This blog is part of a series wth all my music notes. In the Series page you can find all the posts of the series.

Introduction

In this blog I summarize all the tuning systems discussed so far. I think all of them should cover all of practical application for music and beyond. The notation I am using is provided next table.

Symbol Description
$\mathcal{S}$ Single pitch space1
$p(f)$ Frequency-to-pitch mapping1
$G$ Group of the Group Action2
$\lambda_g(x)$ Function of the Group Action2
$F$ Set of pitch’s frequencies
$\mathcal{T}$ Pitch class2

Tuning system table

Tuning system $\mathcal{S}$ $p(f)$ $G$ $\lambda_g(x)$ $F$ $\mathcal{T}$
Positive-real3 $\mathbb{R}$ $f/f_0$ $(\mathbb{R}_{>0}, \cdot, 1)$ $gx$ $\mathbb{R}_{>0}$ $\mathbb{R}_{>0}$ or $\mathbb{R}$4
Real3 $\mathbb{R}$ $b\log_a(f/f_0)$ $(\mathbb{R}, +, 0)$ $g+x$ $\mathbb{R}_{>0}$ $\mathbb{R}$ or $\mathbb{R}_{>0}$4
Equal ratio3 $\mathbb{R}$ $f/f_0$ $(a^{\mathbb{Z}b^{-1}}, \cdot, 1)$ $gx$ $a^{\mathbb{Z}b^{-1}} f_0$ $a^{\mathbb{Z}b^{-1}}$ or $\mathbb{Z}$4
Equal temperament3 $\mathbb{R}$ $b\log_a(f/f_0)$ $(\mathbb{Z}, +, 0)$ $g+x$ $a^{\mathbb{Z}b^{-1}} f_0$ $\mathbb{Z}$ or $a^{\mathbb{Z}b^{-1}}$4
$p_n$-limit3 $\mathbb{R}$ $f/f_0$ $(\mathbb{Z}^n, +, 0)$ $\prod^n_{i=1} p^{g_i}_i x$ $\prod^n_{i=1} p^{g_i}_i f_0$ $g \in \mathbb{Z}^n$ $\mathbb{Z}^n$
Octave-equivalent positive-real5 $\mathbb{C}$ $e^{i2\pi\log_2(f/f_0)}$ $(\mathbb{R}_{>0}, \cdot, 1)$ $e^{i2\pi\log_2(g)} x$ $\mathbb{R}_{>0}$ $\mathbb{T}$
Octave-equivalent real5 $\mathbb{C}$ $e^{i2\pi\log_2(f/f_0)}$ $(\mathbb{R}, +, 0)$ $e^{i2\pi g} x$ $\mathbb{R}_{>0}$ $\mathbb{T}$
n-equal tone ratio5 $\mathbb{C}$ $e^{i2\pi f/f_0}$ $(2^{\mathbb{Z}/n}, \cdot, 1)$ $e^{i2\pi n\log_2(g)} x$ $2^{\mathbb{Z}/n} f_0$ $U_n$ $\mathbb{Z}/n{\mathbb{Z}}$ $\mathbb{Z}_{\bold{mod}\it n}$ $C_n$
n-equal tone temperament5 $\mathbb{C}$ $e^{i2\pi\log_2(f/f_0)}$ $(\mathbb{Z}, +, 0)$ $e^{i2\pi g/n} x$ $2^{\mathbb{Z}/n} f_0$ $U_n$ $\mathbb{Z}/n{\mathbb{Z}}$ $\mathbb{Z}_{\bold{mod}\it n}$ $C_n$
Octave-equivalent $p_n$-limit5 $\mathbb{R}$ $f/f_0$ $(\mathbb{Z}^n, +, 0)$ $\prod^n_{i=2} p^{g_i}_i x$ $\prod^n_{i=2} p^{g_i}_i f_0$ $g \in \mathbb{Z}^n$ $\mathbb{Z}^{n-1}$

Disclaimer

Disclaimer: THE "KNOWLEDGE" IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE "KNOWLEDGE" OR THE USE OR OTHER DEALINGS OF THE "KNOWLEDGE".

Expect conceptual errors and constant updates to the whole blog series. Be skeptical and critical of this series content. Ultimately, you are responsible for the information you consume.

References

  1. Simgle pitch space ↩ ↩2

  2. Group actions and tuning systems ↩ ↩2 ↩3

  3. Simple tuning systems and Tuning systems are homogeneous spaces ↩ ↩2 ↩3 ↩4 ↩5

  4. By exponental mapping. ↩ ↩2 ↩3 ↩4

  5. Tuning systems with equivalence of the octaves. ↩ ↩2 ↩3 ↩4 ↩5