h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

also, i obviously need to make this clearer in the paper, but just wanted to point it out here: i was actually refering to the differance in frequency between one note and the next in eastern scales, not to three quarter notes. the notes are 3:2 times further apart in those scales than in what we are familiar with. (or at least me )

this means that when we go up or down three notes in our scale, the change in frequence is the same as if we went up or down two notes in the eastern scale.

It amounts to the same thing. The three-quarter-tone interval has a freqency ratio which is 3/2 times that of the semitone.

But Arabic scales use three or four different intervals in their scales, one of which is the three-quarter-tone but they use semitones, whole tones and minor thirds as well. Their scale is not composed only of these 3/2 intervals, as the entry seems to suggest.

this has been revised, and many other suggestions were adressed

Hi, I've just got a small thing to ask...

It may be that the mathematic curriculum in Norway differs from that of Great britain, but isn't the Golden Ratio actually 1.618? That's what my mathbook says, at least...

Otherwise, this entry is very interesting...

That's the golden thing about the golden section: the reciprocal of 1.618 is 0.618! So they both express the relation, each way round. So 0.618:1 = 1:1.618.

Thanks Vanamonde for linking to my entry; could I make another request or two:

". . . the ratio of C to E is 5:4, the ratio of E to G is also 5:4"

-- the ratio of C to E is 4:5; to keep it the way you mentioned in the list, you should say "the ratio of E to C is 5:4

-- but the ratio of E to G is 5:6. Put so as to match with the first pair, you should say "the ratio of G to E is 6:5". Old Hairy was right to point out the error here.

". . . chords are groups of notes which, when played together, compliment each other"
-- compliment each other is not the phrase. Complement is closer but still not right, because they are not incomplete to start with. You could say "dovetail" or "fit comfortably" or "reinforce each other" or some such phrase. "Compliment" sounds a happy choice, because it sounds like "complement" (complete) with the added flavour of "making pleasant sounds" -- but it is a tad poetic to use it here, as the meaning is "pass favourable remarks about". My knowledge of history might complement your knowledge of sport on a quiz team, but to compliment you I can simply say how well you look.

". . . this concept has been explaned more thoroughly in "Sol-Fa - The Key to Temperament", another entry on h2g2."
-- typo in "explained"; and the best format is:

this concept has been explained more thoroughly in Sol-Fa - The Key to Temperament.

This will come up in your entry with the name of my entry in blue functioning as a direct html-type link. The POPUP="1" part is optional, it makes the sol-fa entry pop up in a new window. I would omit "another entry on h2g2" altogether, as linking there will show that that's what it is.

You've given up on Beethoven? It struck me that with the exposition repeated (as it must be) the movement is not 502 but 626 bars long. The start of the coda (bar 372) becomes the 496th bar we hear, making it four-fifths of the way through. Even as a proportion of 502, 372 is more like three-quarters than 0.618. Where did you get the figures? Perhaps something can be salvaged, but I can't see what at the moment.

Wait -- if we count "the coda" as being the very last statement only, the last 25 bars, and count the bars to there we get 601. And 372 with the exposition repeated is the start of the recap (marked bar 248, add 124 for the repeat), a formally significant point. This is indeed 0.618 of 601 (well, 0.6189). But I can't find anything special at 0.382 of the way along; it would fall within the expo repeat.

What is salient is that the development and the coda are almost exactly as long as the exposition; so we get:
124 bars exposition (bars 1-124 inclusive)
124 bars repeat (1-124 again)
124 bars development (125-248) (248 is the 372nd bar heard)
126 bars recapitulation (248-373) (497th)
129 bars coda 374-502 (626th)
which is anything but golden. As I said first, omitting the last bit of the coda to make the dramatic recap fall at 0.618 of something makes the whole coincidence less thrilling.

Any old or hairy persons checking these figures must bear in mind the peculiarity of counting bars inclusively; and the fact that bar 248 is both the last bar of the development and also the first bar of the recap. I've tried to make the figures behave but may have committed some bloomers. Overall the point is made, I believe.

Oh, right! Of course...

My apologies...

Vanamonde:

That looks and reads better now. I'm still bothered by the wording of "They are named this because they involve a cross over eight notes, five notes, and four notes in our scale, respectively."

"involve a cross over" may be hard to visualise; how about "They are so called because they are the relations of the first to the eighth, fifth and fourth notes of our scale respectively."

Only come at this late, when many learned comments have been given, and suggestions acted upon.

I'd like to see this in the Edited Guide , but have to agree that there are a number of improvements that could be made to it before it ought to be recommended, to avoid leaving a tough job for the sub-editor.

I tried to grasp the description of the CEG chord, but the 'simple ratios' these notes have to each other just doesn't seem obvious to me from the ratios given.

One minor suggestion is with the link formatting, I'd suggest changing what you have in the following sentence to:
" Pythagoras and his followers are forever remembered for their (sometimes misled</LINK) contributions to the field of mathematics, but their contributions to the field of music have also been very rich. Several of these will be outlined later. "

The only general difficulty I have with the entry as it now stands is the assumption of mathematical purpose in music, when what exists is a system amenable to mathematical modelling. As with the pinecones you mention in relation to the Fibonacci sequence, the fact that something can be described or predicted mathematically does not imply that the pinecone was designed with the sequence in mind.

I would prefer it if the tenor of the entry more clearly reflected that the world is (fortunately) amenable to mathematical modelling on many levels - mathematics can help us understand and predict the world, including music - not that the mathematics is dependant on the world or vice versa. This last is just my opinion mind you, so feel free to disagree

Pimms

OK, Vanamonde, let us know when you consider this entry complete.

" . . . the minor hexatonic scale, which has a pattern of 3-2-1-1-3-2 where each digit indicates the number of half steps up (or down) the scale. Therefore the hexatonic scale for A would be A-C-D-Db-E-G-A."

If I follow your method of counting, that Db should be a D# or Eb. I've never heard of the hexatonic scale, I've always regarded the blues scale as pentatonic if anything (A-C-D-E-G with A major accompaniment). That's the way Leonard Bernstein and the like used to explain blues. But rock and blues are even more difficult than other genres to write about for the very reason that they present themselves as "counter-culture" and push against the conventions of the establishment.

I would hesitate to say "Most rock music and folk music is written in major scales and major keys, and therefore it is hard to pinpoint certain scales that sound like one of these specific genres (as they are differentiated by style instead of ratio)." Are you pointing out the problem that rock and folk music is primarily unwritten, and that the writing (in major keys) falsifies it? That could be put better; you are in danger of falling between judging the music *as written* and judging it as traditionally presented. I dealt with some of the problems of written music assuming supremacy over all other kinds, in A1921114 "The Three Ages of Music".

Perhaps it is enough to point out that the pitch relationships in rock and folk music are less important than the rhythmic features that mark the various genres unmistakeably apart.

Rhythm is also a branch of the mathematics of music; but perhaps hard to write about in a short entry like this?

Where would you put modal folk music in this, Recumbentman?

Modes don't upset the maths at all. I think that's another question. Rameau (the outstanding 18th century theorist) got a bit upset to find that the major scale derives from the subdominant (Fa) better than it does from the tonic (Doh!) -- implying that the natural mode should be the Lydian rather than the Ionian (=major scale). Too bad for Rameau, but such embarrassments don't hurt at all now.

There is a reason for this

Take E in key C - The ratio is nominally 5:4 to C

G# to E (in key E) is equivalent to E:C in key C therefore also 5:4

C to G# is equivalent to C to Ab in theory and therefore also 5:4

But 5:4x5:4x5:4 = 125:64, quite a considerable way from C' to C which is 2:1

The reason is that the ideal pitch of a note varies according to key. E to C is 5:4 in key C. In key G it is the same, equivalent to A to F in key C =(5:3)4:3) = 5:4. But in key F, E to C is equivalent to B to G in key C = (17:9)3:2) = 34:27 !

These discrepencies get even more exaggerated as we get away from C major so that for "equal tempered" instruments such as the piano, each key has a distinct flavour. Playing Greig's A minor concerto a tone up doesn't only affect pitch - Greig's concerto in B minor sounds downright perculiar.

My own instrument was the trumpet, and we had the facility to bend notes up or down a little with lip tension to compensate for the "equal tempered" tubing. Hence I'd "lip up" hard to play a G# in A major, but "lip down" to get the right pitch for Ab in C minor.

The point needs to be made that the ratios cannot generally be multiplied as this effectively changes the key and therefore the applicable ratio.

Vanamonde gave the figures in equal temperament and made it clear that the 5:4 (etc) ratios are all approximations.

The Rameau point is different though. All the notes of the natural C major scale are derivable from multiples of two, three, and five and the fundamental frequency of F.

Say 88 Hz is an F (it is, approximately).

2*88 is another F
3*88 is C
5*88 is A (440 Hz)
3*3*88 is G
3*5*88 is E
5*5*88 is B
3*3*3*88 is D

This gives the "Lydian" scale on F: FGABCDE.

To derive the "Ionian" (normal major) scale you need to *divide* 88 by 3 for a Bb. To put it another way, Bb never occurs as an overtone of F.

Isn't it curious that a trumpeter lips G# up and Ab down; in simple harmonic proportions the G# should be flatter than the Ab (as it is on a concertina that has separate G# and Ab keys). Three major thirds of 5:4 make *less* than an octave, as your figures show. However the practice of sharpening leading-notes goes back as far as the 19th century at least: the violinist Joachim said he sharpened all leading notes. The triumh of function over harmony.

(For non-muso lurkers, the leading note is the note below the keynote; in sol-fa it is the note Te. So G# is the leading note of A.)

Seth of Rabi -- Don't you hate the unintentional smiley response?

You mean to write (5:3) : (5:4) and you get a sad face from the : (

Anyway I don't really buy your contention that "for "equal tempered" instruments such as the piano, each key has a distinct flavour. Playing Greig's A minor concerto a tone up doesn't only affect pitch - Greig's concerto in B minor sounds downright perculiar."

There have been experiments made with pianos where a player played (in full view) a piece in G (say) transposed to Ab. The onlookers, all musically knowledgeable, said the piece sounded different from their familiar G major version. However unknown to them the piano had been retuned, or the keyboard rejigged, so that though they *saw* it played in Ab, they actually heard it in G.

Both Rameau and Bach came round to the view at the end of their lives that the different "colours" of equally-tempered keys are an illusion derived from the "interweaving of modulations". The case is still moot, and a composer friend of mine has definite colour-associations with each key. But though all agree that D is bright, there is a lot of disagreement about other keys.

A number of people have absolute pitch, and to them Grieg's Piano concerto would sound weird in B minor. But not because the note *relationships* are different; they aren't, in equal temperament.

I hear what you say Recumbentman, and yes, there can be a lot of pretence in musical circles on these matters. All I can say is that as a performer, some keys were more difficult to play 'in tune' than others. I'll discuss trumpets because that's what I know best, but the same points apply to most instruments other than keyboards, percussion and fretted instruments (which effectively play in a fixed key).
Most trumpets are tuned to orchestral Bb (trumpet C) giving for the open harmonics (ignoring the unmusical fundamental)
C (2:1) (=orchestral Bb)
G (3:1)
C (4:1)
E (5:1)
G (6:1)
Bb (the notoriously flat 7th harmonic)
C (8:1)
D (9:1)
E (10:1) etc

These simple ratios are what the ear hears as naturally harmonious, and music and written in C major/minor and the related keys (G,F,A,E) are relatively tuneful, the notes being on or very near the natural harmonic series without the need to 'bend them in'. In other keys, the instrument tuning becomes increasingly aberrant giving an altogether darker colour to the sound. Not to say it's bad - just different. Brass instruments sound bright in C, but threatening in C# fact - both aesthetically and mathematically.

For reference, an 'equal tempered scale' often is based on the following ratios:

261.626 1.000 C
277.183 1.059 C#
293.665 1.122 D
311.127 1.189 D#
329.627 1.260 E
349.228 1.335 F
369.994 1.414 F#
391.995 1.498 G
415.305 1.587 G#
440.000 1.682 A
466.164 1.782 A#
493.883 1.888 B
523.251 2.000 C

This is a compromise scale for keyboards, percussion etc, where each semitone is 2^(1/12) higher in frequency from the previous. Here, every interval in every key is constructed from the same fixed pitch ratios, and keys can be freely transposed without changing colour. (This confirms your comment about retuned pianos) - but this scale IS an unnatural one with its own particular colour.

It is remarkable however, how closely this 'mathematical artifice' fits the natural harmonic series. And yes, the unintentional smiley is annoying

The only smiley I find appropriate at this point in the thread is

Pimms

- o This should have been put more simply

I just question why the 'equal tempered' scale - a right old compromise if ever there was one - is given more respect than the 'just tempered' scale based on simple and natural harmonic ratios.
C 264.00
D 297.00
E 330.00
F 352.00
G 396.00
A 440.00
B 495.00
C 528.00
Is entirely musical.

Aside from that the articles fine (except the ratio for the leading note should be 15:8 rather than 17:9).

I think 17:9 arose from the equal-temper origin of Vanamonde's figures. 15:8 is more harmonious, but . . . as you say E.T. is a compromise and the remarkable thing is how close it does approximate to the "natural" figures.

In A1339076 "Sol-Fa - The Key to Temperament" I argue that even the natural scale calls for some compromise. There isn't such a thing as "the" natural scale, as leading theorists such as Rameau and Zarlino give it different values (generally the disagreement is over whether to make La agree with Soh or Fa).

For the interested onlooker: the figures Seth of Rabi gave a few posts up are two tables: one for the frequency (in Hertz or "cycles-per-second") for the equal-tempered scale c' to c" and on the right the multiple of the C frequency, rising each step by the twelfth root of two. The twelfth root of two (1.0594630943593 . . . )is the figure that, multiplying a given number twelve times over, will double the original number: which in musical frequencies raises you one octave.