[Ninth Week, Thurs/Fri Zoom Class]

Quadrivium Assignment


Link to In-Class Handout


Boethius-Pythagorean Hammers Handout


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I have guitar pics, more bridges, and cork feet to hand out in Thursday's class.

Bring your monochords to class.



Bard Diversity and Equity Campus Climate Survey

This is not a requirement, but if you could search your email for

"Bard Diversity and Equity Campus Climate Survey" and take it.

They are collecting data on diversity at Bard.



Summary of Assignments: [Links below]


1)  Macrobius reading with GoogleDoc quotes and comments.

2)  Boethius reading.

3)  Ptolemy reading.

4)  9b exercises.


Summary of things to be deposited into your Homework Folders


-  GoogleDoc with Macrobius quotes and comments.


-  9b Exercises.


Detailed Assignment: Your assignments for class 9b in a bit more detail.

All tolled this may take about 2 hours.


1)  Read the PDF.  Book II, Chapters 1-4 [pp. 185-200] of Macrobius' Commentary on Cicero's "Dream of Scipio"


- Find 2 quotes from the reading [One from chapters 1 or 2, and other from chapters 3 or 4.


- Copy them into this GoogleDoc-Macrobius Quotes at the appropriate location and in the proper location relative to other student selections.  I'd like the GoogleDoc to be in order.


- Identify where your quotes came from at the beginning of the quote [include not only the page number, but also the book:chapter:section.  E.g. p190, Book II, Chapter 2, Section 8].


- Use the comment feature to make a short comment on each of your quotes.


- Use the comment feature to make one short comment on another student's quote.


2)  Read a section from Boethius's Fundamentals of Music. [I reposted this pdf.  It includes a few new notes by me.]  

- Read Book I, chapters 4-11 [pp. 12-19].


3)  Read: Ptolemy's Harmonics.  Solomon Translation.  Sections I.5 and I.8. [pp. 16-19 and 24-27]  In section 1.8 do on your monochord what Ptolemy describes on line 19.1 [p. 27] for making the various consonances. 


4) Do this short exercise sheet and deposit it in your folders: 9b-Exercises.  It's a PDF-form.  Hopefully it will work.


Terms used in Pythagorean Tuning:  These terms are used throughout the above readings.


Primary Consonces

diatessaron: across 4, the fourth, sesquitertian, 4:3 or 4/3

diapente: across 5, the fifth, sesquialter, 3:2 or 3/2

diapason: across all, the octave, duplex or duple or double, 2:1 or 2/1.  [Ptolemy states that is is "nearly isotonic"... equal tone.  We see this in our modern scale.  A and the octave above A is also called A.]


Compound Consonances

diapason plus diatessaron: octave and a fourth, 8:3 or 8/3

diapason plus diapente: octave and a fifth: 3:1 or 3/1

double diapason: two octaves, 4:1 or 4/1


Primary Non-consonant intervals (dissonances)

whole-tone: tone, tonus, 9:8 or 9/8

semi-tone: semitonus, 256:243 or 256/243



Back to Frontis











Notes to myself:

9b In-class order of events

1st 1/3









Discussion of Macrobius.

2nd 1/3








Everybody tune to A.

Discuss how you find the half of an interval.

9/8 x 9/8 = 81/64   (24", 21.33" and 18.96") 

or... 4/3 x 4/3 = 16/9  (24", 18" and 13.5")

    24/18 x 18/13.5 = 432/243 = 16/9


or... 3/2 x 3/2 = 9/4    (24", 16" and 10.6667")

Problem with half of 9/8 or 2/1 


3rd 1/3








Pythag. Hammer Story

16, 12, 9, 8, 6, 4

1) Hit hammers and ask class to distinguish pitches

2) String tension experiment

3) pipe length

4) glasses of water

5) length and thickness of strings







From Gaffurius' Theorica musicae (1492/6)

Note: IV-BAL is Jubal, the biblical founder of music.