And soon thereafter I added:Are there established names for these mathematical operations?
1) f(x) = a^(abs(log_a(x))
2) f(x) = a^(frac(log_a(x))
In both cases, we temporarily shift into logarithmic space to perform a familiar operation and conclude by shifting back out, but in the former case the familiar operation is absolute value, and in the latter case it is taking the fractional or decimal part (in other words, modulus by 1).
I ask because I'm implementing them in code for a musical project, and I'd prefer to use accepted names if available. The first operation is useful for comparing interval sizes; the second operation is used for octave reduction.
I've been informally calling the 1st operation "absolute ratio", since I found references to that name in a couple places online:
https://math.stackexchange.com/question ... ute-ratios
https://yuvalg.com/blog/2012/04/28/intr ... ute-ratio/
The 2nd operation I just gave the placeholder name "period reduction" but I really want something better.
These both seem like the types of common, simple, and useful operations that would have found a name for themselves by the mathematical community at some point in history, so I'm surprised that I haven't been able to uncover official names for them yet. If there actually aren't any established names already, though, perhaps we could call them "logarithmic absolute value" and "logarithmic modulus", respectively?
Illustrations of these operations in use:
5/4 = 1.25 and 4/5 = 0.8. Which one is the larger interval?
Well, which one is further from 1, the home pitch? 1.25 - 1 = 0.25, and 0.8 - 1 = -0.2. We don't care which direction the distance is in, so we take the absolute value of each, resulting in 0.25 and 0.2, respectively. Since 0.25 > 0.2, perhaps 5/4 is the larger interval.
But this has not been the correct approach, since intervals multiply, not add.
So instead we should look at the ratios. 1.25/1 = 1.25, and 0.8/1 = 0.8, of course. Again, we don't care which direction the distance is in, so we take the... wait. We need to take the equivalent of the absolute value, but with respect to multiplication and the multiplicative identity, 1, rather than addition and the additive identity, 0. Basically, just as with taking the absolute value we needed to negate negative values, here we need to reciprocate values less than 1 (subharmonic ratios). So the absolute ratio of each results in 1.25 and 1.25, respectively. Since 1.25 = 1.25, we determine that 5/4 and 4/5 are the same size interval.
f(x) = a^(abs(log_a(x)) is the function which gives this absolute ratio. We can see it at work in this example. The choice of a is irrelevant; it is only necessary that the base of the logarithm matches the base of the exponentiation at the end. So, let's choose a common base as a: 10. log_10(5/4) = 0.09691. log_10(4/5) = -0.09691. We can already see that they are each other's negation. So by taking the absolute value we equivocate them. Raising 10 to this power we get 5/4.
I've got a couple of pitches, 121/5, and 5/11, and my pitch system repeats at the octave, or in other words the period (or interval of equivalence) is 2/1. To understand my pitches better, I'd like to reduce them all to their equivalents which are found between 1/1 and 2/1.
121/5 = 24.2, which is greater than 2, so I need to repeatedly divide by 2 until I first reach a value less than 2: 12.1, 6.05, 3.025, 1.5125.
5/11 = 0.454545 which is less than 1, so I need to repeatedly multiply by 2 until I first reach a value greater than 1: 0.9090909, 1.8181818.
f(x) = a^(frac(log_a(x)) is the function which does period reduction. We can see it at work in this example. Unlike with the function for absolute ratio, the choice of base matters; a is the period we'll reduce to. So, with a = 2, log_2(121/5) = 4.596935, and log_2(5/11) = -1.137504. The fractional parts of these values are 0.596935 and 0.862496 respectively. So the octave reduced form of 121/5 = 2^0.596935 = 1.5125 = 121/80. And the octave reduced form of 5/11 = 2^0.862496 = 1.81818 = 20/11.
And finally I added a fourth question, also related:I also just realized a third concept I've been looking for a name for is in this exact same family:
3) f(x1, x2) = a^(log_a(x2) - log_a(x1)) = x2/x1
Yes, it's just finding the interval from x1 to x2, but it's a particular way of looking at it. "Delta" is a word we use for "difference" in situations when we're looking at a series of ordered values and want the analogous series of differences between them. I would like a word for the equivalent concept, but for the analogous series of ratios between them.
So we could call it the "logarithmic delta"? That's a mouthful and a bit intimidating for such a simple idea, though, so I wonder if anyone has a better one.
On this thread there is some discussion of this concept:
https://math.stackexchange.com/question ... for-ratios
They come up with the name "quotient operator", and "exp Δ log", but neither of those provide a distinct natural language term I could in the same contexts as I would use "delta", such as "here are the deltas".
I could use this term in situations such as this: for a series of pitches 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, the "logarithmic deltas" are 9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15.
I actually realized that I have yet another terminology question in this family: is there a mathematical term that is the equivalent of positive and negative, but for ratios greater than 1 and ratios less than 1? In musical contexts we'd call these otonal and utonal, but is there a mathematical term?
Results after working with Dave
If you wish you may venture into the various subthreads of the original Facebook post linked above where Dave and I hashed stuff out (at one point Dave suggesting we should "get a room" since no one else had chimed in on this theoretically public post in the past 100-something exchanges ).
Or you can just skip to the results here, as I had prepped them for the Untwelve mailing list:
Some cosmetic adjustments were made to the above to make it work well on the forum here, but otherwise that's how it went. With one notable exception: I sent the above slightly prematurely, as Dave and I were still ironing out some details re: symbols for superratio and subratio operators, which we eventually decided were too ambiguous and confusing to be worth it, so while the original post included those, I cut them out from this version.Dave Keenan found my post about this topic on Facebook and gave me a ton of super cool suggestions! After quite a bit of discussion, this is what we worked out.
I'll be addressing them out of order, beginning with our results for the fourth of my original four questions, as those results are immediately handy when addressing the first of those four questions.
4) superratio & subratio
We propose superratio as the name for ratios* greater than 1 and subratio as the name for ratios less than 1.
My original best attempt had been 'logarithmically positive' and 'logarithmically negative'. Our final proposal has the advantages of being:
We further propose super and sub as adjectives, filling the same need that the words 'positive' and 'negative' do for numbers greater than 0 and less than 0, respectively. We could say, for example, "exclude the sub ratios", or "if the result is super".
- immediately obvious, and
- as a bonus in music theory contexts, evokes the related 'superparticular' and 'subharmonic' concepts.
Also, just as we often use 'positive' as a noun (short for 'positive number'), we can do a similar thing with super and sub: e.g. "only the supers", or "the sub with the largest Tenney height".
The words superratio and subratio are already defined in various fields like finance, chemistry, physics, philosophy, and even some mathematical contexts, but the likelihood of name collision is low.
* We figure our terminology should work for negative numbers too, although these won't be useful musically. Therefore, technically the definition of a superratio would be a ratio whose absolute value is greater than 1, and a subratio would be a ratio whose absolute value is less than 1.
1) undirected value
We propose naming this operation the undirected value, such that the word "undirected" plays an analogous role to the one "absolute" plays in the absolute value operation.
Just as the absolute value returns the positive equivalent of the input, so that positives and negatives can be compared, the undirected value returns the super equivalent of the input, so that supers and subs can be compared.
mathematical notation: ̲̅4̲̅/̲̅5 = 5/4 ̲̅5̲̅/̲̅4 = 5/4 function notation: und(4/5) = 5/4 und(5/4) = 5/4 mathematically read as: "the undirected value of four over five is five over four" "the undirected value of five over four is five over four" musically read as: "the pitch four over five has an interval of four to five" "the pitch five over four has an interval of four to five"
Dave and I suspect that the formatting of the horizontal lines above 4/5 and 5/4 in the mathematical notation may not look correct on all systems.
The crux of this naming is that the '/' operator is directed, i.e. 4/3 means something different than 3/4, while on the other hand the ':' operator is undirected, i.e. 4:3 means the same thing as 3:4. In music, an example of a directed ratio would be a pitch ratio, and an example of an undirected ratio would be an interval ratio.
My original best attempt had been 'absolute ratio', which also shared one word in common with 'absolute value', since I had always been keen on evoking that closely related existing operation. However, we decided to switch to directed value for a couple of reasons:
I'd like to call particular attention to the mathematical operator for undirected value. It makes a powerful visual analogy with the absolute value operator's vertical lines, |x|, just instead using two horizontal lines, x̲̅. These horizontal lines also evoke the vinculum (the horizontal line between the numerator and denominator of ratios), as if to say "this operation lets us compare values agnostic to which side of the vinculum they are on".
- It's a bit more natural to say in context. Compare "let's take the absolute ratio of the ratio 3/4" to "let's take the undirected value of the ratio 3/4". Primarily we're concerned that taking a particular type of ratio of a ratio is confusing and awkward, while taking a particular type of value of a ratio is natural. This same point applies to the name 'directed ratio', which we entertained but eventually rejected.
- The absolute value operation has an partner operation, sign (which returns 1 for positives, 0 for 0, and -1 for negatives), which when composed with it cancels it out, i.e. abs(x)⋅sign(x) = x. The directed value too can have a partner operation which when composed with it cancels it out. We propose direction, which returns 1 for supers, 0 for 1**, and -1 for subs, and which when composed with the undirected value cancels it out, i.e. und(x)^dir(x) = x. Naming the direction operation with the same etymological root as the undirected value operation clarifies their relationship; dir extracts the direction of a value and und gives you the value agnostic to what was extracted, just as sign works with respect to abs.
And now for a few auxiliary points:
** And for -1, again, in order to support negative numbers.
- In the original phrasing of my question, I had expressed this operation as a composition of logarithmiation, taking the absolute value, and exponentiation: e^abs(log(a/b)). While that is valid — and helpful for appreciating the sense in which the undirected value is a logarithmic absolute value — a clearer way to articulate the operation exists: max(a,b)/min(a,b).
- Astute readers may have noticed that we read our result as either “five over four” or “four to five”; the former reading corresponds to writing 5/4, the latter to writing 4:5. But why write 4:5 differently — why write it backwards? Well, Dave pointed out an inconsistency in many music theorist's convention for writing dyads versus triads and beyond, namely, that only dyads are written in descending order (i.e. we typically write 5:4, even though we write 4:5:6 and 4:5:6:7, etc.). Dave prefers the consistency of always writing ratios in ascending order, and he convinced me to prefer this too. Regardless, it may still be confusing that both the undirected ratio 4:5 and the directed ratio 5/4 are both valid results of taking the undirected value. So here it may help to draw a distinction between a ratio and its value. Directed ratios are always the same as their values, but the value of an undirected ratio is always the undirected value, which is always the value of the super ratio, not the sub ratio.
- In situations where it is difficult to realize the combination of overline and underline for x̲̅ (such as on a computer, without the appropriate utilities installed and configured), Dave had an earlier notation suggestion that would work too: infixing and outfixing with colons, evoking the ':' operator of undirected ratios, i.e. :4/5: = 5/4.
We propose naming this operation reduce, as in 'octave reduction'.
mathematical notation: red₂(20/3) = 5/3 red₃(5/7) = 15/7 function notation: red(20/3, 2) = 5/3 red(5/7, 3) = 15/7 mathematically read as: "red-two of twenty over three is five over three" "red-three of five over seven is fifteen over seven" musically read as: "the octave-reduction of twenty over three is five over three" "the tritave-reduction of five over seven is fifteen over seven"
My original best attempt had been 'logarithmic modulus', with the function notation logmod, but Dave pointed out that conventionally speaking logmod would mean the logarithm of the modulus, not the exponentiation of the modulus of the logarithm as I wanted it to mean. He provided an alternative terminological example of RMS, or the root-mean-squared operation, which squares a set of numbers, takes their mean, then takes that value's square root. So next I suggested we could name the operation expmodlog, but that we would typically (as with RMS) use its abbreviation, EML. Dave counter-suggested: why not just call it 'reduce'? I had indeed considered that previously, but I'd been too concerned that such a name would be too general. We agreed, however, that the name is good enough to be worth the risks of concept collision and vagueness, and that the meaning of the reduce operation will usually be clear in the contexts it's found.
I'd like to call particular attention to the mathematical reading "red-two" for octave-reduction, by analogy to the abbreviation of "base two logarithm" to "log-two", as well as the subscript 2 in the mathematical notation, similarly analogous to notation for logarithms.
3) Ϙ, 'qoppa'
Dave suggested I should call these types of quotients qoppas and use the Greek capital letter qoppa Ϙ, as a Q standing for quotient, to represent them. This is by analogy with calling these types of differences deltas and using the Greek capital letter delta Δ, as a D standing for difference, to represent them.
I find this definition clever, cool, and fun. It's certainly much preferable to referring to this information is 'ratios between successive pitches', or 'logarithmic deltas'.
Dave suggested that if you're not ready to hop on the qoppa train, 'step ratios' could suffice (but why use two words when you could use one?)
That's what we've got for now. Let me know what y'all think!
@Dave Keenan, let me know if you disagree with any of the conclusions here. Or who knows, maybe having a couple months for this to simmer, other things have occurred to you. Of course there is no rush.
For what it's worth, while working on something Sagittal-related this morning, I had practical use for a "delta" column and a "qoppa" column in the same table
By the way, nobody on the Untwelve mailing list ended up letting me know what they thought of the above, save AKJ, who said, "This is a huge post...will have to dig deeper when I have time to digest it." The sharing of this result, however, did prompt several subscribers to request the creation of a separate thread about math for us weirdos.