First Published 2018 December 3
One of the classes I’m enrolled in while abroad is Creative Cartography.1 The goal of the class is to explore the different ways that maps can express information and authorship. For a final project, we were assigned the task of creating our own unique map of London. So, being a musical child like I am, I decided to make my end product something auditory.
While discussing with my professor, we had many potential ideas. But, what ended up seeming most fruitful was a one hour walk, where I would follow a series of strangers, logging myself with a stopwatch. Every time that I made a turn, I would start a new lap on the watch. By the end of the hour, I’d accumulated 426 times, so 425 turns. Along with that, I had GPS data tracking my movements and a heart rate monitor that was linked to that GPS.
To create the music, I then needed to find a way to convert the raw data into something musical. I decided to create a melody that could be played solely on the white notes of the piano, rather than other scale options, for a variety of reasons. Had I chosen a pentatonic scale, the data would be harder to express. Had I chosen a chromatic scale, the final piece would have sounded incredibly atonal.
So, I assigned the time in seconds that I had lasted before turning as the basis for the melody. If I walked for less than a second, I would make the melody note an A, for 1-1.99, B, and so on through the scale. Of course, I knew that some of my turns would last seven or more seconds, so I would then repeat. That is, If it took 7-7.99 seconds, I would still have an A. But, in doing so, I would obscure a lot of data.
To combat this, I decided that I would use multiple voices in the final piece. I chose ninth chords, because they have an interesting lack of resolution to them that I thought might allow my music to feel more natural, especially as they tend to have forward movement that is needed when imagining a walk. If the turn would last for less than seven seconds, it would be the root of the chord, seven to 13.99, the third, and so on through the ninth. I assumed that I would then be able to uniquely identify each number of seconds. But, I ended up having ten turns that lasted more than 35 seconds. I decided that I would simply deal with that problem by ignoring it, looping at 35 seconds.
I then had to choose the way that each ninth chord would be constructed. I decided that I didn’t want diminished fifths from the bass, because I didn’t feel like it. My watch was accurate to the .01 seconds, so I kept that in mind.
To decide whether the third would be major or minor, I looked at the fractional time. If it was less than .5, I would make it minor. Otherwise, the third would be major.
The fifth was perfect.
The seventh was major if within .25 seconds 0. That is, from 0-.24 fractional seconds, it would be major. .25-.74 was minor, and .75 and up was again major.
The ninth was determined by the final digit. If the fractional second was even (.00,.02,.04...), it would be major. Otherwise, it would be minor.
By doing this, I constructed a lot of chords that don’t exist in the diatonic world. Of course, then I needed to find a way to automate the calculation of the data, because manually computing 426 data points sounded like a horrible idea, especially since I wouldn’t have the time. So, I automated the production of each note of the chord. I then plotted the melody and bass line, deciding that each chord would be root position. What was left was four voices. Using the principle of parsimonious voice leading,2 I crafted each of the lines by hand. I then uploaded the midi data to a synthesizer, where I mixed the levels a bit until I was satisfied. I did nothing with dynamics or fading because that seemed to detract from the music itself. For tempo, I chose 120 as that is the typical person’s cadence. For voicing, I made sure that the bass was always the lowest and that the soprano was always the highest. The other voices, because of how I wrote them, progressed steadily higher throughout the piece. It creates an interesting effect of growing movement. Happily, it resolves on a CMaj7,9 chord, which fits the C diatonic scale.
If you want to listen, it’s currently hosted on Soundcloud.
One of the classes I’m enrolled in while abroad is Creative Cartography.3 The goal of the class is to explore the different ways that maps can express information and authorship. For a final project, we were assigned the task of creating our own unique map of London. So, being a musical child like I am, I decided to make my end product something auditory.
While discussing with my professor, we had many potential ideas. But, what ended up seeming most fruitful was a one hour walk, where I would follow a series of strangers, logging myself with a stopwatch. Every time that I made a turn, I would start a new lap on the watch. By the end of the hour, I’d accumulated 426 times, so 425 turns. Along with that, I had GPS data tracking my movements and a heart rate monitor that was linked to that GPS.
To create the music, I then needed to find a way to convert the raw data into something musical. Although I’d initially thought of using either a chromatic or pentatonic scale, as those each have their own benefits, I ended up deciding to use the C Ionian4 mode as my basis for melody. There has to be a better way to say that. Maybe: I decided to use the diatonic scale with no accidentals. I decided to create a melody that could be played solely on the white notes of the piano. That’s better.
So, I assigned the time in seconds that I had lasted before turning as the basis for the melody. If I walked for less than a second, I would make the melody note an A, for 1-1.99, B, and so on through the scale. Of course, I knew that some of my turns would last seven or more seconds, so I would then repeat. That is, If it took 7-7.99 seconds, I would still have an A. But, in doing so, I would have obscured a lot of the data.
So, I decided that I would have multiple voices. I chose ninth chords, because they have an interesting lack of resolution to them that I thought might allow my music to feel more natural. So, if the turn would last for less than seven seconds, it would be the root of the chord, seven to 13.99, the third, and so on through the ninth. I assumed that I would then be able to uniquely identify each number of seconds. But, I ended up having ten turns that lasted more than 35 seconds. I decided that I would simply deal with that by ignoring it, looping at 35.
Then, I had to choose the way that each ninth chord would be constructed. I decided that I didn’t want diminished fifths from the bass, because I didn’t feel like it. My watch was accurate to the .01 seconds, so I kept that in mind.
To decide whether the third would be major or minor, I looked at the fractional time. If it was less than .5, I would make it minor. Otherwise, the third would be major.
The fifth was perfect.
The seventh was major if within .25 seconds 0. That is, from 0-.24 fractional seconds, it would be major. .25-.74 was minor, and .75 and up was again major.
The ninth was determined by the final digit. If the fractional second was even (.00,.02,.04...), it would be major. Otherwise, it would be minor.
By doing this, I constructed a lot of chords that don’t exist in the diatonic world. Of course, then I needed to find a way to automate the calculation of the data, because manually computing 426 data points sounded like a horrible idea, especially since I wouldn’t have the time. So, I automated the production of each note of the chord. I then plotted the melody and bass line, deciding that each chord would be root position. What was left was four voices. Using the principle of parsimonious voice leading,5 I crafted each of the lines by hand. I then uploaded the midi data to a synthesizer, where I mixed the levels a bit until I was satisfied. I did nothing with dynamics or fading because that seemed to detract from the music itself.
If you want to listen, it’s currently hosted on Soundcloud.
those of you who’ve read through the archive may know that I’ve written about an assignment for this class before↩︎
minimizing movement between chords↩︎
those of you who’ve read through the archive may know that I’ve written about an assignment for this class before↩︎
or D Dorian, E Phrygian, F Lydian, G Mixolydian, A Aeolian, or B Locrian↩︎
minimizing movement between chords↩︎
First Published: 2018 December 2
Luke 21:27 “And then they will see the Son of Man coming in a cloud with power and great glory.”
Happy first Sunday in Advent, New Liturgical Year, and First Night of Hanukkah! Advent has always been my favorite season in the Church. It helps that I have really 5 options: Advent, Christmas, Lent, Easter, and Ordinary Time. But, Advent has always been nice because it’s the period where we wait for a wholly positive event to occur. Unlike Easter, which saves our souls at the cost of the Lord’s life, Christmas is nothing more or less than a celebration of the birth of our Savior.
And yet, the Gospel today is not about Christ the meek child being born to a virgin, surrounded by shepherds. Instead, it’s a reference and prophesy about the Lord’s next coming. It tells us that we must stay awake and vigilant, and not be snared by the problems of our life. And really, that’s what Advent is about. It’s a preparatory season where we assess ourselves and begin to hope that we could be worthy to welcome the baby Jesus into the world.
It’s interesting reading today’s readings, because they fall really nicely in a chronology. The First Reading talks about the first coming of Jesus, the season we celebrate in just a few weeks. The Psalm and Second Reading tell us the message Jesus often spoke, that we are to be holy and loving through all that we do. Then, the Gospel talks about the second coming. It’s a great reminder of the way we are to view Advent, which is a celebration of the Lord’s birth, life, and second coming.
First Published 2018 December 1
As I mentioned yesterday, today we’re talking about crystals. Polymers are known as semi-crystalline, because they have both crystalline and amorphous sections. Some might wonder why polymers can’t be wholly crystalline.1 As I2 have mentioned before, polymers are weird because more or less all of their strength comes from the inter-molecular forces that hold them together, as opposed to normal matter, which I have no clue how it works apparently. Anyways, if a polymer is fully crystalline, it doesn’t have any entanglements with other polymer chains, and so will become a powder, because each polymer is wholly self contained. So, in short, they can it just isn’t useful.
Semi-crystalline3 polymers are especially useful when used in applications where the Heat Distortion Temperature (HDT) is useful, because for most crystalline polymers, the HDT is far larger than the Tg. Another way to increase the HDT of a polymer is through addition of glass particles. Those increase stiffness, and effectively just shift the modulus of the material directly upwards. Since amorphous polymers lose strength so quickly around Tg, the addition of glass does nearly nothing. However, crystalline polymers lose strength more slowly, so the addition of particles is helpful.
Crystals in polymers form as lamellae, the ordered regions within a polymer. Lamellae clump together into spherulites, which are what looks like the crystal in a polymer. In order for crystals to form, the polymer must not be atactic.
Polymers can only crystallize when below Tm, because that’s what Tm means. When below Tm, polymer crystallization rate is affected by two factors: nucleation and growth.
Nucleation is the formation of the spherulites. It occurs more quickly the further below Tm you get, until Tg is reached, because then the polymer can’t move. Other ways to increase the rate of nucleation include strain/shear stress. In a wholly homogenous polymer, the polymer will spontaneously crystallize, but that is slow. Most polymers have a nucleating agent added to them, commonly sorbitol. But, almost anything in a polymer, including dyes, other polymers, and glass particles can also act as a nucleation point. In general, the more nucleation points there are, the smaller the final size of a spherulite.4 Small spherulite size can be beneficial, especially if it can be made smaller than the wavelength of light, so a bottle can be see through.5 The rate of nucleation increases as T drops until Tg.
The other factor, growth, is exactly what it sounds like. Once nucleated, the spherulites grow. Their growth rate is fastest somewhere between Tm and Tg, though closer to Tm. This is because as T increases, the flexibility of the polymer chain increases. That makes it easier to align, but also easier to unalign. So, by plotting the two rates together, you can find the optimal temperature to cool to in order to let a polymer crystalize.
If, after cooling below Tg, you realize the polymer needs to be more crystalline, then you can anneal it. That is, you can raise it above Tg so that the polymer can form into crystals.
Fun fact: although incredibly regular, the chain stiffness of polycarbonate is such that it is not crystalline in consumer uses because of the time required.
First Published 2018 November 30
Prereading note: I actually looked in my notes for this one!
Today we’re talking about the factors that affect Tg and Tm. Factors that affect Tg will affect all polymers, while the factors that affect Tm will only affect semi-crystalline thermoplastics.
There are four main kinds of factors that affect Tg: intramolecular forces, intermolecular forces, chain length, and timescale.
The intramolecular forces that affect Tg are: chain stiffness, side groups, and cross link density. As chain stiffness goes up,1 Tg goes up. This is because as the amount of energy it takes to rotate along a bond2 increases, so too does Tg. As side groups become bulkier, they increase Tg, for the same reason. As cross-link density goes up, so does Tg, This is because it’s hard to rotate around a link, because the side group is suddenly large. Also, if cross-link density isn’t 0, there is no Tm.
The intermolecular forces that affect Tg are: side group dipole moments,3 side group chain length, and plasticizers. As the side groups become more polar, Tg goes up, because it draws the different polymer chains closer together. For that same reason, as side groups become longer, Tg goes down, because it keeps them further away. For that same reason, plasticizers also tend to lower Tg.
Third, assuming the molecular mass is less than Me,4 the increase of molecular mass will increase Tg. Above Me, there is effectively no difference in Tg for chain length.
Finally, everything in polymers is time dependent. This makes sense when you think about how free energy diagrams work. At any temperature above 0K, there’s some energy. Given enough time, any bond can rotate, because it’ll at some point be able to overcome its energy barrier. Heating it just makes that happen faster. So, Tg decreases as the time-scale you view it from increases. This also works in reverse, so a polymer that’s being stressed at high rates will remain solid at higher temperatures than one that is not.
In order of importance, the intramolecular forces are more important than the intermolecular forces are more important than the other two in governing Tg.5
So, assuming cross-link density is 0, eventually the polymer will become viscous.6 This is affected mainly by the same factors that influence Tg, which is why they have the relationship I talked about in Polymer Review 2. As interactions between the chains goes up, Tm goes up. As chain stiffness goes up, Tm goes up. As branching7 goes up, Tm goes down. As delta S8 goes up, Tm goes down. To explain this, Tm can also be expressed as delta H9 divided by delta S10. Turning a crystal, which is highly ordered, into a liquid, which is not, increases the entropy of the state a lot. So, somehow that also works in reverse? This part is where the physics goes beyond me.
To lower delta S, you can orient the polymer more. This leads well into crystals which is almost certainly the topic of tomorrow’s post.
First Published 2018 November 29
Prereading note: as with the rest of these posts, it’s written mainly from memory, and will1 be consulted and fixed for accuracy before examinations.
Corrections (since I looked at some notes):
Below Tg, the molecule is only able to vibrate. Above Tg, the molecule is able to rotate at each of its bonds. Above Tm2 or Tv3, the molecule is able to move, and flows through a process known as reptation.
Tm is double Tg for symmetric polymers,4 and 1.5 for the rest.
Polymers with crosslinks are thermosets, which are like epoxies and what most rubbers are, and elastomers, which are less tightly cross-linked thermosets.
Tm is where crystallization can begin in a polymer. I’ll probably think about how that works tomorrow.
Polymers are super cool, for a variety of reasons. The one that I’m thinking of today is that they take all of the things that you get told in chemistry class exist but don’t matter, and then suddenly make them matter. The two examples of that I’m thinking of are the fact that the main force holding polymers together is their Van Der Waals interactions, which is the force I’ve5 always been able to ignore as non-significant for a material. The other is that polymers treat all energy as energy. That is, the temperature that a polymer needs to be at to behave as a liquid is lower if mechanical energy is supplied.
Speaking of, there are a few phases that polymers exist in. I’ll briefly discuss them from lowest to highest temperature.
At the coldest, a polymer is in what’s known as the “glassy” state. Here, it is solid, brittle, and the chains are held rigid.
When you heat up, you’ll eventually reach the glass transition temperature, Tg. This is the temperature range where the polymer would be described as “leathery.” It also corresponds to a quick drop in modulus.6 This is due to the fact that below Tg, the energy needed to break the Van Der Waals bonds is higher, because the polymer is moving less. Once you reach Tg, however, the polymers move enough on their own that the Van Der Waals forces effectively disappear. The only thing that gives the polymer strength then is the entanglements, which we talked about yesterday as Me. Above Tg, you’re in the rubber phase. This is where a polymer behaves a lot like what we think rubber does. And, right near Tg, the material exhibits a lot of damping.
After Tg, you reach what’s known as Tm, which is the temperature where the polymer stops acting like a solid, and begins acting like a fluid. Apparently it’s only melting when crystals do that, so that isn’t what happens to many polymers. One interesting tidbit is that polymers that are crystalline7 will have a Tm equal to 1.5x the Tg in Kelvin.
The information about Tm only applies to thermoplastics, polymers that don’t have chemical crosslinks. If a polymer has crosslinks, it never reaches the viscous phase. Instead, it stays at the “rubber plateau” for modulus until the material ultimately burns.8
The other thing to note that is different about amorphous and semi-crystalline polymers is that the drop at Tg is much larger in amorphous polymers than in semi-crystalline, because Tg is where amorphous regions fail, while Tm is where the crystalline sections fail. So, since a semi-crystalline polymer has a much smaller composition of amorphous polymer, it loses less strength.
First Published 2018 November 28
Prereading Note: I’ll apologize in advance for what will likely be most of my posts until the end of the semester. Of all the classes I’m in, only one has a final, my class on polymers. So as a way to review, I’ll be describing different concepts here, if only so that I know that I know them.
I was wrong on calculating Mw.1 To calculate Mw, you do as I said, and take the chains, 10 of 10 and 10 of 100. Multiply the number of chains by their mass, and sum them (total: 1100). Divide the mass sums of each chain length by the total mass (1/11,10/11). Multiply these numbers by the sum mass of each length (1/11*100+10/11*1000) We end up with a Mw of approximately 920, which now that I look at it, seems much more reasonable.
Mn just is the arithmatic mean of the masses. To find it, you sum the mass of all of the chains and divide by the number of chains that are measured.
To find Mw, you can also sum each chain multiplied by the square of the mass and then divide by the sum of chains multiplied by masses.
Mw
Me is the molecular mass required for entanglements to form, as well as the molecular mass between the entanglements apparently, which is really weird and I should reconfirm this with the textbook.
Today, I’ll be discussing the different ways of measuring the mass of a polymer. As far as I remember, there are only four that are relevant to me for the final: Mn, Mw, Mz, and Me. I’ll discuss each one in turn, first with how you calculate it, and then what it is used for.
So, first is Mn. Mn is the number average molecular mass of a polymer. As such, like Mw and Mz, it’s only useful for thermoplastics. To calculate Mn, you divide the mass of each polymer chain by its numerical representation, then sum all of those. So, if you have 10 chains with a mass of 10, and 10 with a mass of 100, it’s 10/2+100/2 = 55 for Mn. Since Mn2 is the molecular weight most concerned with the number of short chains, it’s good for telling you about how a polymer will yield, especially once it begins to crack.
Next is Mw. Mw is the weight average molecular mass, and is the most common/useful of them. To calculate Mw, you divide the mass of each polymer chain by its mass representation, then sum all of those. So, in the initial example, 10 chains of 10 and 10 chains of 100, you would get 100+1000=1100, and then 100/1100+1000/1100 = around 92. So, we immediately see that Mw is more easily measured, because it’s easier to measure the mass of a polymer than the number of chains. Mw is most relevant for the processability of the polymer, especially its flow in the molten state.
Mz is weird. Like Mw, it weights the mass, but it takes it the next step, and does the square of the mass. Mw is relevant mostly in governing die swell, also known as melt-elasticity.
Finally, we have Me. Me is the measure of the mass between entanglements. Off the top of my head3 I don’t know how this is measured. But, it’s mostly useful for knowing how strong the polymer will be in the rubber phase.4
So, yeah. There’s four major ways to say the mass of a polymer. Oh! Also, the difference between Mn and Mw is known as the Molecular Weight Distribution, MWD. That tells you how much strain softening will occur. A larger (broader) MWD will lead to more strain softening.
In conclusion, polymers are weird. Hopefully by the end of these posts you’ll understand why if you don’t already.
First Published: 2018 November 27
Tonight, I had the wonderful opportunity to see Bat Out Of Hell at the Dominion Theatre. It was an odd show for a lot of reasons.
Plot wise, it definitely felt like Peter Pan. There were some fun fourth wall breaks, including the orchestra coming out with broken instruments. It was really fun, especially as a sing-along.
First Published: 2018 November 26
Today, I had the wonderful fortune of watching “Pinter at the Pinter,” a series of Pinter shows at the Harold Pinter Theatre. It was confusing. Apparently I don’t really understand absurdism. But, such is life.
The stage was fantastic, and spun. The lighting was minimal, but well executed. The ukulele playing was mediocre, which was sad. All of the other sounds were beautifully done, whether as subtle or blatant effect
First Published: 2018 November 25
Revelations 1:8 “‘I am the Alpha and the Omega,’ says the Lord God, ‘the one who is and who was and who is to come, the almighty.’”
Today is the last Sunday in the1 year, the Feast of Christ the King.2 It’s a great way to end the year, especially because we get a line from my3 favorite book, Revelations.
We also get to use today as a way to reflect on the year we’ve had, and whether we’ve lived this Year of Grace4 as if Christ was “King of the Universe.”5
Since I’m preparing for the end of my semester and time abroad, the ending of the year is the first of a lot of endings I’m going to have to be ready for. But, that’s neither here nor there.
Today is the last Sunday in the6 year. It’s the feast of Christ the King.7 So, today is the day we celebrate the idea of Christ as the “one who is and who was and who is to come” as a part of the Trinity.
And, we get a reading out of the book of Revelations, which is always nice in a reading. But, today, as the last day of the year, so it’s time that I think about endings. My liturgical year has almost ended, and so the Year of Grace has too. I wonder what Pope Francis will decide next year will be.
First Published: 2018 November 24
I’ve realized lately that one of the biggest problems I’m beginning to have in my life is that everything means more than one thing. So, for instance, the title of this post, “rolls”, could refer to one of two things, both of which are applicable to today, which is odd.
The first and most obvious of these is the food kind. Small little loaves of bread. Today, I decided to try making rolls out of normal bread dough. But, since I measured exactly nothing, I can’t give the recipe.
The oven was set to its hottest temperature, probably 2501 and I kneaded the dough around 5 times over 3ish hours. It baked until crunchy, and was then drizzled with butter.
The other kind of roll is the musical roll. Although wikipedia defines music roll differently, in the Celtic Folk tradition, a roll is a way to break up notes. As you probably know about bagpipes, they can’t play staccato or tongued notes.2 So, to differentiate between a half note and two quarter notes, supplementary notes are played between them. A roll is where you play the note, the note above it, and the note below it, in a quick fashion. There’s more to it than that, especially in terms of spacing, timing, and etc. but that’s a basic3 summary. I worked on making my rolls on the penny whistle4 today.