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

Entry: Sol-fa (2): the key to temperament - A1057736
Author: Recumbentman (keeper of solmization syllables) - U208656

Doh my dears, part two, woohoo

Improved and expanded.

Gazes out of window while humming and tapping fingers on desk . . . funny thing about temperament, nobody wants to know.

I've just read both of your articles. As someone who knows just about as much about music theory as can be learned from music lessons up to grade 4 and then from books with titles like 'Blues Guitar Solos for Beginners', I have to admit that I scarcely understood it. However, I was fascinated to learn about the history and mathematics of tuning, and also that in fact my guitar isn't really in tune with itself when I tune it using the fifth and fourth fret technique.

Hello there Daniel! Thank you for your kind remarks. This is just the kind of reader I had in mind.

I tend to write things as short as possible, but I hope that if you re-read the first article it will make increasing amounts of sense; or at least it will arm you for dealing with other stuff about sol-fa and notation that you might come across.

~Andrew

I hate to say it, but I don't find this entry easy to read. I knew most of that stuff already so I do understand what you are saying, but I feel that it would be unintelligible to someone who doesn't already understand it. I think you should simplify it.

I wrote an entry A428852 on this the day I joined h2g2, but I never submitted it to Peer Review, so you are welcome to borrow from it if you want to.

On the subject of guitar tuning, I remember going to a 'Simon and Garfunkel' concert in the early 80s. Some people recognised the songs when they started singing the words. Other more discerning listeners recognised the songs from the introduction. My wife and I found we could recognise the next song from the way that Paul Simon tuned his guitar.

Hi Gnomon, welcome back to sunny Ireland! Hope you didn't get too soaked in Greece

Fascinating to see your early article -- why did you never submit it? Waiting to polish it up? I've just trawled through your entire list of entries to get it up on a different window to paste bits here -- japers you've written a lot! One every few weeks since 2000 it looks like!

I have a few comments: I tried (foolishly perhaps) to introduce as few terms as possible in my entry, without at least a word of explanation (eg "pitch class"). So if my article is hard to follow, is it just too laconic, or is it trying to cover too much too fast?

Your introduction to the syntonic comma is beautifully simple, as is the introduction to equal temperament, "Since adding intervals involves multiplying frequency ratios, we don't just divide 2 by 12. Instead, we the take 12th root of 2" -- wonderfully succinct. Alas non-mathamatical people tend to leave the room at this stage, no matter how elegantly you put it!

I amicably question the following:

"The Equal Temperament Scale was devised about 200 years ago"

--There can be little doubt that equal temperament was what Aristoxenus recommended; I can direct you to a quotation of his (fatally flawed) geometrical proof that (in our terms) Ab = G#; it is fascinating, but at the end it relies on the evidence of "how it sounds" which suggests to me that he did a demonstration lecture, tuning strings or the like, and sneaking the temperament in undetected as he went along.

--Equal temperament was also mentioned by Zarlino in 1588, by which time experimental compositions had already begun going full circle round the keys, and fretted instruments used a rule of thumb that gave a 99-cent semitone. Newton actually worked out the figures for equal semitones based on the twelfth root of two to many places, around 300 years ago; and isn't it over 250 years now since Rameau championed it?

". . . a perfect fifth. The frequencies of the two notes are in the ratio 3:2. The final chord we need is do - mi. This is called a major third. The frequencies are in the ratio of 5:4."

--You have given the ratios in the reverse order to the notes. I would suggest 2:3 and 4:5. I see why you do it, because you like to use a figure greater than one later on when expressing the ratio as a decimal; but it is a possible source of confusion.

"Because there are two different sizes of whole tone, we find that a note in one key is not exactly the same as the equivalent note in a different key."

--As you point out in the next paragraph, the discrepancy arises already in the very first hexachord (re-fa and re-la are incompatible) before you even begin modulating. Students always say "temperament is needed if you want to change keys" but it's worse than that, Jim.

"It is interesting to note that string players in orchestras tune their strings in fifths, so that they are tuned to the notes of the Pythagorean scale. Cellists, for example, tune their top string to A and then tune the other strings in fifths below this."

--We are on shaky ground when we talk about what musicians actually do: as you said, Paul Simon tuned differently for each song! Part of the trouble is that musicians can't generally tell you what they are doing; they don't even know unless they are Malcolm Proud. Orchestras are incorrigible; the strings do their best to go a little higher than the wind, for extra edge! This leads to new wind instruments being needed, at 442, 444, even 446. Even the pure fifths may be unconsciously tempered; it is stretching it to say that "When you sing a major scale, you subconsciously use a scale called the Just Intonation Scale" -- people tune the way they have learned!

"The lowest string, C, will be slightly out of tune with the highest, even though the fifths in between are perfect."

--This I can't quite follow. The A will not be La in tune with Fa, but it will be La in tune with Sol!

"Mean Tone Scale. This has one size of whole tone and one size of semitone"

--Hm. The diatonic semitone will be one size, the chromatic semitone will be another. Fair enough, you are limiting your discussion to the diatonic scale. Otherwise we appear to contradict one another.

"The Equal Temperament Scales plays equally well (or equally badly) in any of the 12 possible keys. With its introduction, composers were free to write music in any key they liked. This led J S Bach to compose "The Well-Tempered Clavier", in which he wrote a prelude and a fugue in each of the 12 keys."

--Alas modern theorists veer away from equating "Well-tempered" with "equal-tempered". It seems unlikely we will ever know for sure; but Grove lists four or five possible contenders for Bach's preferred temp., some of which (French Ordinary for instance) are hotly denounced by some of those who study these things, as utterly unthinkable.

Talk about cans of worms.

One thing I want to find again is the description I read somewhere of an organ built in the early 20th century in Norway, that automatically adjusts itself as you play, to purify all chords. The oddest thing is, nobody likes the sound of it! It's described as "oily". Temperament is apparently a welcome spice. John Potter (Hilliard Ensemble and others) is a singer of medieval and renaissance music that pushes the temperament most expressively; so do some if not all jazz musicians. Astrud Gilberta always sings consistently flat, to very sultry effect.

As to which entry is easier to comprehend, I wonder is there an innocent reader who will actually read either or both of them and tell us?

OK, I never said mine was entirely right, and I'm sure you know more about this particular can of worms than I do. I just feel that my style is a bit easier for beginners to understand. (OK, the twelfth root of 2 is not something I would spring on an unsuspecting reader in an Edited Entry - I never did submit this).

I don't want you to be me - but I think you could benefit from aiming at a slightly less knowledgeable reader.

Sorry for the overkill. As you said, that wasn't sent for PR!

I do like your article though, and it seems there are plenty of educated and scientific people in the world of hootoo that can take the twelfth root of two in their stride.

We could amalgamate, produce a combined effort -- think it could work? Trouble is, I get so fond of an entry when I'm writing it; I become like a Feis mother. [Note to other readers: Feis in this connection = Eisteddfodd = Music Competition]

I'd like your opinion on A1060552 - "How to win at gambling" whenever you catch up with work enough to have the leisure for that. It's short.

An interesting pair of entries.

So that's where the Wolf Tones got their name!

Let us know what you intend to do with these entries, Recumbentman!

Hi ZSF -- I don't think I'll try an amalgam; I've made some changes and improvements to this entry and I feel that this is more or less how it'll stay. It's not everyone's cup of tea, but any mathematically minded reader should follow it all right. If you have any reservations or requests I would be glad to try and accommodate them.

Story from RL: checking my college emails one morning a few weeks back I discovered I was to present a short paper to my fellow graduate music students -- that morning. No panic -- I printed out my h2g2 entry on sol-fa (that was before I split it into two) and presented that. Utterly painless.

Quite a bummer there, and I like big entries!

I think you can also profit from A473528 and the WW thread F57153?thread=90606 to get the maths into some simpler form. For Jeremy, this would have the benefit of reducing A473528 to a simple 'Brass Instruments' and thus make it more easily digestible too.



Bossel

That's a really interesting pair of Entries you wrote here. I love those pieces of work that dig deeper into the realms of music theory than I have ever dared to dig myself ...

If you would like to use some parts of my Entry about tuning and harmonics, please go ahead and take whatever is useful for you. Gnomon has already written an excellent piece about brass instruments, so if you can include the maths part of my Entry into a bigger context, that's fine for me

Jeremy

This entry is quite difficult to read. Some of the commentary here goes to the first article, too.

The problem with understanding is that you seem to be unaware you are bringing up new concepts without defining them. Because I know so little, but just enough to chorally sing and play piano, when I get to the point in your article where my mind finally goggles, it's a chore to go back and figure out just where you've done it.

Here are some things I took notes on:

On what's traditionally concordant: this needs to have the word 'modern' added to it, so no-one is surprised later by your Greek footnote. I know in medieval chant, a second is considered perfectly fine. What is and isn't supposed to be concordant is hard for me to comment on, because for some reason I never acquired the western ear and like gamelan and Indian tunings just fine.

I can't make heads or tails of the "temperament issues" sentence.

I don't understand what is meant when you refer to a note in the pattern: letter-solm-solm. I have tried to cross reference the notes on your diagram to see if it's helpful, and sometimes it works in your original entry, but it definitely doesn't work in this one. It's not clear to me why some note would have to be capital and some are not.

There are a lot of numbers after notes, and I am not sure if they are proportions, or declarations of absolute hertz. Also, humans usually aren't capable of understanding more than 4 to 7 things in a list, so the long lists of notes followed by numbers needs to be pared somehow. Humans aren't capable of intuiting complex ratios, and only the ones divisible by twelve are easily understood and memorized. There's no point in including all the really abstruse ones at all, and you can just say "they don't match" "this is a little to small to some ears" or "this offended a people with a fine sense of what constituted perfect music of the heavens" etc.

No commentary at all about fictas and the 'devil in the music'?

Sea Change, these are good points, and it is valuable to me to see how this entry comes across to someone not already familiar with sol-fa.

I cut corners a bit, being an economy-of-words freak; "C Fa Ut" for instance is how they named the note C, reading across the table of three hexachords, where C is Fa in the hard hexachord and Ut in the natural hexachord. Higher up, c is Sol in the soft, Fa in the hard, and Ut in the natural hexachord, and so they called middle C "c Sol Fa Ut". This is not essential to the entry, and it may be worth cutting it out.

The capitals were used by Guido and his successors for notes in the low octave, small letters for the high one; I mention in a footnote where I abandon this usage for the rest of the entry (both entries).

I tried to have everything mentioned once only, so that puzzles could be unravelled by re-readings. I would rather be too brief than long-winded. However I gather that many h2g2 researchers are mathematicians or other academics, and I value the completeness of lists.

I'll look over it again soon, with your comments in mind. Thanks

Aha! You have caught me in mid-goggle, so now I can be more helpful.

I did a word search for 'hard hexachord' on both entries. 'Hard hexachord' is never defined, and isn't in this particular entry at all.

Your footnote explains that something is happening in which some notes get capitalized and others not. Why some might be capitalized or not may be in your articles, but I didn't find the explanation, and it's certainly not anywhere near where you've inserted the footnote.

Is 'middle C' what is meant by 'low octave' or 'high octave'?

I'm sorry, I really thought that was clear enough.

"Hard Hexachord" is used first as a heading in the table in both entries, introduced by "here are the medieval hexachords" or suchlike. It's just a name, I didn't think any more was needed. The words for hard and soft remain in use in French and other langauges, but in English they have become "sharp" and "flat".

The early theorists used Gamma for the note G now found on the bottom line of the bass clef; then capitals A - G, then small letters a - g, then aa - gg. Middle C is therefore a small c, in the middle octave. I thought that the footnote "From here on in this entry the mediæval distinction between capital and small letter-names is abandoned, and all members of any pitch class (that is, all the octaves of any note) are taken as equivalent" was enough about that; it's not particularly important.

I *can* make the entry longer, but I'd hate to do it!

In entry (1) I say "They were called the ‘hard’, ‘natural’ and ‘soft’ hexachords, starting respectively on G, C and F." Is more needed?

About ficta and the diabolus in musica -- that all belongs in another entry altogether, about harmonic/contrapuntal theory. so far we're only dealing with the scales.

I'd like to know which sentence about temperament issues you found confusing; or was it all of them?

A lot of musicians don't want to know about temperament, or if they do, usually a rule-of-thumb is plenty ("flattened major thirds sound sweeter"). After all, all important musical facts are learnt by ear. This kind of analysis is academic.

I should have looked them up just using the word hard, perhaps that would have gotten the definition. Since footnote 4 happens much later in your article, I was trying to see how the hard hexachord could possibly start on G, since it starts on Gamma in your diagram.

Here is the sentence in it's entirety: Temperament issues arise from the incompatibility of notes within and between hexachords.

Just below it is a table, which starts another boggle for me. I got from your definition that hard hexachord was a specific title of a specific hexachord. Since I thought it was medievals who thought of this name, and that they thought of each note as unique enough to get it's own symbol, this didn't seem farfetched to me, though it could be wrong. Yet, there you have listed another hard hexachord.

Is it a matter of where the percieved semitone is in each hexachord, or is it a literal matter of mathematics, or is the system extended past singable notes into playable ones?

(knows we've got some number loving folks on H2G2 and still puzzling on how to make an intelligible and felicitous compromise)