Literature:Elements of Harmony: Difference between revisions

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'''Wackersalish''' is the proto-language for !Zoom.
'''Elements of Harmony''' ([[Windermere/Classical|Classical Windermere]]: ''Yămyămał clisăyfäl'') is a textbook on just intonation authored in [[Windermere/Classical|Classical Windermere]] by physicist, mathematician and composer Tsăhongtamdi covering elementary number theory, acoustics, and just intonation music theory.


{| class="wikitable"
==Contents==
|-
*Book 1 discusses mathematical results:
! colspan="2" rowspan="2" |  
**Prime factorization
! rowspan="2" | [[Bilabial consonant|Bilabial]]
**Continued fractions and mediants
! colspan="2" | [[Alveolar consonant|Alveolar]]
*Book 2 discusses basic acoustics (don't mention frequencies)
! rowspan="2" | [[Postalveolar consonant|Post-<br>alveolar]]
**monochord; building it
! rowspan="2" | [[Palatal consonant|Palatal]]
**Mersenne's Laws?
! colspan="2" | [[Velar consonant|Velar]]
**harmonic series
! colspan="2" | [[Uvular consonant|Uvular]]
**intervals as rational string length ratios (given equal thickness and tension); these can be written as tuples by unique factorization
! rowspan="2" | [[Glottal consonant|Glottal]]
*Book 3 discusses just intonation scales built from notes taken from overtone and undertone series.
|- style="font-size: x-small"
**odd- and prime-limit
! [[Central consonant|central]]
**chord voicings
! [[Lateral consonant|lateral]]
**The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals.
! plain
! [[Labialisation|labial]]
! plain
! [[Labialisation|labial]]
|-class=IPA align=center
! rowspan="2" | [[Nasal consonant|Nasal]]
! style="font-size: x-small;"| plain
| m
| n
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ŋ&nbsp;/ɴ/
| &nbsp;
| &nbsp;
|-class=IPA align=center
! style="font-size: x-small;" | [[glottalic consonant|glottalized]]
| mʼ
| nʼ
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ŋ&nbsp;/ɴʼ/
| &nbsp;
| &nbsp;
|-class=IPA align=center
! rowspan="2" | [[Plosive consonant|Plosive]]
! style="font-size: x-small;" | plain
| p
| t
| &nbsp;
| &nbsp;
| &nbsp;
| (k)
| kʷ
| q
| qʷ
| &nbsp;
|-class=IPA align=center
! style="font-size: x-small;" | glottalized
| pʼ
| tʼ
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| kʼʷ
| qʼ
| qʼʷ
| ʔ
|-class=IPA align=center
! rowspan="2" | [[Affricate consonant|Affricate]]
! style="font-size: x-small;" | plain
| &nbsp;
| c&nbsp;/t͡s/
| &nbsp;
| č&nbsp;/t͡ʃ/
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|-class=IPA align=center
! style="font-size: x-small;" | glottalized
| &nbsp;
| cʼ&nbsp;/t͡sʼ/
| ƛʼ&nbsp;/t͡ɬʼ/
| čʼ&nbsp;/t͡ʃʼ/
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|-class=IPA align=center
! colspan="2"|[[Fricative consonant|Fricative]]
| &nbsp;
| s
| ɬ
| š&nbsp;/ʃ/
| &nbsp;
| &nbsp;
| xʷ
| x̣&nbsp;/χ/
| x̣ʷ&nbsp;/χʷ/
| h
|-class=IPA align=center
! rowspan="2" | [[Approximant consonant|Approximant]]
! style="font-size: x-small;"| plain
| &nbsp;
| &nbsp;
| l
| &nbsp;
| y&nbsp;/j/
| &nbsp;
| w
| &nbsp;
| &nbsp;
| &nbsp;
|-class=IPA align=center
! style="font-size: x-small;"| glottalized
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| yʼ&nbsp;/jʼ/
| &nbsp;
| wʼ
| &nbsp;
| &nbsp;
| &nbsp;
|}


{| class="wikitable" style=text-align:center
==Full text (Classical Windermere)==
|-style="font-size: 90%;"
 
!colspan=2 rowspan=2|
==Full text (English)==
!rowspan=2| [[Bilabial consonant|Bilabial]]
 
!colspan=2| [[Alveolar consonant|Alveolar]]
==Sketch==
!rowspan=2| [[Palatal consonant|Palatal]]
<!--
!colspan=2| [[Velar consonant|Velar]]
'''Motivation for log''': Assume strings 1 and 2 have equal tension and thickness, and have lengths $L_1$ and $L_2$ respectively. Find a function $\log$ that, given the ratio between $L_1$ and $L_2$, measures the corresponding subjective difference in pitch.
!colspan=2| [[Uvular consonant|Uvular]]
 
!rowspan=2| [[Glottal consonant|Glottal]]
$\log(\frac{L_1}/{L_2})$ gives the subjective intervallic change when moving from string 1 to string 2.
|-
 
! <small>[[Central consonant|central]]</small>
Thus: if $r_1$ and $r_2$ are string length ratios, then we want
! <small>[[Lateral consonant|lateral]]</small>
 
! <small>plain</small>
$\log(1) = 0;$ $\log(a) > 0$ when $a > 1;$ $\log(r_1 r_2) = \log(r_1) + \log(r_2).$
! <small>[[Labialization|labial]]</small>
 
! <small>plain</small>
The area $A(t)$ under the hyperbola $y = \frac{1}/{x}$ from $x = 1$ to $x = t$ satisfies the desired properties: $A(1) = 0; A(tu) > A(t) + A(u)$. (Give a geometric proof)
! <small>labial</small>
|-
Ts-T derives using calculus that the desired function is well-approximated by Taylor polynomials: namely, $log(x+1) = A(x+1) ≈ \sum_{n=1}^{N} (-1)^n \frac{x^n}/{n}$, when $|x| < 1$.
!rowspan=2| [[Nasal consonant|Nasal]]
-->
! <small>plain</small>
| {{IPA|m}}
| {{IPA|n}}
|
|
|
|
|
|
|
|-
! <small>[[Glottalization|glottal.]]</small>
| mˀ
| nˀ
|
|
|
|
|
|
|
|-
!rowspan=2| [[Stop consonant|Stop]]
! <small>plain</small>
| {{IPA|p}}
| {{IPA|t}}
|
| {{IPA|tʃ}}
| {{IPA|k}}
| kʷ
| {{IPA|q}}
| qʷ
| {{IPA|ʔ}}
|-
! <small>[[Ejective consonant|ejec.]]</small>
| {{IPA|pʼ}}
| {{IPA|tʼ}}
|
| {{IPA|tʃʼ}}
| {{IPA|kʼ}}
| kʷʼ
| {{IPA|qʼ}}
| qʷʼ
|
|-
!colspan=2| [[Fricative consonant|Fricative]]
|
| {{IPA|s}}
| {{IPA|ɬ}}
|
| {{IPA|x}}
| xʷ
| {{IPA|χ}}
| χʷ
| {{IPA|h}}
|-
!rowspan=2| [[Approximant consonant|Approxi-<br>mant]]
! <small>plain</small>
|
|
| {{IPA|l}}
| {{IPA|j}}
|
| {{IPA|w}}
|
|
|
|-
! <small>glottal.</small>
|
|
| lˀ
| jˀ
|
| wˀ
|
|
|
|}

Latest revision as of 12:52, 9 December 2019

Elements of Harmony (Classical Windermere: Yămyămał clisăyfäl) is a textbook on just intonation authored in Classical Windermere by physicist, mathematician and composer Tsăhongtamdi covering elementary number theory, acoustics, and just intonation music theory.

Contents

  • Book 1 discusses mathematical results:
    • Prime factorization
    • Continued fractions and mediants
  • Book 2 discusses basic acoustics (don't mention frequencies)
    • monochord; building it
    • Mersenne's Laws?
    • harmonic series
    • intervals as rational string length ratios (given equal thickness and tension); these can be written as tuples by unique factorization
  • Book 3 discusses just intonation scales built from notes taken from overtone and undertone series.
    • odd- and prime-limit
    • chord voicings
    • The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals.

Full text (Classical Windermere)

Full text (English)

Sketch

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