Musical questions and answers thread

[vinteuil]

I've probably asked this before.

With our tradional, Western division of the octave into twelve semitones comes a difficulty in tuning the smaller intervals.

If we had chosen instead a larger number of stopping points on our way to the upper octave, could we have avoided uncomfotable compromises like equal temperament?

... if you wish to construct a series of notes based on the idea of a principal and then perfect fifths etc above it, and work this out thro' all available octaves up and down, and then perfect fifths etc above the perfect fifths you created first time around, et ainsi de suite, and don't do any tempering to make nearly-right notes coincide - you're going to end up with a helluva lot of notes squeezed into each octave!

.

[MrGongGong]

I agree that most of us have come to expect the compromises of equal temperament, certainly not to find them unbearable.

But that doesn't answer my question.

I don't know the real answer
but suspect it is NO


and (predictably) I do think it's all about context, I never find Gamelan music sounding "compromised".

[ferneyhoughgeliebte]

With our tradional, Western division of the octave into twelve semitones comes a difficulty in tuning the smaller intervals.

If we had chosen instead a larger number of stopping points on our way to the upper octave, could we have avoided uncomfotable compromises like equal temperament?

Possibly - but there might then have been other "uncomfortable compromises" that arose, such as the number of keys per octave on keyboard instruments.

The thing is that the distance between a pitch and the one an octave higher isn't really a series of steps (as inherent in the very word "scale") but a continuous glissando, which could be stopped at any point in between to create innumerable different "scales"/modes using the pitches in between the semitones used in usual practice Equal Temperament. As MrGG suggests with his reference to Harry Partch, some Musicians have based entire pieces of Music exploring such sounds; but even sticking with more traditional scales, there is a subtle difference between the pitch F# when used in G major, and when it is used in A major or between Bb in F major and Bb in G minor when sung or played on a violin or bassoon, for example. These differences aren't readily noticeable (which is how keyboards have got away with it for centuries) but if played together, the "beat" of the differences make them clear. So, to cope with this, a keyboard would need two different keys for "F#" - and a third for E major (and that's not going down the difference between F# and Gb) - or to retune the instrument for each piece in a different tonality.

As for the Upper Octaves - it's not just a matter of the division of the octaves that creates the "difficulty in tuning"; how the human ear registers high pitches, and reognizes the difference between them also comes into play. At the very highest frequencies, (IIRC*) a "correct" tuning (created by an electronic instrument using mathematically correct divisions of the octave) sounds "out-of-tune", and it sounds "better" slightly flatter than "correct". (This can also be a source of contention between woodwind and string players playing in the octaves needing ledger lines above the treble clef - Flute players getting the "correct" pitch, Violinists the "corrected" one.) {*This paragraph is based on recollections of a series of lectures I attended at the University of Leeds in 1979, so I'd be grateful if anyone else can confirm the accurasy of my recollections, or, better, correct any inaccuracies. }

Equal Temperament is a compromise; one that resulted in an enormous repertory of fabulous works, but also imposed itself on other fabulous Musics that are better suited to other tuning systems. With electric keyboards, and developments in microtonal instrument manufacturing, I think that the time has come to expand the possibilities for exploring such different systems available to composers.

[Pabmusic]

I agree that most of us have come to expect the compromises of equal temperament, certainly not to find them unbearable.

But that doesn't answer my question.

Here's a passage from my book on Butterworth, dealing with the problem - the Pythagorean comma:

Comparison of fifths seems to have been the basis for many early systems of tuning. But it contained a paradox that limited its usefulness to only a handful of keys.

Starting on the note C, the sequence of fifths produces : [illustration I can't reproduce here]

Each subsequent note differs from its predecessor by being either five notes (a fifth) higher or four (a fourth) lower : C-G-D-A-E-B-F♯-C♯-G♯, etc. After twelve such intervals we return to a C. The problem becomes apparent in this example from the 10th note, because the sequence should really be : D♯-A♯-E♯, ending not on a C but on a B♯.

To make things more easily understandable, we can convert these notes to E♭-B♭-F, returning to C. But the interval between bars 9 and 10 – G♯ to E♭ – can no longer be called a fifth, strictly speaking. But surely this is only a question of musical semantics, since D♯-A♯-E♯ and B♯ are the same as E♭-B♭-F and C?

Aren’t they?

Well, no they aren’t. For one thing any singer, string player, timpanist (and at least some brass players) will confirm that B♯ and C are in fact quite different notes, and that much of their art lies in making small compensating adjustments to pitch throughout a performance. For another, the strict application of Pythagorean tuning (that is, by fifths in the constant ratio of 3:2) gives a thirteenth note that overshoots the octave, and is sharp in the ratio of 1:1.01364326477050788125.

This is a miniscule amount, of course, but it is detectable by anyone with a ‘good’ ear. [And from this note, the whole of the next sequence ("octave") willbe sharp by this amount, ad inf.]

And it does not stop there, for the possible tonalities that are implied by each note of the scale generate other anomalies. For instance, the first and fifth notes in the above example are C and E – a major third – but the interval is not the ‘pure’ one that would be generated by the harmonic series, it is an artificial one generated by Pythagorean mathematics. The practical effect of all this is to limit the system’s usefulness for building harmonic structures to a narrow range of tonalities based (in Greek times) on a limited scale of five or six notes, and in Medieval times on a scale of eight notes – the octave – all in the same tonality. There was no idea of key, no notion of changing key, because there was no ability to do so without some notes sounding ‘wrong’.

We are familiar with the solution, of course. Often called ‘equal temperament’, it is a system of tuning compromises that arose largely in the 17th and 18th centuries. Simply put, by playing scales with fixed intervals (in other words, by putting up with slightly odd intonation) any one scale could be treated exactly as any other ; and from there, any one ‘key’ might be indistinguishable from another, except for the actual pitches involved.

Over several centuries, any tuning based upon the harmonic series came to sound odd to ears used to the artificial intervals of equal temperament. Composers have exploited this contrast very effectively : Vaughan Williams in A Pastoral Symphony and Benjamin Britten in the Serenade for Tenor, Horn & Strings, for instance.

More importantly, equal temperament allowed for the notions of key, and modulation into new keys. It opened up the range of tonalities available to composers ; we could never have had Mozart or Wagner – let alone Schoenberg, Ives, Messiaen, Boulez or Glass – without equal temperament.

I've left out a load of footnotes, but I hope you see the problem.

[jean]

I do, thank you very much for that.

As some members here may recognise, I have been locked for several years into a fruitless online argument with someone who seems to think 'our Western scale' is a given, and it is our failure to understand the science behind it that has given rise to a system of notation that doesn't work.

And now, perversely, we are (he thinks) ignoring Helmholtz and spurning all the much better notation systems that we might use (including his own, of which he won't give any details.)

From my own experience singing early music unaccompanied, I think we make the necessary harmonic adjustments without even thinking about it. We do not even depart from the 'written note', because( as I think of it) each note as written has a sort of fuzzy area around it.

[Pabmusic]

I do, thank you very much for that.

As some members here may recognise, I have been locked for several years into a fruitless online argument with someone who seems to think 'our Western scale' is a given, and it is our failure to understand the science behind it that has given rise to a system of notation that doesn't work.

And now, perversely, we are (he thinks) ignoring Helmholtz and spurning all the much better notation systems that we might use (including his own, of which he won't give any details.)

From my own experience singing early music unaccompanied, I think we make the necessary harmonic adjustments without even thinking about it. We do not even depart from the 'written note', because( as I think of it) each note as written has a sort of fuzzy area around it.

Well, there's something in what you say. The Western scale is one of the great achievements - only with it can we have an idea of key or tonality. But it is a compromise.

[jean]

...The Western scale is one of the great achievements...

...but I am rather keen on the modes, myself!

And then, what of the microtonal intervals used is non-Western scales? I don't like the idea that they are not so great. Perhaps they are just differently great.

.

[ferneyhoughgeliebte]

...but I am rather keen on the modes, myself!

- yes, when you've heard one major scale, you've heard 'em all! (Which is why Equal Temperament was so useful for modulating and enabling contrast and conflict between Tonalities of the different degrees of the scale.)

I'm impressed that you continue your online argument, jean - I'd've made my polite excuses and left long before now: it's like "arguing" with a member of the Flat Earth Society (which organization, I'm told, advertises itself as having members "across the globe"!).

[ferneyhoughgeliebte]

There's a scientist on a respected internet site who claims that "there's no such note as B#" - which rather beggurs up the "Moonlight Sonata".

[Pabmusic]

...but I am rather keen on the modes, myself!

Indeed. The chapter in which that passage occurs deals with how conposers could use folk modes 'out of context' to extend their harmonic range. For instance, with our equal temperament hats on, B-flat and F-sharp don't belong to the key of C. But They fit confortably into modes centered around C. The B-flat is the flattened 7th of mixolydian C, the F-sharp the 6th of dorian C, etc. So new tonal vistas open up. Butterworth was rather good with this sort of thing.

[ferneyhoughgeliebte]

Indeed. The chapter in which that passage occurs deals with how conposers could use folk modes 'out of context' to extend their harmonic range. For instance, with our equal temperament hats on, B-flat and F-sharp don't belong to the key of C. But They fit confortably into modes centered around C. The B-flat is the flattened 7th of mixolydian C, the F-sharp the 6th of dorian C, etc. So new tonal vistas open up. Butterworth was rather good with this sort of thing.

- as were RVW, Sibelius, and Shostakovich, amongst many others. But the difference between the Modes (Dorian, mixolydian, etc) that they used (based on playing them on the equally-temperamented tuning of a piano) and those (with the same names and "spelling") that (say) Tallis would have heard should also be borne in mind.

[Pabmusic]

There's a scientist on a respected internet site who claims that "there's no such note as B#" - which rather beggurs up the "Moonlight
''Sonata".

Sorry to point this out, Ferney, but you can't spell 'buggers'.

[Pabmusic]

- as were RVW, Sibelius, and Shostakovich, amongst many others. But the difference between the Modes (Dorian, mixolydian, etc) that they used (based on playing them on the equally-temperamented tuning of a piano) and those (with the same names and "spelling") that (say) Tallis would have heard should also be borne in mind.

Of course. But t the folk singers I have in mind would rarely, if ever, have heard a piano.

[Bryn]

Of course. But t the folk singers I have in mind would rarely, if ever, have heard a piano.

Let alone an 'in tune' one.

[MrGongGong]

From my own experience singing early music unaccompanied, I think we make the necessary harmonic adjustments without even thinking about it. We do not even depart from the 'written note', because( as I think of it) each note as written has a sort of fuzzy area around it.

I seem to remember there is a bit about this in here (and lot's of research into the whole psychology of tuning)

http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198521280.001.0001/acprof-9780198521280