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A few questions:

1) Is there a preferred or standard tuning system (e.g., even-/just-tempered, etc.) for modern western music or, perhaps a more appropriate question, what is your ideal tuning system?

2) Is it possible to formulate a mathematical expression for the intervals in any given tuning system, or are they set a little more arbitrarily?

3) How do intermediate intervals change when using the fractional definition of a fifth interval (3/2)?

Looking at the wiki page it says that equal temperament "...is the most common tuning system used in Western music, and is the standard system used as a basis for tuning a piano", but I have read elsewhere that this is not the best option.

I'm currently trying to code an open-source tuning app and have stumbled so far into the nether zone that I'm now more confused about what I'm trying to achieve than how I will achieve it.

Any advice/opinions would help a great deal.

closed as primarily opinion-based by Dom Jul 13 '18 at 18:00

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    There have been endless complete books written about tuning systems - often by who are more inclined to mathematics (or more precisely, complicated arithemetic) than music, though. If you are just doing this as a programming exercise, equal temperament is as good as anything else for the first option to implement. Whether you ever implement more than one option is your choice! – user19146 Jul 22 '17 at 21:55
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The best tuning varies with the situation. For keyboards and fretted instruments, equal temperament seems good, however, even keyboards are voiced (small tuning adjustments) for each instrument. Similarly for organ voicing and for adjustment of guitar and other fretted instrument tuning.

The problem is that now power of 3 equals a power of 2 (except the zero power of both which equals 1) so nothing ever comes out right. Ya pays your money and ya takes yer choice." The wiki on temperament (which you referenced) does have some discussions of the trade offs.