Dream closure

(There's no point to this story – it's just an unusual pair of dreams for me.)

A while ago – a year-ish perhaps, but it wasn't so remarkable at the time that I noted it down anywhere – I had a dream, possibly a recurring one, that I'd gone back to university. Studying for a coursework Master's, I was enrolled in a geostatistics course, the sort of course that I was lecturing in real life. And I was totally overwhelmed by it. I couldn't keep up with the assignments, and as the weeks went by, it became clearer and clearer to me that I was going to fail. I dropped out, my self-confidence crushed, having lost what is perhaps my best skill, namely passing maths-heavy university courses.

Those feelings of total inadequacy passed soon after waking up, and I hadn't thought back to those dreams since long after waking up after the last of them. Then last night I dreamt that I'd taken on tutoring a third-year maths course in neural networks, despite only knowing about one sixth of the lecture material. (I'm working through a book on the subject over at my .com.) I was sure that I'd be able to learn the content well enough as the course progressed to teach it.

I can't remember if I was awake or still dreaming when I recalled my dreamt coursework failure of a year ago, though given the strength of the emotions involved, it was surely while still dreaming. It was as though, by becoming a tutor for material that I didn't yet know and being sure that I'd do a good job at it, I'd redeemed myself for giving up on the geostats course. A small moment of strongly felt triumph, a heavy weight lifted from my shoulders. I woke up happy.

Brief notes on partially franked dividends

On a recent trip to Brisbane I was having dinner with a group of old friends, and I realised that we had become incredibly boring people who talked about superannuation, the stock market, and so forth. In this post, I set out some notes on the taxation of partially franked dividends; I think it is relevant to at least two people, author included.

(I worked this algebra out today, not financial advice, may only apply to Australia, may not even be correct though I think it is, etc.)

You own shares in a company. The company pays out a dividend to its shareholders, without having paid company tax on it first. In this case, we call the dividend unfranked, and you pay tax on it according to your marginal income tax rate (probably 32.5% or 37%). If your marginal income tax rate is ti, and the dividend is D, then you'll owe the tax office ti*D.

The more complicated case is if the company pays its company tax on the dividend before giving it to shareholders. If the company tax has been paid in full, then the dividend you get is fully franked. The key points are:

  • The ATO taxes, at your marginal income tax rate, the imputed full value of the dividend before the company tax was paid on it (the grossed-up dividend).
  • You get credits for the tax that the company has already paid on the grossed-up dividend, so that the tax isn't paid twice.


Algebra should make this clearer (if not, there are heaps of explanations on Google). Let tc be the company tax rate (30%). Let Df be the franked dividend, i.e., what you receive. Let Dg be the grossed-up dividend, i.e., what the tax is being imposed on. We have

(1 - tc)*Dg = Df.

More generally, the dividend may be only partially franked. Let f be the fraction of the dividend which is fully franked. Then

(1 - f*tc)*Dg = Df.         (*)

The total amount of tax the ATO wants is ti*Dg. The company's already paid f*tc*Dg, so you owe

tax_owing = ti*Dg - f*tc*Dg = Dg*(ti - f*tc).

(If this quantity is negative, then the tax office owes you money, and this can either turn into a tax refund or offset some other tax.)

It is more useful to use (*) to work out how much tax you owe as a function of the dividend that you receive:

tax_owing = Df * (ti - f*tc) / (1 - f*tc).

Plugging some numbers in: if the dividend is fully franked, then f=1. Say ti = 37.5%, and tc = 30%. Then you owe the ATO (37.5% - 30%) / (1 - 30%) = 10.7% of the dividend you receive.

In the last couple of years, Vanguard's VHY fund has been giving distributions (which I gather is called a different term to 'dividend' because the distribution comprises lots of individual dividends from all the companies in the fund) around 70% franked. Plugging in f = 0.7, we get (37.5% - 0.7*30%) / (1 - 0.7*30%) = 20.9%. So about a fifth of the distribution you get will go to the tax office.

(This remains the case even if you set up an automatic re-investment plan; the ATO treats it as though you received the income in the form of partially franked dividends, then bought more shares with it. The income is taxed.)

We can use this to estimate the effective returns from a fund. From the above link, VHY's growth since inception has been an annualised average of 13% p.a., comprising 7.1% p.a. growth in the unit price and 5.9% p.a. in distributions. Since about a fifth of the latter is eaten up by tax, the effective return has been a bit under 12% p.a. rather than 13%.

My eyeballing of the table in that PDF file suggests that taking a percentage point off the returns is a decent rule of thumb for working this out, at least if you're in my tax bracket. The forecast growth is 8%, so I'll interpret that as around 7%. (And remember that as a 95% confidence interval, the forecast for a single year is more like (8 +/- 25)%.)

Fixing relative notation in music

I've been learning a tiny little bit of music theory – major scales and chords and so on – and I would like to change everyone's use of relative notation.

The usual way of writing relative notes in a scale is 1, 2, 3, 4, 5, 6, 7. In C major, this would correspond to C, D, E, F, G, A, B. Chords are usually written in Roman numerals, with capital letters for major chords and lower-case letters for minor chords: I, ii, iii, IV, V, vi, viio for C, Dm, Em, F, G, Am, Bdim.

The first problem I see with this is when we want to describe secondary chords like V/V, "five of five": we temporarily go to the major scale of the 5, and extract the V chord from that scale. In C major, the V chord is a G major; in the G major scale, the 5 is a D, so the V/V is a D major chord.

I expect that people who work with these things regularly work these out as quickly as I can do my times tables, hopping between scales with ease. But I would like to work things out in terms of modular arithmetic. In the case above, things appear to work out: there are eight notes in an octave, 5 + 5 = 10, and 10 (mod 8) = 2, and D is the 2 in the C scale.

But this breaks down if we want the V/ii chord: 5 + 2 = 7, but the ii is D, and the 5 in the D scale is an A, not a B.

The mathematically-inclined may have already spotted at least one of the mistakes in the above reasoning. There are seven different notes in the scale, so we should be working modulo 7, not modulo 8. The second mistake is with the notation: the scale should start with zero, not 1.

To do the calculation properly, we need to first subtract 1 after doing the addition. So, V/V is 5 + 5 - 1 (mod 7) = 2. And V/ii is 2 + 5 - 1 (mod 7) = 6. It works.

Having to subtract the 1 is really annoying though, and the special case of ending on a 7 (e.g., V/iii which becomes zero mod 7) needs to be handled. A better scale would be

0, 1, 2, 3, 4, 5, 6

with chords
O, i, ii, III, IV, v, vio.

Then the normal-notation V/V becomes a IV/IV, and can be calculated as 4 + 4 (mod 7) = 1, the D major chord. And a normal-notation V/ii becomes a IV/i, and can be calculated as 1 + 4 (mod 7) = 5, the A major chord.

This is much cleaner, and the only minor issue is a roman numeral for the zero , which I wrote above as a letter O.

Taking into account how little music theory I know, I figure my proposal is somewhere about as optimistic as my suggestion for question marks.

Diamantina Drover

(A longer post than is likely warranted for not hearing lyrics correctly, but perhaps it's worth it if overseas readers (both of you) haven't heard the song before.)

I can't specifically remember it, but I think I first heard Diamantina Drover in the form of John Williamson's cover version on his album Mallee Boy. My parents had a few Williamson albums and that was definitely one of them. I recall later learning the lyrics in a primary school music class; I can't remember actually singing it, but we must have. Whether we sang something closer to Redgum's original or Williamson's more compactly arranged cover, I don't know, but certainly it's Williamson's version which remains one of my favourite songs (of any genre, and certainly within Australian folk).

The song's narrator tells us about how he moved from Sydney a decade ago to become a cattle drover. The first verse and chorus end with "I won't be back till the drovin's done." The last verse ends with "I won't be back when the drovin's done", a change kept in the final chorus as well.

I was at the YouTube video of the Williamson version of this song, and started reading the comments.

Musically i like Johns version but he should have stuck to the red gum lyrics. Changing that one little word at the end takes out all the impact and kind off the whole point to the song.

?????

I knew when reading this that the commenter was referring to the till/when switch, saying that Williamson had sung 'till' in every case instead of changing to 'when'. And I straight-up didn't believe this, until I played through the YouTube video, hearing "till the drovin's done" always and never hearing "when the drovin's done". I figured I couldn't have misheard the lyrics so consistently over so many years, so I checked the album version that I own... and I had misheard.

I'd obviously been taught the Redgum lyrics, and they'd stuck with me through many dozens of plays of John Williamson's version. Perhaps, many years ago, I noticed that John didn't make the till/when switch – having written this post I now recall noticing this sometime around 2000, listening to the song on cassette in Dad's car. But I'd long forgotten about it, if that is actually a genuine memory and not something I'm inventing for myself.

Someone not hearing lyrics right is hardly earth-shattering news, but this one feels much more interesting to me than others.

Having thought about this, I think a compromise would actually improve the lyrics even further. The final verse should switch to 'when', as in the original. But the final chorus should stay as 'till' – the final verse then would have that brief moment of raw honesty, before the narrator slips back into the lie that one day he'll move back to Sydney.

I V vi (iii) IV (I IV V)

In November 2006, Rob Paravonian posted his Pachelbel Rant to YouTube, and it quickly became popular, and now has 12 million views.



At the 2009 Melbourne International Comedy Festival, the Axis of Awesome played their Four Chords Song, and it became really popular, with that video having over 30 million views, and their 2011 official music video (with a slightly different set of songs) a tick under 20 million.

If you read the comments on the Pachelbel Rant video, you get things like this:

Was disappointed when the group "AxesofAwesome" completely ripped it off with "Four Chords".

I just realized that Axis of Awesome completely stole the concept of this video.


And OK, maybe those are the only two comments accusing the Axis of Awesome of plagiarising the concept. But I want to respond to them here anyway, because yesterday I discovered this video by Benny Davis (keyboardist for the Axis of Awesome) singing an early version of the Four Chords Song in November 2006.

So there we go, independent (re-?)discoveries of a piece of musical comedy.

YouTube comments

When Google made commenting on YouTube go via Google Plus, it created a loud chorus of online protest. News and tech sites ran with these stories, no doubt hoping to attract lots of eyeballs of angry YouTube commenters who didn't want to use Google Plus.

Left largely unremarked during the controversy, but generally known, was that YouTube comments sections were typically a cesspit featuring the absolute dregs of humanity. The switch to G+ comments has improved the quality of comments tremendously. You'll still see the occasional hundred-post-long flame war on Israel-Palestine on a video about ducks or whatever, but the percentage of non-offensive and even useful comments is much higher today than it used to be. I often read the comments, and occasionally I even find them useful – perhaps pointing me to an interesting related video, or raising some background information that I can go away and verify.

There's one exception to this general rule that I came across tonight. In the pre-G+ era, the saddest place I ever saw on YouTube was the comments of Mariah Carey's One Sweet Day. Almost all of the comments – literally 95% or more – were RIP messages to lost friends or family. Page after page of people finding some comfort from the song and leaving a little personal message. I don't know what motivated anyone to express their grief in the form of a YouTube comment, but the memory of those comments makes me tear up even now.

There's still some of that in the G+-style comments to One Sweet Day, enough to make me sad if I scroll through enough of them. But people posting the song to Google Plus are often not leaving a comment at all, or perhaps snarking about the evolution of Carey and pop music more generally since the 1990's.

A little bit of good Internet has been lost.

In which I write about myself

When I was little, perhaps eight or nine years old, I experienced a certain phenomenon for the first time. Perhaps it's a common thing that I just haven't read about. I don't know what it's called, and I doubt I can even describe it well enough in prose, let alone in terms that Google might understand. This first occasion experiencing it was so long ago, I don't even remember whether it was an external sound, or something purely imagined. It might not have even been an auditory thing, but I'll pretend that it was regardless.

Imagine a sort of beat, but the sound isn't something sharp like a drumbeat. Maybe the sound of walking on gravel, or flicking the bristles on a toothbrush. This sound repeats regularly, roughly once a second. In my brain, it is as though these sounds follow a positive feedback loop (alternatively, like a resonance), rising in volume. Eventually (maybe after a few seconds, maybe a few tens of seconds) the imagined noise in my head is intolerably loud. That evening when I was eight or nine, I burst into tears at it. As I recall it, the news was on the TV, and Mum thought that I was reacting to the footage of the war zone (Yugoslavia?), trying to comfort me accordingly.

Over the years, this same sort of feedback-loop-thing repeated itself occasionally, ending more calmly than wild crying. My memory is that it was something I could almost command at will: imagining those regular, quiet sounds, and having them dominate the apparent noise inside of my brain. But it was long ago (how long ago? When did it stop? I don't know – in my teenage years maybe? Early adulthood??), and perhaps not as common or as controlled as I remember. I tried summoning those regular sounds just now, and I can only experience what feels like a faded ghost of what I remember: my brain stubbornly whirring away normally, saying only (in some metaphorical sense) "I know what you're trying to do, the sound used to get loud like this,", but this isn't actually overpoweringly loud.

Sometimes it wasn't sound-based. In what I associate more (though not exclusively) with dreaming was having the size of a ball (or a spherical rock) get larger very quickly. Perhaps the rock was on one end of a seesaw, and, without wanting to, I would imagine it rapidly getting many times larger than the seesaw. It would destroy any hope of imagining what I wanted to imagine about that ball or rock.

I was reminded of these old memories today. I'd been reading a discussion about photons and coherent states, and I pondered, as I occasionally do, how little I understand about quantum mechanics. What's an "observable" and why should it be a Hermitian operator in a Hilbert space? (Real eigenvalues, whatever, my main confusion is on representing something physically measured as a matrix. Or why non-commuting operators should exist. Turning Poisson brackets into commutator brackets just because. Totally weird stuff, though perhaps within the realms of "spend a few weeks looking at your old uni notes, in particular representing things in quantum by wavefunctions rather than kets, and you'll work it out".)

I went on to think of how, more generally, I don't understand things. Why does particle physics have Lie algebras in it? Why does anything exist? At the latter question, I imagined galaxies and the Big Bang and atoms and gravity and I had one of those weird positive-feedback-loop-things, my brain getting totally flipped out over the existence of anything at all, matter, energy, physics. It only lasted a second or so, but it was a powerful force in my head for that second, as though it was driving me fast towards a sort of existential madness*. Then it ended. The existence of the universe and physics is still really weird, but it's something that my brain can consider calmly and stably.

*Whichever meanings or connotations of 'existential' apply here, those are the ones that I mean.

I think this "getting briefly and excessively weirded out over the existence of anything" thing has happened to me before. Whether it belongs in the same category as the regular beats that made me cry when I was eight, I don't know, but it feels very similar.

Statement questions

"Why would anyone do this."

Sometimes people ask a rhetorical question, and deliver it as a sentence, without the upwards inflection at the end that we see in genuine questions. Not all rhetorical questions are delivered in this way, but it's an important enough difference in tone that when writing, the question mark is often replaced by a full stop.

That is how I used to write these statement-questions, but a friend objected, finding the full stop mentally jarring having just processed the words as a question. Did I misread? Did he mistype? And so I started putting a question mark after the full stop in these cases: "Why would anyone do this.?"

It's a solution that's just satisfying enough for me to keep using it – for people who get what I'm doing, it makes the statement-question unambiguous, and should hopefully reduce the mental jarring. But even assuming that people know what I'm doing, it is unsatisfactory: I've tried to internalise the full-stop-question-mark over a period of years, but that question mark still makes me start upwardly inflecting the end of the sentence that I've just typed specifically to not get upwardly inflected.

It occurred to me this evening that a better solution would be to borrow and then modify from Spanish. Upward-inflecting questions would get an inverted question mark ¿ at the start and a regular question mark at the end ? . As soon as the question starts, the reader would prepare to upwardly inflect at the end.

Statement-questions, on the other hand, would only get the final question mark.

Upwardly inflect: ¿Why would anyone do this?
Don't upwardly inflect: Why would anyone do this?

So, the final question mark merely confirms that gramatically, the words just written are a question. The presence or otherwise of the inverted question mark primes the reader for the appropriate inflection.

As practical suggestions go, this is somewhere beyond the "utterly useless" end of the spectrum that covers everything I've ever suggested before. It would require the internalisation of a different punctuation system by all English speakers, forgetting entirely the cues that come with the single question mark at the end (cues that would live on in all the centuries of books written before my idea becomes standard). But it seems an elegant solution, and I thought it was worth documenting, albeit in a post which I'll sneakily upload to LiveJournal in the dead of night Australian time and won't link to elsewhere.