2788: Musical Scales

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Musical Scales

Title text: In the Hall of the Mountain King was accidentally composed on log/log paper.

Explanation

In standard Western musical notation, the horizontal position of a note indicates its relative temporal position in the piece, and the vertical position of a note denotes its pitch; but the pitch is really a logarithm of the note's frequency (every octave/seven named notes/12 semitones/13 named notes including accidentals equals a doubling of frequency), so it's a semi-log plot of sorts. The comic thus explores what a notation would look like if the horizontal axis behaved this way instead. Likewise, the vertical axis has been rendered linearly by frequency, with the normally equally-set lines on a normal musical staff stretching to compensate for the increasing jumps between pitches.

The nonstandard music notation depicted in the comic is the opening five measures of "Simple Gifts" (listen.)

Randall may have mistakenly assumed that the lines of a music stave represent a linear increase in pitch and thus an exponential increase in frequency. He has thus mapped them onto a logarithmic scale by doubling the space between each successive line of the stave - one space between the bottom two lines, then two spaces, then four, then eight. In fact, the lines and the spaces between them correspond to notes in the C Major scale, which have unequal gaps between them. It is also possible that he is aware of this small inaccuracy and chose to ignore it in the name of humor.

The title text purports to explain how In the Hall of the Mountain King, which progressively increases in tempo/speed and intensity from an initially subtle start into a rapid hustle towards a series of crescendos at the end, was written on log-log paper, that features nonlinear expansion in both its axes (in order to render various exponential graphs linear, often for the purposes of ease of understanding). Rather than these features being a deliberate composition decision, this says that they're only the result of how it was written down, or thenceforth read. The accident would have to be that music was originally written as though for log-log paper, using shorter notes as the piece went on, intended to have consistent actual durations throughout the piece—and then interpreted on a linear time scale, such that the later notes actually had shorter durations, speeding up the piece.

Transcript

[The sheet music for "Simple Gifts" on a distorted staff.]

[Caption below the panel]:

When transcribing music, remember to put frequency on a log scale and time on a linear one, not the other way around.

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Discussion

I'm wondering wether the lineage of in the hall... is worth mentioning. ie Grieg composed it for an Ibsen play. 172.70.175.178 23:06, 12 June 2023 (UTC)

Now I have to re-listen to In the Hall…; I think there are some errors here.172.71.146.212 01:23, 13 June 2023 (UTC)

Can someone make a midi of Hall of the Mountain King but with an exponential time scale to "compensate" for the log transform? I want to hear a version that both starts and ends at 200 bpm. Is there any music that actually uses mathematically varying tempos? Quantum7 (talk) 06:35, 13 June 2023 (UTC)

I can't make a midi but I can make an mp3: https://voca.ro/17QJDbYxNnlh Viliml (talk) 20:25, 13 June 2023 (UTC)

Ein belegtes Brot mit Schinken, ein belegtes Brot mit Ei...(Germans will understand.) 198.41.242.94 06:50, 13 June 2023 (UTC)

das sind zwei belegte Brote, eins mit Schinken und eins mit Ei. Bischoff (talk) 07:28, 13 June 2023 (UTC)

und dazu eisgekühlter Bommerlunder, Bommerlunder eisgekühlt. 172.69.33.163 05:29, 17 June 2023 (UTC)

But what does that have to do with dead pants?? 172.71.26.254 13:36, 13 June 2023 (UTC)

I don't really understand much German, and my Dutch is too weak to compensate, so I used my translator... :) (NOW I recognize some words, LOL!) I suspect that's the German equivalent to the English saying "Six of one, half a dozen of the other", :) Basically "Eh, either way works". NiceGuy1 (talk) 04:30, 17 June 2023 (UTC)

It is a song by Die Toten Hosen that gets quicker and higher for each repeat until you can't sing anymore, https://www.dietotenhosen.de/diskographie/songs/eisgekuehlter-bommerlunder

Why "mistakenly"? Sure there are some-half notes in there, but it's generally linear in the sense that every 7 steps correspond to a doubling of the frequency no matter where you start from 172.68.51.197 (talk) 07:30, 13 June 2023 (please sign your comments with ~~~~)

For the line spacing it doesn't matter much. A true linear-scale staff which takes half-steps into account would have spacings of 0.9, 1.8, 4 and 8. The one glaring discrepancy is that on a true linear scale, the note E5 (659 Hz) would be closer to F5 (698 Hz) than to D5 (587 Hz). Rick4 (talk) 14:29, 13 June 2023 (UTC)

Most sheet music is not truly linear in time to begin with. It's pseudo-logarithmic but in the sense that the shorter notes (8ths and 16ths and heaven forbid 32nds for us da**ed drummers) are given MORE space relative to the (fixed) size of the note heads compared to quarter, half, and full notes. This then affects the on-page length of measures: measures with faster notes are longer (as measurable with a small ruler) than those with longer/slower notes, even though -- assuming a fixed tempo -- their play speed (time duration) stays the same. And then you get modifiers like "rit(ardando)", "rall(entando)", "accel(erando)", "piu mosso", "meno mosso", and the like which modify tempo and throw the whole page-space-to-time relation out the window as if the page of sheet music itself (or the audience) sped to near-light speeds. Randall's going off the deep end trying to make this insane notation fit into fixed science rules; best to leave it to us crazy musicians and just enjoy the music. 172.69.65.46 10:44, 13 June 2023 (UTC)

"Randall seems to have mistakenly assumed" what? no, the entire point of the comic is that Randall knows standard staves do *not* represent a linear increase in frequency. A treble clef is centered on G4, which has a frequency of 392 Hz, F4 has a frequency of 349, and E4 has a frequency of 330. The drawn stave has one line between E4 and F4, corresponding to a jump of about 19 Hz. Two lines between F4 and G4, and we're assuming a linear scale, so that's about right to get to 392. The size of the games grows geometrically, as you expect. Again, this is the entire point of the comic.

That's not what it says, though. It says he may have assumed it's a linear increase in *pitch*, and therefore a *exponential* increase in frequency.172.70.86.2 08:40, 14 June 2023 (UTC)

is that not correct? Doesn't an equal temperament scale exactly mean that it is a linear increase in pitch? 172.70.114.240 16:11, 14 June 2023 (UTC)

An equal temperament scale means there's a linear increase in pitch by half-step/semitone, correct. However, any Western scale or mode regardless of quality will only include 7 notes, while there are 13 when including accidentals; the quality is determined by which notes are adjacent or separated by an accidental, or in other words if they are separated by one half-step or two. *However*, since standard notation uses other symbols to indicate deviation from the expected frequency and not separate lines, I'd argue that such would be the approach taken in a world where this sheet music was used. Therefore, I'd say this whole paragraph is unnecessary and misguided. Randall is taking a standard piece of sheet music and warping the scales of the axes, nothing more. He's not trying to make a valid, coherent new system of notation, he's making a graph joke. 162.158.159.125 17:27, 14 June 2023 (UTC)

But the point is that not only is his 'mistake' version 'wrong' (non-standard), but his remedy would result in notation that was 'wrong' as well, on both 'axes'.172.70.85.49 08:37, 15 June 2023 (UTC)

I'm not following. Don't the lines in a staff indicate equal spaced whole steps (between consecutive lines) or half-steps (between lines and spaces)? What is the "mistake" that randall is alleged to have made, and have we agreed that it is infact a mistake or not? 162.158.158.58 04:01, 17 June 2023 (UTC)

They do not. Each line in a staff is either a half step or whole step above the space below it, depending on the key. For example, C major has the notes C, D, E, F, G, A, B with no sharps or flats. Each one gets a line or a space between lines on the staff. But the interval between C and D is a whole step, while the interval between E and F is a half step. In equal temperament tuning, every half step has the same ratio, so the ratio in a full step is its square. On a log plot, that would mean the full steps would be twice as wide as the half steps, but they are in fact the same width. It gets even more complicated when you consider accidentals. For instance, D and D♭ are a half step apart but occupy the same line. Meanwhile, B and C♭ are enharmonically equivalent (i.e. the same pitch in equal temperament tuning), but they occupy different parts of the staff. And of course, double accidentals just make things worse. EebstertheGreat (talk) 15:12, 17 June 2023 (UTC)

All this musical theory talk is giving me a headache. I think I'll go put on some Zappa, RUSH, and Tool albums to relax. These Are Not The Comments You Are Looking For (talk) 02:25, 19 June 2023 (UTC)

The first 1:40 of this song: https://www.youtube.com/watch?v=t-Lt5fRNgO8 162.158.222.37 08:03, 22 June 2023 (UTC)