Talk:Identity (music)

Disambig

First, the disambig at Identity is completely mistaken that "identity" is a musical term coined or exclusively used by George Perle. Secondly, I hardly think we need stub articles for terms which are nearly if not identical to the meanings of the mathematical terms from which they were borrowed. See: Musical set theory, please. Hyacinth 23:39, 5 Aug 2004 (UTC)

Incomprehensible

I know something about music theory and something about universal algebra, but I find the article basically incomprehensible. The examples given seem not to be identities but fixed points of certain transformations. Lambiam 13:59, 18 February 2006 (UTC)

What is incomprehensible? Hyacinth 08:57, 19 February 2006 (UTC)

First off, there is no notion of "identity" in universal algebra; there is a notion of identity in specific algebras, such as monoid algebras. It is not clear how that would apply here. The second sentence mentions identity function, which is a well-known notion in general mathematics. However, a function $f:A\longrightarrow A$ is an identity function (and the identity function on A) precisely when for all elements x in A we have f(x) = x. In words, this function does "nothing"; it just passes on its input without change. This does not fit the description at all.

Second, the article never states what an identity in music is. The disambiguation page at least states that it is "a transformation of pitches". So apparently in the article the reader is assumed to identify "identity" in the caption and first sentence with "identity function" in the second sentence. That is not at all an obvious thing to expect a reader to do.

Third, is it any pitch transformation? Is transposing all pitches upwards by a minor third an identity (function)? Speaking mathematically, it transforms the class of all pitches into itself. Presumably it is not a coincidence that all examples given are fixed points, as noted above. So is an identity a fixed point of a transformation? (In that case everything is an identity, since everything is a fixed point of the identity function.) Or is it a transformation together with a fixed point? It is all not quite clear to me.

Fourth, if the retrograde operation "transforms a pitch class", then why not the operation of replacing any sequence of tones by 0123... (which, when applied to 0123..., produces the same thing!)? Then it would appear that any process of "melody in – melody out" fits the description.

Finally, in the last part, I don't know what is meant by "interval-4 family" (tones that are a major third apart?), by "sum-2" family, or (in this context) "axes of symmetry" and "family of symmetrically related dyads". I see that the next two lines show chromatic ascending and descending scales starting from D, but what they have to do with anything discussed before or after escapes me. I cannot discern a relationship with any of the surmised notions of identity given above. Lambiam 19:23, 19 February 2006 (UTC)

So by "incomprehensible" you mean under explained, since you understood that the subject was a part of music theory and its terms if not their meanings where borrowed from some field in mathematics. Hyacinth (talk) 09:11, 26 December 2009 (UTC)

See Trivial (mathematics). Hyacinth (talk) 09:44, 26 December 2009 (UTC)

Universal algebra currently reads: "After the operations have been specified, the nature of the algebra is further defined by axioms, which in universal algebra often take the form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z. The axiom is intended to hold for all elements x, y, and z of the set A." If there is no concept of identity in universal algebra then the "Universal algebra" article needs to be corrected. Hyacinth (talk) 21:40, 25 October 2019 (UTC)

I can see where Lambiam is coming from. This article appears to be written largely by one person (Hyacinth) and is a reflection of their understanding of a mathematical concept, it's terminology and how it applies to music. As such, terms from mathematics are getting blurred together into a vague shadow of how they're are used by mathematicians. In mathematics, these terms refer to very specific abstractions. As an example, the current version of the article uses the word "identity" how mathematicians would use the word "symmetry": Somewhat formally, this is the property of being a fixed point of a function with usually a simple, intensional definition such as rotation about some point by 180 degrees. In contrast, "identity" could mean any of a number of specific things depending on context: 1. A specific element of an algebraic structure that acts on or combines with other elements to produce those same elements or 2. a function that maps ever element of a set to that same element or 3. an equation that holds over all possible values for it's variables. It's entirely possible that music theory has started to assimilate the language of mathematics with imperfect transcription and I share in the dismay that Lambiam seems to feel for a situation that needn't be made any more confusing than it already is.Nbrader (talk) 23:07, 19 August 2021 (UTC)

Interval vs. Sum

Judging by the quoted example, it looks like the figures are swapped - that is, the figure for Interval-4 actually depicts the Sum-4 family and vice versa. I didn't check whether the midi files are also switched. - Nonstandard (talk) 19:49, 1 April 2013 (UTC)

You appear to be correct. See: Talk:Common tone#F sharp major. Hyacinth (talk) 09:41, 16 July 2017 (UTC)