 10 = 1.100000000000000 with error 0.040536905000000 12 / 11 = 1.090909090909091 with error 0.031445995909091 13 / 12 = 1.083333333333333 with error 0.023870238333333 14 / 13 = 1.076923076923077 with error 0.017459981923077 15 / 14 = 1.071428571428571 with error 0.011965476428571 16 / 15 = 1.066666666666667 with error 0.007203571666667 17 / 16 = 1.062500000000000 with error 0.003036905000000 18 / 17 = 1.058823529411765 with error 0.000639565588235 53 / 50 = 1.060000000000000 with error 0.000536905000000 71 / 67 = 1.059701492537313 with error 0.000238397537313 89 / 84 = 1.059523809523810 with error 0.000060714523810 107 / 101 = 1.059405940594059 with error 0.000057154405940 196 / 185 = 1.059459459459460 with error 0.000003635540540

The dividers on the MK50240 produced a scale in which three of the notes are more than one cents flat (compared to the mean line - compared to the tonic it was even worse).

But though that's easily audible to folk with ear training, it's the LF beats that are notably missing; hence the extensive use of tremolo in a weak attempt to make up for it. You can clearly hear that in "Light My Fire". It's a classic sound, so we like it; but doesn't resemble the sound of physical resonators.

Clifford Heath.

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** Hammond are the best known example of tone-wheel organs and had no frequency dividers - just an amplifier.

The teeth on the wheels were shaped to create different waveforms.

..... Phil

Sure there were. Like this:

I had a 1969 model for many years.

--
Regards, Joerg 

http://www.analogconsultants.com/

codename the POS :)

there's also this

-Lasse

There's also this (our first car):