Tolerance Tuning Tables

[Tony Deff]

In the realm of Electronic Music, we have occasion to ensure that our note generators are not only in-tune with the outside world, but also with each other (in-scale). This is especially true of so-called "free-phase" instruments, which have multiple banks of oscillators.

A conventional frequency look-up table (FLUT) gives the tuner little notion of the limits of acceptable accuracy: is a 0·5 Hz. error at 440 Hz. acceptable? The table here may not give the exact tolerance that you require, but it does allow you to judge the accuracy.

The decimal resolution required depends both on the frequency and what degree of key-colour change (variance from a precise scale, so that a change of key produces a slight change of interval accuracy) is acceptable. Often you have little control over how your hardware (oscillators and frequency meter) reacts. A tolerance table gives a much better indication of the degree of tuning errors.

For the lowest notes, the decimal resolution of frequency meters is usually inadequate. Some meters provide an inverse indication as a time period, measured in milli­seconds or microseconds. Conversely, for the higher notes the decimal resolution of time periods is inadequate, and display of frequency is to be preferred: this table provides that choice.

The table is produced using “spreadsheet technology”, avoiding type­setting errors found slavishly duplicated in some older publications.
(It also allows tables to be produced for other absolute pitches, for example an older standard where C = 256 Hz. If you need such a table, just ask). The maths involved has already been published in the article Harmonic Logarithms, so you can also produce your own tables.

Octave Designations

There are many confusing, contradictory and arbitrary notations for the designation of octave pitches (organ "footages"). The first system to be widely-accepted was proposed by Baron Herman Ludwig Ferdinand Helmholz (1821-1894) in a publication of 1863.

Since then , there have been different national and arbitrary systems in use. The MIDI standard does not specify octave notation, so C5 or C3 might be used in documen­tation to represent middle C (note #60), and some books may mix systems without remark.

The system incorporated here is that proposed by the Acoustical Society of America in London in July 1939, viz. Scientific Pitch Notation, where C0 marks the nether-region below which sound is perceived more by the intestines than the ear. A4 refers to A=440·0 Hz. (above middle C4).
(If an international meeting in 1939 can meet to agree the pitch of A = 440Hz, rather than being concerned with trivialities such as preventing catastrophic global war, then it must be extremely important!)

Scientific pitch notation is an unambiguous specification of a note as a fixed frequency. Conventional pitch notation may specify a written and an actual compass of a transposing instrument.

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[Rykhaard]

Thank you GREATLY for posting THIS!

I was JUST searching for note pitches to know what I need to tune my voices to, in my CMOS String Machine that I'm building, from scratch. This will come in majorly handy for the equal temperament scale!

(Now, I'd like to find the same thing for at least the Even Temperament tuning standard and other 12 note tuning systems. Shall search.)

[JovianPyx]

Wow, that's very interesting technical information.

I read an article (can't cite a URL, sorry but I'm sure Google can help) that said that most humans don't perceive a badly tuned scale if the notes are all within 5 cents of "perfect". Given that, I would say that plus or minus 2 cents would be at least good enough. Even if pitches could be perfectly computed for concert pitch tuning, there will be phasing heard with intervals. A simple 5th is slightly off from a true perfect 5th, you hear a nice phasing beat sound. A bit of tuning inaccuracy will speed or slow that effect but if you're within +/- 2 cents I wouldn't hear anything I'd call "sour".

In my own synth designs, I use rather wide phase accumulators and phase increment registers (usually 32 bits or more). The sample rates are usually 200KHz to 1.0MHz with one or two designs as low as 65KHz. It is my understanding that wider phase accumulators and phase increment arithmetic allows for better tuning accuracy. Such wider arithmetic is supported by higher sample rates. I compute my tuning tables with the usual K*2^(N/12) formula.

The bottom line for me - my ears. I tune everything by ear and get the tuning effects (phasing) I want by using my ears.

I hope that helps.

[Rykhaard]

When I was setting up the tuning for pairs of CD40106 oscillators, I found that using a 100k summing node to a non-inverting 100k feedback, the closer that the 2 'oscillators' were to the exact required pitch, the greater the chances were of completely cancellation of amplitude.

In working to find where they were both clearly oscillating audibly, with the 100k summing, I needed a detuning range for the 2 x C#5 (554.365hz) notes of about 551hz to 557hz ...... wassat? About 6 to 8 cents between them overall?
This gave me a chorusing effect at about 8 to 12hz / second. My goal for the oscillator pairs for each 'voice' is about 5 to 8hz / second. I'm going to try what's done for the SSM2044 chip's 2 audio inputs - a difference of 3dB that they specified for it, to avoid cancellation between 2 oscillators.
With square waves, I'm hoping it'll work. If not - I'll have to stick with the wider chorusing effect.

[Tony Deff]

[Rykhaard]