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2.2.3: major and minor tones
As a brass-player knows, the 7th note isn't a proper e-flat at all, but is quite a lot flatter than that, and for most purposes is quite unusable: and the situation is actually that the intervals in the natural series do get smaller the whole time, at a constant rate. Thus, the interval from the 6th to the 7th note in the series is smaller than that from the 5th to the 6th, and the distance 9-10 in its turn is smaller than 8-9.
But that means that the tone step from g to a is smaller than the tone step f to g: how can that be right? - in school we learned that the intervals of the scale were tone-tone-semitone-tone-tone-tone-semitone, without any distinction. Well, that doesn't appear specially systematic, does it? And in fact it's quite an oversimplification, applying only to one system of tuning: in the natural scheme of things there are in fact two sizes of tone, one greater (Lat. major) and one smaller (Lat. minor), and the western scale was originally expressed as 'majortone-minortone-semitone' twice:
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ex.8: major tones (T) and minor tones (t)
* there will be sound examples here soon *
If you can reflect the difference between major and minor tones when you sing and play - you make sure your third note is in tune with the first, and then, make the step from the first note to the second a tiny bit bigger than the step from the second to the third - you get a result which is satisfying, harmonic, close to the natural order of things: but if you can't readily vary your pitches - or can, but don't, like most wind-players today - you have a problem if you want to use, for instance, the scale starting on f and the scale starting on g, without retuning.
When you start on f, the first step, f-g, is larger, a major tone, while the second, g-a, is smaller, a minor tone: but when you start on g, the step g-a is the first step, a major tone - so it is an observable fact that if you use the same notes for the F-major scale and the G-major scale, you must be playing out of tune.
Then the question is, how much out of tune is it, and is it worth bothering about? The answer to the first part is 'it is enough out of tune for baroque musicians them to distinguish between them as the normal thing (Grassineau, 277); and the answer to the second part is, 'it depends', as the ancient Greeks recognised. Pythagoras' pupils considered Reason to be the sole judge, and said that the difference between major and minor tones must be observed: Aristoxenus (a pupil of Aristotle) said the ear must decide, and that it could not hear the difference: Ptolomy taught that the Senses and Reason were inseparable companions who must agree. We can say that it depends how much of the listener's attention is focused on the tuning, it depends on what instruments or voices are used, it depends on the acoustics of the room, it depends on the kind of music: but you can always be sure that the more aspects of tuning you get sorted out, the more comfortable you and your listeners will feel. At the same time, no-one will feel comfortable if a musician is worrying about something he can't cope with, so it must be better to decide to ignore such a problem than to let yourself become inhibited by your inability to resolve it. But singers, recorder-players and string-players particularly can solve many worrying, half-recognised enigmas if they recognise these subtle differences - and the enigmas become more amenable to improvement as soon as you start working with them.
But with fixed-pitch instruments, of course, people have accepted that a compromise, a temperament, is necessary, and have chosen a tone which is neither major nor minor, but halfway between - the middle, or 'mean' tone, hence the modern expression 'mean-tone tuning'. Many of the old teachers did not mention this aspect specifically when they were giving hints on tuning, because they were simply concerned with telling you what to do, and you get mean tones anyway if the tempered 5ths are all equally flat.
In the same
way, contrary to what we learned in school, there can be a bigger half and a
smaller half - a semitone major and a semitone minor - just as it's scarcely
possible for the two 'halves' of an apple to be exactly the same size: the expression
'semi-' in Latin has basic meaning of 'incomplete', as it can in English
too - like when you're half-awake, for instance. Thus the interval a-b is bigger
than b-flat-b-natural: between two notes having different names (a-b-flat, b-natural-c,
e-f) the step is a major semitone, between two notes with the same name (b-flat-b-natural,
e-e-flat, f-f# the step is a minor semitone. Indeed, earlier musicians thought
of b-flat and b-natural not as different notes, but as two forms of the same
note, one 'softer', one 'harder': and they reflected that in the way of writing
the letter-name, the soft one written round, 'b', also written the hard one
written square, 
, also written
'h' , 
and #.
The various
forms of the letter b were later transposed and used in front of other notes
than b to give the same effect. The form
meant only 'hard' until around 1700, when it started being used to neutralise
either a previous soft b or a hard #.
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The reason for using mean-tone tuning, as we have seen, is that the commonly-used intervals are very well in tune; the limitation is that, given only twelve notes to an octave, you have only three sharps and two flats, so you can't play in E-major or E-flat-major without retuning: and that pieces which are very chromatic sound awful when you have to use an e-flat as a d#.
It is clear that many people accepted this awfulness, and far later than has been generally thought to be the case; Quantz recommends that when you have to play an e-flat on the harpsichord together with a violinist's d#, you at least play it in a different octave, because then it's less disturbing: and there is evidence that many organs were tuned to mean-tone tuning well into the 20th century.
Other musicians preferred to tune the note between g and a to a pitch somewhere between g and a - you end up with neither, but it's just about usable as both.
Roger North:
"Some very good tuners will help a little by robbing Peter to pay Paul; as by making #G over sharp. But then E, a more usuall note, will suffer in its 3rd, which will hurt the musick in A key; and for that reason they call that note the wolf, which may neither be held, nor let goe. But since Musick for the most part is keyed upon the ordinary accords, it is best to hold them as perfect as may be, and not to corrupt them by sallving others seldome used." (211)
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