What is a 'temperament', in the most general sense?

Question

In Does this scale have a name: 1, 15/14, 15/13, ..., 15/8?, some commenters to the question opined that the OP was looking for a 'temperament', not a scale.

This surprised me as I've while I've heard (and used) the word 'temperament', I'm not seeing quite how it makes sense used in that way.

Usually I hear 'temperament' used in the sense of 'Meantone temperament' or 'Well temperament'- which are as Wikipedia says, tuning systems that "slightly compromise the pure intervals of just intonation to meet other requirements." In other words, they are 'tweaks' to, or 'specifically-pitched versions of', an already-defined scale such as the Major / diatonic scale. This is the sense used in questions such as On Tuning: What's in a Temperament? (Well-Temperament vs. Equal Temperament), and most other questions tagged 'temperament' on this site..

Another sense in which I've seen the word 'temperament' used is when referring to e.g. 15-TET or 53-TET - where it's basically used as a synonym for 'EDO', rather than really referring to any 'compromise' to another set of pitches (as per Do the EDO and TET acronyms mean the same thing?).

Neither of those meanings make sense to me in the context of the linked question, where the OP is definitely not asking about an already-defined scale or a version of the diatonic, and also doesn't seem to be basing his pitches on a concept of EDO.

So, as per title: what is a 'temperament', in the most general sense? Or, what other senses does it have other than the ones mentioned?

Answer 1

From Wikipedia "a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements." Here "a tuning system is the system used to define which tones, or pitches, to use when playing music. In other words, it is the choice of number and spacing of frequency values used."

The difference from a scale ("any set of musical notes") is that tuning systems (and temperaments) usually want to "pack" many scales into the keys or strings as possible. In particular, the standard equal temperament tuning packs 12 different major and minor scales all in the tuning. To do this it uses two tricks:

1. It only approximates the major/minor scales. (Of course, some might say that the major scale in equal temperament is the correct major scale; but if we assume it must contain the true overtones, then equal temperament only contains an approximation.)
2. It uses more than 7 notes. Naturally, the more notes you include in your tuning, the easier it is to pack many scales (which usually only contain 5-7 notes.)

In the question you mention, Dom posted the following link: http://www.dolmetsch.com/musictheory27.htm#calculator which contains a lot of different tunings, but not that many scales. (Unless you consider the entire set of notes in a tuning a scale, but that seems to be untypical.)

In his book The Mathematical Theory of Tone Systems, Jan Haluska lists a large number of different scales. Some classical, some exotic and some really weird. He uses a combination of ratios and "cent notation" to define them:

On the Xenharmonic wiki, there is also a large index of different scales, grouped by type: http://xenharmonic.wikispaces.com/ScaleIndex Some of these are in ratios and cents, while others are in H/W notation or letters. The later seems a particularly useful notation, when the scale is a subset of a standard 12-note tone system.

Answer 2

First off the first comment mentions tunings and temperaments. The reason for the distinction is importation as they are very similar topics.

Tunings and temperaments both are systems used to define pitches used. Nothing more, nothing less.

So for example both Equal Temperament and Just Intonation (which is a tuning) have 12 initial intervals constructed from the root, but the pitches in each are not defined the same. Here is a comparison of the differences in cents between them.

The distinctions between tunings and temperaments is as follows:

A tuning is laid out with nothing but pure intervals, leaving the Pythagorean or ditonic comma to fall as it must.

A temperament involves deliberately mistuning some intervals to obtain a distribution of the comma that will lead to a more useful result in a given context.

Solutions can be grouped into three main classes:

• tunings (Pythagorean, just intonation)

• regular temperaments where all fifths but the wolf fifth are tempered the >same way; note: regular meantone implies that all major thirds are identical

• irregular temperaments where the quality of the fifths around the circle changes, generally so as to make the more common keys more consonant Source.

So in the other question, the OP defines a set of notes by ratios which neither come up in Just Intonation nor Equal Temperament. So to answer the question, we need to figure out the tuning or temperament that defines all ratios inside it.

If it didn't matter, we could just answer the question in letter names and would be the scale consisting of the notes C, C♯, D, E, F, G, A, B (which could be looked at as a mode of the D full minor scale).

It is possible (though unlikely) that the ratios listed make up a full temperament or tuning and thus the temperament/tuning and scale will represent the same set of notes in which case a lot of this discussion is moot. The more likely case is that whatever the temperament or tuning is this is only a subset of the notes of it which makes the question harder because you have to identify a subset of the tones in the temperament or tuning (if it exists) and then determine scale that it represents based on that (if any).

Answer 3

Usually I hear 'temperament' used in the sense of 'Meantone temperament' or 'Well temperament'- which are as Wikipedia says, tuning systems that "slightly compromise the pure intervals of just intonation to meet other requirements." In other words, they are 'tweaks' to, or 'specifically-pitched versions of', an already-defined scale such as the Major / diatonic scale.

This "in other words" restatement seems to equate "the pure intervals of just intonation" with "an already-defined scale." But there is no reason why a scale, taken alone, should necessarily have any reference to just intervals. Just intervals are useful as harmonic intervals, while scales may be concerned only with melodic intervals. This is one reason why so many scales from various musical traditions comprise intervals that are not in just ratios: when a scale is not used harmonically, it does not matter whether its notes are pleasant when sounded at the same time.

Temperament is the compromise that arises when one tries to create a scale that works within certain restrictions. When different contexts call for the same note to be played at a different pitch, but the nature of the instrument dictates that the pitch of the note cannot actually be varied, compromise is necessary.

In European classical music, the restrictions are commonly harmonic. This is usually illustrated with the impossibility of closing the circle of fifths, because (3/2)¹² is 129.746 while 2⁷ is 128; the circle of fifths ends up 23.46 cents sharp.

Ïn 5-limit just intonation, however, which calls for "pure" major thirds in a 5:4 ratio, the problem arises much more rapidly. In fact, not really possible to define the major scale in just intonation if the notes of the scale will be used harmonically.

Consider a keyboard with just seven pitches, c, d, e, f, g, a, and b. We want to tune it so the perfect fourths and fifths are in the ratio 4:3 and 3:2, respectively, and the major thirds are in the ratio 5:4. Our fundamental C is 264 Hz. Here are E and G:

Desired | Source | Source
Pitch   | Pitch  | Frequency | Ratio | Result
--------|--------|-----------|-------|--------
E       | C      | 264 Hz.   | 5:4   | 330 Hz.
G       | C      | 264 Hz.   | 3:2   | 396 Hz.

There is the C major chord. Next, the dominant, working from G=396. Remember that the keyboard has only seven keys, so the D will be a fourth below the G at a ratio of 3:4 (instead of a fifth above at a ratio of 3:2):

Desired | Source | Source
Pitch   | Pitch  | Frequency | Ratio | Result
--------|--------|-----------|-------|-------
B       | G      | 396 Hz    | 5:4   | 495 Hz
D       | G      | 396 Hz    | 3:4   | 297 Hz

Let's check something here: when we defined B, we didn't only create a major third with G, but also a fifth with E. (We don't need to check the D against the G because we derived the D from the G to start with.) Are the B and the E in the correct relationship with one another?

Higher | Lower | Higher    | Lower     | Expected | Actual
Pitch  | Pitch | Frequency | Frequency | Ratio    | Ratio
-------|-------|-----------|-----------|----------|-------
B      | E     | 495 Hz    | 330 Hz    | 3:2      | 3:2

Fortunately, they are. Next, the subdominant, working from C=264:

Desired | Source | Source
Pitch   | Pitch  | Frequency | Ratio | Result
--------|--------|-----------|-------|-------
F       | C      | 264 Hz    | 4:3   | 352 Hz
A       | F      | 352 Hz    | 5:4   | 440 Hz

We don't need to check the F against C, because we derived it from C, but let's check the A against D:

Higher | Lower | Higher    | Lower     | Expected | Actual
Pitch  | Pitch | Frequency | Frequency | Ratio    | Ratio
-------|-------|-----------|-----------|----------|-------
A      | D     | 440 Hz    | 297 Hz    | 3:2      | 40:27

Oops! If we divide 40:27 by 3:2, we get 80:81. To get the perfect fifth above 297 Hz, we can multiply 440 Hz by 81:80, yielding 445.5 Hz, which is of course also the result of multiplying 297 by 3:2. The lower A at 440 Hz in fact beats quite unpleasantly against the D at 297 Hz.

The problem here is not the inconsistency between twelve fifths and seven octaves, but that between four fifths and one major third (plus two octaves). Since that difference is the syntonic comma, we can flatten each of the four fifths by a quarter of a syntonic comma to yield a pure third. This gives quarter-comma meantone temperament, and we didn't even need to think about other keys or chromatic notes before we found a need for it.

On the other hand, if you will never play the A at the same time as an F or a D, none of this is in the least problematic, and temperament would be quite unnecessary.