Reflections on Today’s Gospel

First Published: 2022 September 1

Luke 14:14 “blessed indeed will you be because of their inability to repay you. For you will be repaid at the resurrection of the righteous”

Draft 1

The common theme this week I saw was a fairly straightforward exhortation to seek the Lord our G-d. The best way to do so, as the Gospel points out, is through corporal works of mercy. That is, seeing Christ in the poor and needy, and giving to them.

As I thought about the Gospel, something else came to mind as well. We can never repay the Lord for any of the good that He gives us. In that way, striving to be like the Lord means doing good to those who cannot do good to us.

The Second Reading is also powerful this week, especially since I’m currently reading the part of the Bible referenced in it. In this reading, St. Paul reminds us of the difference between seeking Christ and seeking the Father without Christ. Without Christ, the Lord’s presence is too awesome for us to behold.

24 for Twenty Four

First Published: 2022 August 22

Draft 8: 3 August 2023, added final notes and removed some extras!

This year I chose not to participate in a yearly tradition titled n for n.¹ While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,² I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”³ together, since each is really only half of a new skill.^{4 5}I’ve also started playing with a band now, so together it’s for sure at least one skill.⁶

2. I performed an original song at an open mic!⁷

3. I learned how to do basic accordion repair!⁸

4. I was able to run five miles! ⁹

5. I played bagpipes at a friend’s wedding!¹⁰

6. I learned how to add dragon-motifs to my Celtic knots

7. I successfully did NaNoWriMo and successfully finished a Writathon!

8. I started self-publishing a webnovel!¹¹

9. I wrote two books at once!¹²

10. I published a whole book.¹³

11. I passed my candidacy exam!

12. I’ve been towed now!¹⁴

13. I gave double red blood!

14. I learned how to do Tunisian crochet!¹⁵

15. I presented my research at the state capital building!

16. I took a Cumbiya class!

17. I taught religious education!

18. I went on a blind date!

19. I went swimming again!¹⁶

20. I started using a fountain pen and writing letters to friends.

21. I went on a service trip!¹⁷

22. I went to Michigan!¹⁸

23. I juggled three balls successfully for 100 tosses!¹⁹

24. I went to an improv show!²⁰

Thanks for following me on this journey! I think that I revisited it a little too infrequently for the hope of intentionality to seep too far. I did feel motivated to try a few more things these past weeks. Now that I’ve completed the list, I think this is something I want to try again with each following year of my life. There are an infinite number of things that I can do, which means there’re an infinite number of things I haven’t done yet.

Draft 7: 26 July 2023, added new notes

This year I chose not to participate in a yearly tradition titled n for n.²¹ While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,²² I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”²³ together, since each is really only half of a new skill.^{24 25}I’ve also started playing with a band now, so together it’s for sure at least one skill.²⁶

2. I performed an original song at an open mic!

3. I learned how to do basic accordion repair!

4. I was able to run five miles!

5. I played bagpipes at a friend’s wedding!

6. I learned how to add dragon-motifs to my Celtic knots

7. I successfully did NaNoWriMo!

8. I started self-publishing a webnovel!²⁷

9. I successfully finished a Writathon!²⁸

10. I wrote two books at once!²⁹

11. I passed my candidacy exam!

12. I’ve been towed now!³⁰

13. I gave double red blood!

14. I learned how to do Tunisian crochet!³¹

15. I presented my research at the state capital building!³²

16. I took a Cumbiya class!³³

17. I taught religious education!³⁴

18. I went on a blind date!³⁵

19. I went swimming again!³⁶

20. I started using a fountain pen³⁷

21. I went on a service trip!³⁸

22. I went to Michagin!³⁹

23. I published a whole book.⁴⁰

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁴¹ have far too few entries, I’ll use some time to branch out and try new things.

Note 26 July: I’ve apparently done twenty three new things this year, which is basically where I want to be. I need to do a single new thing before I turn twenty five, which should be doable.⁴² If I do more things that feel notable, I suppose that I can get rid of some items I don’t feel strongly about/combine them.⁴³

Draft 6: 3 June 2023, added new notes and revised opening text

This year I chose not to participate in a yearly tradition titled n for n.⁴⁴ While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,⁴⁵ I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”⁴⁶ together, since each is really only half of a new skill.^{47 48}I’ve also started playing with a band now, so together it’s for sure at least one skill.⁴⁹

2. I performed an original song at an open mic!

3. I learned how to do basic accordion repair!

4. I was able to run five miles!

5. I played bagpipes at a friend’s wedding!

6. I learned how to add dragon-motifs to my Celtic knots

7. I successfully did NaNoWriMo!

8. I started self-publishing a webnovel!⁵⁰

9. I successfully finished a Writathon!⁵¹

10. I wrote two books at once!⁵²

11. I passed my candidacy exam!

12. I’ve been towed now!⁵³

13. I gave double red blood!

14. I learned how to do Tunisian crochet!⁵⁴

15. I presented my research at the state capital building!⁵⁵

16. I took a Cumbiya class!⁵⁶

17. I taught religious education!⁵⁷

18. I went on a blind date!⁵⁸

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁵⁹ have far too few entries, I’ll use some time to branch out and try new things.

Note 3 June: I’ve apparently done eighteen new things this year, which is two or three⁶⁰ fewer new tasks than I would like to be at now. My standards are absolutely dropping here, but that is probably ok. After all, I am generally ok with who I am as a person, so remaining myself isn’t anything too bad.

Draft 5: 15 May 2023, added new notes

This year I chose not to participate in a yearly tradition that was titled n for n. While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,⁶¹ I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go. Thus far I really only have one, but I’m hopeful to add more.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”⁶² together, since each is really only half of a new skill.^{63 64}I’ve also started playing with a band now, so together it’s for sure at least one skill.⁶⁵

2. I performed an original song at an open mic!

3. I learned how to do basic accordion repair!

4. I was able to run five miles!

5. I played bagpipes at a friend’s wedding!

6. I learned how to add dragon-motifs to my Celtic knots

7. I successfully did NaNoWriMo!

8. I started self-publishing a webnovel!⁶⁶

9. I successfully finished a Writathon!⁶⁷

10. I wrote two books at once!⁶⁸

11. I passed my candidacy exam!

12. I’ve been towed now!⁶⁹

13. I gave double red blood!⁷⁰

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁷¹ have far too few entries, I’ll use some time to branch out and try new things.

Note 15 May: I’ve apparently done thirteen⁷² new things this year, which is five⁷³ fewer than I would like to be at now. Either I need to start lowering my standards for what counts as new or I need to start doing more.

Draft 4: 1 December 2022, added new notes

This year I chose not to participate in a yearly tradition that was titled n for n. While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,⁷⁴ I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go. Thus far I really only have one, but I’m hopeful to add more.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”⁷⁵ together, since each is really only half of a new skill.^{76 77}I’ve also started playing with a band now, so together it’s for sure at least one skill.⁷⁸

2. I performed an original song at an open mic!

3. I learned how to do basic accordion repair!

4. I was able to run five miles!

5. I played bagpipes at a friend’s wedding!

6. I learned how to add dragon-motifs to my Celtic knots

7. I successfully did NaNoWriMo!

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁷⁹ have far too few entries, I’ll use some time to branch out and try new things.

Note 1 December: I am now slightly behind which is fine. Now it’s time to start doing things outside of my comfort zone I guess.

Draft 3: 26 October 2022, added new notes

This year I chose not to participate in a yearly tradition that was titled n for n. While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,⁸⁰ I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go. Thus far I really only have one, but I’m hopeful to add more.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”⁸¹ together, since each is really only half of a new skill.^{82 83}I’ve also started playing with a band now, so together it’s for sure at least one skill.⁸⁴

2. I performed an original song at an open mic!

3. I learned how to do basic accordion repair!

4. I was able to run five miles!

5. I played bagpipes at a friend’s wedding!

6. I learned how to add dragon-motifs to my Celtic knots

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁸⁵ have far too few entries, I’ll use some time to branch out and try new things.

Note 26 October: I’m currently even more⁸⁶ slightly ahead of schedule for skills which is fun! I haven’t had to use this list as motivation yet, but probably soon.

Draft 2: 1 October 2022, added new notes

This year I chose not to participate in a yearly tradition that was titled n for n. While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,⁸⁷ I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go. Thus far I really only have one, but I’m hopeful to add more.

1. As I’ve alluded too before, I’ve started playing at open mics. I’m going to put the skills of “playing with a partner” and “regularly doing an open mic”⁸⁸ together, since each is really only half of a new skill.^{89 90}I’ve also started playing with a band now, so together it’s for sure at least one skill.⁹¹

2. I performed an original song at an open mic! (added 1 October)

3. I learned how to do basic accordion repair! (added/did 1 October, post to follow later)

4. I was able to run five miles! (added 1 October)

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁹² have far too few entries, I’ll use some time to branch out and try new things.

Note 1 October: I’m currently slightly ahead of schedule for skills which is fun! At some point I may want to trim down and condense skills, but that’s a problem if I do too well only.

Draft 1

This year I chose not to participate in a yearly tradition that was titled n for n. While what that initially meant is unimportant, what’s important is that I’ve decided to reincorporate the name. Instead of meaning whatever it used to,⁹³ I’m going to try to learn 24 new skills while I’m 24. These can be actual skills, like an instrument or language, or things I just had never done and so had no way to do them. I’ll also include skills I once had but no longer do, such as picking up an instrument I’ve forgotten or doing some athletic feat that I needed to retrain for. I’ll keep this list updated with them, and may remove them depending on how things go. Thus far I really only have one, but I’m hopeful to add more.

Thanks for following me on this journey! I’m hopeful that this will let me reflect on the passage of time more intentionally. I’m also hopeful that it will mean that next summer, when I⁹⁴ have far too few entries, I’ll use some time to branch out and try new things.

Reflections on Today’s Gospel

First Published: 2022 August 22

Psalm 117:1 “Praise the LORD, all you nations!

Extol him, all you peoples”

Draft 1

Oh boy is there a lot to talk about this week. The theme common to all three readings was that of the Lord as Father, especially Father of All.

However, there are also other pieces to talk about. The Second Reading was written by Paul to the Jewish people. In it, he reminds us that, just as children may not understand the reason for punishment or discipline from a just parent, so too do we sometimes misunderstand the reasons for suffering in our own lives.

Board Games

First Published: 2022 August 22

Draft 1

This was supposed to come out Saturday, but life got in the way.

This past Saturday I spent more or less the entire day playing board games. It was really incredible.

The day began with a 6 hour DnD session.¹ It was really fun to try an officially published encounter, for all that it lacked the organic feel of homebrew.

After that, I went to a board game night. There I found out that original rules Risk goes significantly faster. I assume this is due to the three following reasons:

1. Starting territories are randomly assigned²

2. There’s no initial reinforcement phase³

3. The number of units gained from reinforcements grows quickly⁴

More or less, this meant that I had 15 soldiers⁵ to take over Australia. Once I did, I then gained 15 troops from eliminating someone, which spiraled out of control quickly. I think the fact that all of us played an aggressive game also helped.

After that, we played some Anomia, which is always fun.

I then learned how to play euchre. And, then it had been 12 hours of board games!

Supper Club

First Published: 2022 August 22

Draft 1

This was supposed to come out Friday, but life got in the way.

Anyways, this past Friday I went to my first supper club. It’s apparently a Wisconsin thing. More or less, it felt like a restaurant, with a major difference: you did all the food ordering before getting a seat, only getting a table when your food was ready. It was great to see a friend before they leave Madison!

Sharing a Song

First Published: 2022 August 18

Draft 1

In October of 2018, I apparently wrote about writing a song. Despite the fact that I’ve done a lot of composing, I realize now that most of it happened in the dark ages.¹ As a result, I don’t know that I’ve talked about writing a song at all.

Anyways, I wrote most of a song, which I’ve shared about previously.² I shared the song with the person I play open mics with yesterday, and he liked it.³ It’s really scary to share things you make with people around you, so I’m glad for his reaction. Now I just have to finish it and muster the courage to share it with people at an open mic.

Open Mic

First Published: 2022 August 16

Draft 1

Like my last post on the topic, I missed writing and uploading a post last night because I was at an open mic. Unlike last time, though, I was performing at this one. A friend from my graduate program and I have played at this open mic twice now, and it was a great time.

The vibes are really nice at the place. The crowd is generally supportive but not hyperfocused on the music.¹ The other performers were almost universally better than me, but incredibly supportive.² So yeah! I’m excited to keep going.

Reflections on Today’s Gospel

First Published: 2022 August 14

Jeremiah 38:4 “Then the princes said to the king, ‘This man ought to be put to death. He is weakening the resolve of the soldiers left in this city and of all the people, by saying such things to them; he is not seeking the welfare of our people, but their ruin.’”

Draft 1

Again I start this reflection reflecting on the lack of cheer. All three readings point to a single common element today: the Peace we are promised is not temporal. That is, though living a Christian life will bring us peace, that does not mean we will live without conflict. Each reading points to a different way in which our struggles to live as Christ commands will be redoubled by the world.

In the first reading, we hear Jeremiah being condemned by the nobility in the city. In the context of today’s readings, this reminds us that to live a holy life is to be persecuted for righteousness. Now, this isn’t to say that people being mad at your choices means they’re holy, there are plenty of ways to be persecuted rightly. But, to truly live a life centered around the Lord’s will means going against the powers of the day.

The second reading reminds us that, though our spiritual struggles are painful, we are spared from the worst of it. Indeed, there is no way that we could redeem ourselves through merit alone. It is through the grace of our Lord alone that we have the strength to resist sin, and it is through His sacrifice that we are redeemed. This reading reminds us that, though we can always strive more, we should strive always to model Christ.

The final reading is in many regards the hardest. The Church has always had enemies with temporal power, and hopefully there’s no need to be reminded that Christ suffered for us. Instead, the Gospel today tells us that even the people we love most in this world may be at odds with us in following a holy life. There’s an expression I’ve heard, that all love on Earth is only good in so far as how it orients us to He who is Love, but having it expressed as Christ does showcases the pain. There is a more joyful reading though: the love a parent has for their child is nothing compared to the love that Love has for us.

Book Review of Till We Have Faces

First Published: 2022 August 10

Draft 1

This summer I joined a book club at my parish. The book the parish chose to read was C.S. Lewis’s¹ The story is a retelling of Eros and Psyche focusing on one of Psyche’s sisters.

I really enjoyed the book, especially looking back on it. At the beginning, I found that there was a lot of anger and world-building that I couldn’t quite see the point of. Unlike most of the Lewis books I’ve read, there was no immediate didactic idea I could find.

I really don’t know how to write a review, but I did really enjoy the book. Lewis does a great job taking an unreliable narrator and using the unreliability in the text. Most of what is beautiful about the book to me entirely spoils the reveal of the book, so I’ll leave it there.

The general premise of the book is that in the kingdom of Glome, which is far from Ancient Greece, there lives a king and his three daughters. One of them is beautiful, one shrewd, and one doesn’t really matter.² The first two³ are mentored by a Greek, who they lovingly refer to as the Fox.

Overall, I’d give the book a 4 out of 5. It was really captivating as it continued, and it has a lot of great places for me to see how I can improve as a person. However, the beginning was really slow for me.

Talk Planning,Tuning Theory

First Published: 2022 August 9

Draft 1.1 2022 August 22 (a typo was pointed out)

Yesterday I claimed that I have the ability to freestyle a cogent narrative about tuning theory. To test¹ that claim,² I figured that I should do that here. Halfway through writing this, I already realize that I can speak on the theory, though not in a cogent way.

I hope that I don’t need to motivate what music is, at least informally. Pitch is fairly fundamental to most of Western Music³. The concept of pitch as inherent to a specific body is an ancient thought.

Pythagoras is reported to have begun his theories on tuning when he observed blacksmiths. Regardless of how hard the smith struck an anvil, each anvil rang at its own pitch.⁴ He then allegedly measured them, and found what it was about each anvil that made it produce its specific pitch.

Moving now from the story to reality, tuning theory is based primarily on the harmonic series. The harmonic series is produced⁵ by any resonating body. It’s really easy to visualize harmonic series now that Fourier transforms are easy to perform. Put simply⁶, it’s every⁷ integer multiplication of a base frequency. That is, if your fundamental pitch is at 100 Hz, the first harmonic occurs at 200 Hz, the second at 300, and so on.

The ratio between any two⁸ of these pitches in the series corresponds to a harmonic interval. A one to two ratio is an octave.⁹ A two to three ratio is a fifth. A three to four ratio is like a 2:3:4 ratio, and so therefore a fourth, since stacking a fifth over a fourth is an octave. A four to five ratio is a major third.

The human ear, of course, can’t perfectly hear intervals. If you have something close to a 2:3 ratio, instead of hearing 1999:3001, for instance, your ear is likely to hear a simple 2:3. The smaller the numbers in the ratio, the more likely it is that you will hear when it’s out of tune.

Pythagorean tuning is the earliest Western tuning we have. It’s based on the first three pitches of the harmonic series. To construct it, start with a base frequency, then stack fifths until you get back to your starting note, dropping octaves as needed.

Of course, the mathematically inclined among you may notice that this doesn’t mathematically work. Other than the trivial case of n = m = 0, there is no m and n that satisfies: $$\frac{1}{2^n} = \frac{2^m}{3^n}$$ If you plot how close these numbers are, you will see that the closeness of fifths is optimized at 12 and 24.¹⁰

To balance having few pitches with good octave and fifths, they chose 12. The final fifth is just incredibly out of tune, and called a wolf fifth, because it “sounds like a barking wolf”. Thirds are also somewhat out of tune. That’s because a harmonically tuned major third is a 4:5 ratio, as mentioned above. In Pythagorean tuning, each step is well tuned, being a 8:9 ratio. However, two steps then becomes a 64:81 ratio, which is slightly different than 4:5. That’s not horribly out of tune. A minor third in harmonic tuning is a 5:6 ratio. A minor third in Pythagorean tuning is a a 32:27 ratio, which is different than 5:6.

As a result, as harmonies in the medieval time period started becoming based more around the third than the fourth and fifth, new tuning systems were derived. Among these is quarter comma meantone, where you take the difference between 12 fifths and an octave, the comma, and distribute it to make four intervals slightly worse. Other tuning systems come too, such as the well temperment of Bach fame. One commonality among these systems is that each key sounds different, because the intervals are slightly different.

Now we come to the modern tuning systems of equal temper and stretched octave equal temper. Why have intervals based on simple ratios, when you can instead say only the octave matters? In equal temper, each half step is the twelfth root of two larger than the note below. To discuss the difference between different temperments’ interval size, we use the term “cents”. One hundred cents are an equal tempered half step. Stretch octaves is what pianos are tuned to. Scientists have found that octaves slightly larger than 1200 cents sound more like an octave than a real octave does, so piano intervals are even different. Before this turns into a rant on pianos, I’ll move on.

In retrospect, maybe I should start with the explanation of equal temper so that I can say what the difference between intervals in tuning systems is. Anyways, this is at least some text that I can¹¹ clean up and throw onto a slide deck. I think I cover most of what I want to cover here, though maybe I should tie it into the astronomy? That ties with Kepler well. We’ll see I guess.

Draft 1

Yesterday I claimed that I have the ability to freestyle a cogent narrative about tuning theory. To test¹² that claim,¹³ I figured that I should do that here. Halfway through, I already realize that

I hope that I don’t need to motivate what music is, at least informally. Pitch is fairly fundamental to most of Western Music¹⁴. The concept of pitch as inherent to a specific body is an ancient thought.

Pythagoras is reported to have begun his theories on tuning when he observed blacksmiths. Regardless of how hard the smith struck an anvil, each anvil rang at its own pitch.¹⁵ He then allegedly measured them, and found what it was about each anvil that made it produce its specific pitch.

Moving now from the story to reality, tuning theory is based primarily on the harmonic series. The harmonic series is produced¹⁶ by any resonating body. It’s really easy to visualize harmonic series now that Fourier transforms are easy to perform. Put simply¹⁷, it’s every¹⁸ integer multiplication of a base frequency. That is, if your fundamental pitch is at 100 Hz, the first harmonic occurs at 200 Hz, the second at 300, and so on.

The ratio between any two¹⁹ of these pitches in the series corresponds to a harmonic interval. A one to two ratio is an octave.²⁰ A two to three ratio is a fifth. A three to four ratio is like a 2:3:4 ratio, and so therefore a fourth, since stacking a fifth over a fourth is an octave. A four to five ratio is a major third.

The human ear, of course, can’t perfectly hear intervals. If you have something close to a 2:3 ratio, instead of hearing 1999:3001, for instance, your ear is likely to hear a simple 2:3. The smaller the numbers in the ratio, the more likely it is that you will hear when it’s out of tune.

Pythagorean tuning is the earliest Western tuning we have. It’s based on the first three pitches of the harmonic series. To construct it, start with a base frequency, then stack fifths until you get back to your starting note, dropping octaves as needed.

Of course, the mathematically inclined among you may notice that this doesn’t mathematically work. Other than the trivial case of n = m = 0, there is no m and n that satisfies: $$\frac{1}{2^n} = \frac{2^m}{3^n}$$ If you plot how close these numbers are, you will see that the closeness of fifths is optimized at 12 and 24.²¹

To balance having few pitches with good octave and fifths, they chose 12. The final fifth is just incredibly out of tune, and called a wolf fifth, because it “sounds like a barking wolf”. Thirds are also somewhat out of tune. That’s because a harmonically tuned major third is a 4:5 ratio, as mentioned above. In Pythagorean tuning, each step is well tuned, being a 8:9 ratio. However, two steps then becomes a 64:81 ratio, which is slightly different than 4:5. That’s not horribly out of tune. A minor third in harmonic tuning is a 5:6 ratio. A minor third in Pythagorean tuning is a a 32:27 ratio, which is different than 5:6.

As a result, as harmonies in the medieval time period started becoming based more around the third than the fourth and fifth, new tuning systems were derived. Among these is quarter comma meantone, where you take the difference between 12 fifths and an octave, the comma, and distribute it to make four intervals slightly worse. Other tuning systems come too, such as the well temperment of Bach fame. One commonality among these systems is that each key sounds different, because the intervals are slightly different.

Now we come to the modern tuning systems of equal temper and stretched octave equal temper. Why have intervals based on simple ratios, when you can instead say only the octave matters? In equal temper, each half step is the twelfth root of two larger than the note below. To discuss the difference between different temperments’ interval size, we use the term “cents”. One hundred cents are an equal tempered half step. Stretch octaves is what pianos are tuned to. Scientists have found that octaves slightly larger than 1200 cents sound more like an octave than a real octave does, so piano intervals are even different. Before this turns into a rant on pianos, I’ll move on.

In retrospect, maybe I should start with the explanation of equal temper so that I can say what the difference between intervals in tuning systems is. Anyways, this is at least some text that I can²² clean up and throw onto a slide deck. I think I cover most of what I want to cover here, though maybe I should tie it into the astronomy? That ties with Kepler well. We’ll see I guess.