Werckmeister Temperment
en.wikipedia.org
en.wikipedia.org
I absolutely agree with you about the film - more people should experience his work.
https://search.worldcat.org/title/How-equal-temperament-ruin...
Historically, the older systems were some derivation of Pythagorean or just intonation, which are generated from small integer ratios like 3:2, 4:3, 5:4. If you continue building a tuning system out of these integer ratios, you get weird ratios like 81:80 (syntonic comma) or 531441:524288 (Pythagorean comma); you can then choose to include these ratios in your tuning system (too complicated), stop before you reach them (too simple), or divide up the ratios and distribute them in other parts of your scale. This last approach is called “musical temperament”.
https://en.wikipedia.org/wiki/Musical_temperament
Werkmeister devised a set of musical temperaments in the late 1600s. Alternative temperaments include mean-tone temperament (various flavors) and equal temperament. Mean-tone temperament divides the commas equally apart parts of the octave. Equal temperament divides the commas equally across the entire octave. Werkmeister temperaments divide the commas unequally. This results in some intervals which sound more consonant and some intervals which sound more dissonant, relative to equal temperament. Unlike equal temperament, different keys will have different character.
Just to illustrate a more concrete problem: systems which improve the fifth often require sacrificing the third. Pythagorean has exactly perfect fifths but the thirds are not so good. Quarter-comma meantone goes the opposite direction and gives you exactly just thirds, but the fifths are not so good. Equal temperament is between the two, but much closer to Pythagorean.
If you want to play with it immediately, and you have Logic Pro installed, you can go menu diving into the tuning settings for your project. Other DAWs too. Various synthesizers also have Werkmeister buried in the menus—some brands more than others (I know Yamaha has had them for ages).
Someone with perfect pitch can tell what note you are playing just by listening to that note in isolation. Most people can’t do that. “Absolute pitch” is probably a better name, but it’s usually called “perfect pitch”.
Perfect intervals are the intervals which don’t have a smaller and larger version of an interval with the same name. It’s nothing more than a naming convention for intervals. The 4th and 5th are perfect. So are the octave (8th) and unison (1). The other intervals (2nd, 3rd, 6th, and 7th), rather than being “perfect”, come in minor and major versions.
Basically, every interval either has minor/major, or perfect.
It’s a naming convention for intervals based on the western diatonic scale. Any interval which is smaller than minor/perfect is called “diminished”, and an interval which is larger than major/perfect is “augmented”. For example, C to Gb is a diminished fifth, and Ab to F# is an augmented sixth. Which number you use to name an interval is strictly based on the letter names of the note, ignoring sharps and flats.
Never mind 31TET etc.
I’m a little confused by what you are saying here, because most people who say “perfect pitch” mean that you can tell which note is being played in isolation. That’s what “perfect pitch” means to ordinary people, as far as I can tell.
It’s something you have or don’t have. Like, “I have perfect pitch” or “I don’t have perfect pitch”.
If you mean something other than that definition, maybe you could find some alternative way to express it because I don’t understand what meaning you are getting at. I’m not trying to be pedantic here, but “perfect pitch” is right there in the dictionary and it’s most commonly used to mean that you can recognize pitch without a reference, i.e., absolute pitch recognition.
The "perfect" in "perfect pitch" glosses over the messy reality of temperaments. And "perfect" is a pretty strong word in a cannon of exaltent claims about Western classical music. That's what I find interesting.
Or to explain further, providing a synopsis of tonal harmony in response to my comment is another of the sort of things I find anthropologically interesting.
> The "perfect" in "perfect pitch" glosses over the messy reality of temperaments.
Yeah. So do all the other terms like minor, major, diminished, augmented, second, third, fourth, fifth, sixth, seventh, consonant, dissonant, etc. All of those terms gloss over the messy reality of temperaments.
The tuning system (including temperaments) are, in fact, what establish the possible relationships between these interval names and the actual pitch ratios. For example, a just major third is 5:4, a pythagorean major third is 81:64, and an equal temperament major third is ∛2:1.
> Or to explain further, providing a synopsis of tonal harmony in response to my comment is another of the sort of things I find anthropologically interesting.
I’m happy to answer questions but I don’t like feeling like I’m being treated as an anthropological curiosity.
If I’m providing more detail than you expect, then chalk it up to a property of Hacker News, which is full of pedants. Sometimes my comments here are damn verbose, just to get ahead of the pedants.
Yes, sometimes we use letters to name notes, and the letters refer to a particular tuning system. But perfect pitch can be applied to any tone in any tuning system. If there's a way of naming the note, someone "with" perfect pitch could use that ability to name the note.
If I invented an 18-tone-geometrically-fumigated scale based on octaves of the golden ratio, someone with perfect pitch could name all 18 of my new notes based on hearing them (after learning my new system). Perfect pitch has no relationship to western music, and certainly doesn't exalt it.
It's can be used on any music with discernible pitches.
Perfect pitch means that someone can identify pitches accurately
“Accurately” is doing some psuedo-scientific hand waving. That hand waving what I find interesting.
Pitch is continuous and not discreet. There’s nothing wrong with tonal harmony as a system of making music. Its elevation to an ideology is anthropologically interesting…e.g. the “perfect” in “perfect pitch.”
The "perfect" in perfect pitch refers to a person's ability to recognize pitches, and makes no value judgements on any systems of harmony. Perfect pitch is not even related to harmony at all, as it pertains to a single tone.
Perfect pitch equally applies to all music that's based on pitches. It makes no value judgment about any particular practices, styles, traditions, genres, or tuning systems.
> “Accurately” is doing some psuedo-scientific hand waving. That hand waving what I find interesting.
What? A pitch can be measured as a frequency, which is a number. If someone can correctly identify the frequency accurately, they have perfect pitch. We can define "accurate" to be "within 1%", or whatever error tolerance you prefer. To which pseudo-science do you refer? Note names are used only as shorthand for referring to pitches, but "perfect pitch" doesn't require them. You can demonstrate perfect pitch without any note names or tuning system at all.
I really don’t think there’s much more to it than that, except composers with picky ears should just bite the bullet with 17-TET or something. (I am a guitarist and sometimes I wish I had flatter thirds and tritones without having to bend the strings.) As shown off-handedly in the Werckmeister Harmonies novel and film, composers who get too deep into the weeds of temperament systems often find themselves confusing form with function and waging ideological battles, forgetting that the point of the exercise is create interesting music.
The dominance of 12TET is not only about how well it approximates certain intervals. Equal temperaments in general are specifically about being able to transpose music freely without altering the "character" of the music. On the other hand, specifically having 12 notes in an octave turns out to produce some rather nice music-theoretical properties. Many equal-temperament tunings are regular diatonic tunings (https://en.wikipedia.org/wiki/Regular_diatonic_tuning), but 12TET is unique in that the size (as in, logarithmic pitch ratio) of the semitone is exactly half that of the whole tone.
>composers with picky ears should just bite the bullet with 17-TET or something.
For what it's worth, 17-TET (https://en.wikipedia.org/wiki/17_equal_temperament) has exceptionally "bad" thirds, both major and minor. It has an excellent semitone, since the just chromatic semitone is equal to the difference between the just major third and just minor third.) 19-TET (approximates third-comma meantone, and has an excellent minor third) and 31-TET (approximates quarter-comma meantone, and has an excellent major third) are common choices.
53-TET allows for approximating "zero meantone" - i.e. an almost exact perfect fifth - and a single such step is quite close in size to the syntonic comma. This has fairly dubious value in itself, since it's a lot of notes and the 2-cent difference between a 12TET perfect fifth and a just perfect fifth is not discernible for most people. However, as I understand it, some traditional Indian and Arabic tunings can be understood as subsets of 53TET.
Electronic musicians with an interest in such exotic systems (or just microtonality in general) have a surprisingly complex system of jargon for describing tuning - since many of the systems they use don't depend on traditional music theory. Sometimes these systems are chosen for deliberate dissonance; some don't have pure octaves. (In this world (https://en.xen.wiki/w/Main_Page), equal temperament tunings are often labeled as "edo" - "equal division of the octave" - rather than TET.)
Ideally you want exact numeric ratios between the various note combinations. (E.g. the frequency of a G should be 1.5 times the frequency of a C.)
If you start with a C and set up perfect ratios, you get a smooth sound as long as you play in the key of C.
But if you change key and move pitch patterns up and down the keyboard - transposition - the pitches will be off. Sometimes by a lot.
So you have these different systems that try to approximate a solution in different ways.
Modern Western music uses Equal Temperament, where the frequency ratio between notes is the same all the way up the keyboard (exactly the 12th root of 2.)
This is super-simple and gives close-enough ratios that aren't quite exact but are very usable.
The cost is a hint of sourness in some note combinations, but mostly we just hear those as movement within the sound.
The older alternatives to Equal (mean tone, Werckmeister, Kirnberger, and others) smooth out the ratios more selectively. So some note combinations are less sour, others are more sour.
They sound quite spicy if you're not used to them - somewhat out of tune (ironic...) but most people are so used to Equal Temperament anything else sounds exotic.
So something like this is of historic interest. These tunings are sometimes used for authentic performances of very old ('early') music, and some old church organs were built with them.
It's very rare to find them being used for effect in modern songs. (Does happen, but very unusual.)
The best solution is dynamic tuning which takes into account the surrounding notes and shifts pitches a little up and down for the smoothest effect. Choirs and orchestras tend to do this anyway, but making it happen in software is a much harder problem.
https://support.apple.com/en-gb/guide/logicpro/lgcpa88a63e7/...
> Hermode Tuning automatically controls the tuning of electronic keyboard instruments (…). It retains the pitch relationship between keys and notes, while correcting the individual notes of electronic instruments, ensuring a high degree of tonal purity. This process makes up to 50 finely graded frequencies available per note, while retaining compatibility with the fixed tuning system of 12 notes per octave.
Other DAWs have similar options, e.g. Cubase: https://www.steinberg.help/r/cubase-pro/13.0/en/cubase_nuend...