## Reinventing the Pythagorean Scale

### Pythagorus' Contributions

In this activity we are going to attempt to recreate Pythagorus's exploration of the relationship
between frequences. Pythagorus noticed that certain ratios of frequencies sounded pleasant (consonance) while
others didn't (dissonance). In particular, he identified two ratios that were foundational to his
scale: 2:1 and 3:2. Today we refer to a ratio of 2:1 as an octave and a ratio of (roughly) 3:2 as a
perfect fifth. The octave is particularly important because it is an equivalence relation - i.e.
440Hz, 220Hz, 880Hz are all considered to be the same note (Pythagorus didn't use the same note-naming
conventions we use today, but we would say that all of those tones are the note A.)

### Activity

In your groups, open an Excel or Google Spreadsheet to keep track of your work. First, take some time to play around with
different frequencies and combinations of frequencies. Try to determine whether our perception of changes in pitch is logarithmic or linear. Also, try
to determine the a smallest perceptable change in frequency.

Next, pick a base frequency (something around 200-300Hz is probably good). Then, using only the ratios 2:1, 3:2, 2:3, and 1:2,
construct notes between your base frequency and one octave above that. For example, I could start with 200.

- Call 200Hz N1.
- One octave above N1 is 400Hz, which we also refer to as the note N1.
- Multplying by 3/2, I get the note N2 = 200*3/2 = 300.
- If I repeat the process I get N3 = 300*3/2 = 450. This is greater than the octave, but I can multiply by 1/2
to bring it back within the octave: N3 = 225.

Continue in this way and see what you can come up with. You can input frequencies below to listen to them.

### Discussion

- How many distinct notes can you construct within one octave in this way?
- In your spreadsheet, campare the ratios between successive notes. What do you notice?
- Can you foresee any problems with the Pythagorean tuning system?
- A chord is a collection of two or more notes played simultaneously. Find some chords that sound
"good" using the collection of notes you've constructed. Make note of the relationships between
the notes in your chords.Do you notice any patterns?

### Triads